Recent zbMATH articles in MSC 35https://www.zbmath.org/atom/cc/352021-02-12T15:23:00+00:00WerkzeugRANS/ILES method optimization for effective calculations on supercomuter.https://www.zbmath.org/1452.653022021-02-12T15:23:00+00:00"Savin, G. I."https://www.zbmath.org/authors/?q=ai:savin.g-i"Benderskiy, L. A."https://www.zbmath.org/authors/?q=ai:benderskii.l-a"Lyubimov, D. A."https://www.zbmath.org/authors/?q=ai:lyubimov.d-a"Rybakov, A. A."https://www.zbmath.org/authors/?q=ai:rybakov.a-aSummary: The article discusses the use of the RANS/ILES method for modeling gas-dynamic processes in a combustion chamber, described using an axisymmetric block-structured computational grid. For these calculations, the approaches of the linked border conditions and the fragmentation of the computational grid are used, which allows to effectively calculate this problem on a supercomputer, achieving acceleration by more than two orders of magnitude, using a total of 32 computing nodes.Numerical solution of variable-order fractional integro-partial differential equations via sinc collocation method based on single and double exponential transformations.https://www.zbmath.org/1452.652682021-02-12T15:23:00+00:00"Babaei, A."https://www.zbmath.org/authors/?q=ai:babaei.afshin"Moghaddam, B. P."https://www.zbmath.org/authors/?q=ai:moghaddam.behrouz-parsa"Banihashemi, S."https://www.zbmath.org/authors/?q=ai:banihashemi.seddigheh"Machado, J. A. T."https://www.zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiroSummary: This paper addresses the numerical solution of the multi-dimensional variable-order fractional integro-partial differential equations. The upwind scheme and a piecewise linear interpolation, are proposed to approximate the Coimbra variable-order fractional derivatives and integral term with kernel, respectively. Two new approaches via the Sinc collocation method based on single and double exponential transformations are adopted for the temporal and spatial discretizations, respectively. The convergence behaviour of the methods is analysed and the error bounds are provided. In addition, four test problems illustrate the validity and effectiveness of the proposed algorithms.Nonlocal boundary-value problem for a parabolic-hyperbolic equation of the second kind.https://www.zbmath.org/1452.351142021-02-12T15:23:00+00:00"Urinov, A. K."https://www.zbmath.org/authors/?q=ai:urinov.akhmadjon-kushakovich|urinov.akhmadzhon-k"Okboev, A. B."https://www.zbmath.org/authors/?q=ai:okboev.a-bSummary: In the work, a boundary value problem with non-local condition for the second kind degenerated parabolic-hyperbolic equation, was investigated. The solution of the considered problem was searched in the form of solution of the first boundary value problem in the parabolic domain and also as the solution of modified Cauchy's problem in the hyperbolic domain. The special class \(R^{\lambda}_{00}\) of regular solutions of modified Cauchy's problem for hyperbolic type equation was introduced and the view of the solution simplified. Considered problem equivalently reduced (in the meaning uniqueness and existence of the solution) to Volterra's integral equation of the second kind. Necessary conditions were found for providing continuity of the kernel and the independent term of the obtained integral equation. The identities involving hyperheometric functions were proved by using Laplace transform. In the work, the properties of Bessel functions, hyperheometric functions, Euler's gamma function and Pohgammer's symbol were used.On a problem of heat equation with fractional load.https://www.zbmath.org/1452.352452021-02-12T15:23:00+00:00"Ramazanov, M. I."https://www.zbmath.org/authors/?q=ai:ramazanov.murat-i"Kosmakova, M. T."https://www.zbmath.org/authors/?q=ai:kosmakova.minzilya-t"Kasymova, L. Zh."https://www.zbmath.org/authors/?q=ai:kasymova.l-zhSummary: In the paper, the solvability problems of an nonhomogeneous boundary value problem in the first quadrant for a fractionally loaded heat equation are studied. Feature of this problem is that, firstly, the loaded term is presented in the form of the Caputo fractional derivative with respect to the spatial variable, secondly, the order of the derivative in the loaded term is less than the order of the differential part and, thirdly, the point of load is moving. The problem is reduced to the Volterra integral equation of the second kind, the kernel of which contains the generalized hypergeometric series. The kernel of the obtained integral equation is estimated and it is shown that the kernel of the equation has a weak singularity (under certain restrictions on the load), this is the basis for the statement that the loaded term in the equation is a weak perturbation of its differential part. In addition, the limiting cases of the order of the fractional derivative are considered. It is proved that there is continuity in the order of the fractional derivative.On one solution of a periodic boundary-value problem for a third-order pseudoparabolic equation.https://www.zbmath.org/1452.350832021-02-12T15:23:00+00:00"Orumbayeva, N. T."https://www.zbmath.org/authors/?q=ai:orumbaeva.n-t|orumbayeva.nurgul-t"Keldibekova, A. B."https://www.zbmath.org/authors/?q=ai:keldibekova.a-bSummary: This article is devoted to the study of the solvability of a periodic boundary-value problem for a third-order pseudoparabolic equation with a mixed derivative. Nonlocal problems for pseudo-parabolic equations have been investigated by many authors. Of particular interest in the study of these problems is caused in connection with their applied values. Such problems include highly porous media with a complex topology, and first of all, soil and ground. To solve this problem, new functions are introduced and the boundary-value problem for a third-order pseudoparabolic equation is reduced to a periodic boundary-value problem for a system of hyperbolic equations with a second-order mixed derivative. Based on the equivalence of the boundary-value problem for a system of hyperbolic equations and the periodic boundary-value problem for a family of systems of ordinary differential equations, two-parameter families of algorithms for finding an approximate solution are constructed and the conditions for unambiguous solvability of the problem under study are established.Unique solvability of problems for a mixed parabolic equation in unbounded domain.https://www.zbmath.org/1452.351132021-02-12T15:23:00+00:00"Mamanazarov, A. O."https://www.zbmath.org/authors/?q=ai:mamanazarov.azizbek-oSummary: In the work, parabolic equations with perpendicular time directions considered together and the Gevrey problem with generalized gluing conditions, and a problem with the Bitsadze-Samarskii conditions have been studied. The uniqueness and existence of the solution of the considered problems were proved. The Gevrey problem was equivalently reduced to the two-point problem for ordinary differential equation involving fractional differential operators. The uniqueness of the solution of the obtained problem was proved by the principle of extremum and the existence was proved by the method of integral equations. The theory of the second kind Volterra integral equations and properties of two-parameter Mittag-Leffler function are applied at studying a unique solvability of the problem with Bitsadze-Samarskii condition. Necessary conditions for given functions for solvability of considered problems have been found.Frankl-type problem for a mixed type equation with the Caputo fractional derivative.https://www.zbmath.org/1452.351092021-02-12T15:23:00+00:00"Karimov, E. T."https://www.zbmath.org/authors/?q=ai:karimov.erkinjon-tulkinovichSummary: In this work, we investigate Frankl-type problem with integral conjugating condition for a mixed type equation consisting of sub-diffusion and wave equations. A uniqueness and the existence of formulated problem have been proved using energy integrals and the method of integral equations, imposing certain conditions on given data.Three-dimensional problems for a parabolic-hyperbolic equation with two planes of change of type.https://www.zbmath.org/1452.351122021-02-12T15:23:00+00:00"Islomov, B. I."https://www.zbmath.org/authors/?q=ai:islomov.b-i"Umarova, G. B."https://www.zbmath.org/authors/?q=ai:umarova.g-bSummary: In this paper in infinite three-dimensional domains the analogues of the Gellerstedt problem (Problem \(AG)\) are formulated and studied for a parabolic-hyperbolic equation with two type change planes.Applying the Fourier transform, the considering problem reduces to a plane analogue of the Gellerstedt problem (Problem \(AG_{\lambda})\) with a spectral parameter and with the boundary value conditions. The uniqueness of the solutions of the Problems \(AG\) and \(AG_{\lambda}\) are proved by the aid of new extremum principle for the second order mixed type equations. The existence of solutions of the two Problems \(AG\) and \(AG_{\lambda}\) are proved by the method of integral equations. In addition, the asymptotic behavior of the solution of the Problem \(AG_{\lambda}\) is studied for large values of the spectral parameter. Sufficient conditions are found under which all operations in this work are legal.On a boundary-value problem for a parabolic-hyperbolic equation with fractional order Caputo operator in rectangular domain.https://www.zbmath.org/1452.351112021-02-12T15:23:00+00:00"Islomov, B. I."https://www.zbmath.org/authors/?q=ai:islomov.b-i"Ubaydullayev, U. Sh."https://www.zbmath.org/authors/?q=ai:ubaydullayev.u-shSummary: In this paper we study a new problem for a parabolic-hyperbolic equation with fractional order Caputo operator in rectangular domain. There are many works devoted to study problems for the second order mixed parabolic-hyperbolic and elliptic-hyperbolic type equations in rectangular domains with two gluing conditions with respect to second argument and with boundary value conditions on all borders of the domain. In studying the unique solvability of this problem, it becomes necessary to specify an additional condition on the hyperbolic boundary of the domain. For this reason, the considering problem became unresolved in an arbitrary rectangular domain. In this paper, we were able to remove this restriction by setting three gluing conditions for the second argument.The \(H^1\)-error analysis of the finite element method for solving the fractional diffusion equation.https://www.zbmath.org/1452.652632021-02-12T15:23:00+00:00"Zhang, Tie"https://www.zbmath.org/authors/?q=ai:zhang.tie"Sheng, Ying"https://www.zbmath.org/authors/?q=ai:sheng.yingSummary: In this paper, we study the fully discrete numerical method for solving the time-fractional diffusion equation: \(\partial_t^\alpha u - \operatorname{div}(A \nabla u) = f\), \(0 < \alpha < 1\). This numerical method is based on the \(L1\) difference approximation to \(\partial_t^\alpha u\) and the \(r\)-order finite element discretization on the spatial domain. We first establish a new error bound of \(O(\triangle t^\beta)\)-order for the \(L1\) formula where \(\beta\) (\(1 - \alpha \leq \beta \leq 2 - \alpha)\) depends on the smoothness of function \(u(t)\). Then, we show that this fully discrete scheme is \(H^1\)-unconditionally stable and the discrete solution admits the optimal \(H^1\)-error estimate of \(O(\triangle t^\beta + h^r)\)-order. When the exact solution \(u(t)\) is weakly singular at \(t = 0\), by using the graded time-meshes, we also derive the \(H^1\)-stability and the optimal \(H^1\)-error estimate. Moreover, by using the postprocessing technique, we derive an \(H^1\)-superconvergence estimate of \(O(\triangle t^\beta + h^{r + 1})\)-order. Numerical examples are provided to support our theoretical analysis.Solvability of pseudoparabolic equations with non-linear boundary condition.https://www.zbmath.org/1452.350812021-02-12T15:23:00+00:00"Berdyshev, A. S."https://www.zbmath.org/authors/?q=ai:berdyshev.abdumauvlen-suleymanovich"Aitzhanov, S. E."https://www.zbmath.org/authors/?q=ai:aitzhanov.serik-ersultanovich"Zhumagul, G. O."https://www.zbmath.org/authors/?q=ai:zhumagul.g-oSummary: The work is devoted to the fundamental problem of studying the solvability of the initial-boundary value problem for a pseudo-parabolic equation (also called Sobolev type equations) with a fairly smooth boundary. In this paper, the initial-boundary value problem for a quasilinear equation of a pseudoparabolic type with a nonlinear Neumann-Dirichlet boundary condition is studied. From a physical point of view, the initial-boundary-value problem we are considering is a mathematical model of quasi-stationary processes in semiconductors and magnetics, taking into account the most diverse physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions of tasks boundary conditions of which are linear with respect to the function and its derivatives. Among these methods, Galerkin's method leads to the simplest calculations. In the paper, by means of the Galerkin method the existence of a weak solution of a pseudoparabolic equation in a bounded domain is proved. The use of the Galerkin approximations allows us to get an estimate above the time of the solution existence. Using Sobolev 's attachment theorem, a priori solution estimates are obtained. The local theorem of the existence of the solution has been proved. The uniqueness of the weak generalized solution of the initial-boundary value problem of quasi-linear equations of pseudoparabolic type is proved on the basis of a priori estimates. A special place in the theory of nonlinear equations is taken by the range of studies of unlimited solutions, or, as they are otherwise called, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally intractable: solutions increase indefinitely over a finite period of time. Sufficient conditions have been obtained for the destruction of its solution over finite time in a limited area with a nonlinear Neumann-Dirichle boundary condition.Solution of Cauchy problem for the generalized Gellerstedt equation.https://www.zbmath.org/1452.351102021-02-12T15:23:00+00:00"Berdyshev, A. S."https://www.zbmath.org/authors/?q=ai:berdyshev.abdumauvlen-suleymanovich"Hasanov, A."https://www.zbmath.org/authors/?q=ai:hasanov.a-l|hasanov.akbar-d|hasanov.allahverdi-b|hasanov.amil-a|hasanov.a-a.1|hasanoglu.alemdar|hasanov.a-j|hasanov.anvar-h|hasanov.anvar|khasanov.a-s|khasanov.a-b"Abdiramanov, Zh. A."https://www.zbmath.org/authors/?q=ai:abdiramanov.zh-aSummary: In this paper, the unique solvability of the Cauchy problem for a generalized hyperbolic Gellersted equation with two lines of degeneracy of different order is studied. A modified initial problem is formulated. By using the Riemann method the solution of this problem is constructed in an explicit form inside characteristic triangle. The constructed Riemann function is expressed by special Appell functions and Gauss hypergeometric functions of two variables.On unique solvability of a boundary-value problem for a viscous transonic equation.https://www.zbmath.org/1452.350052021-02-12T15:23:00+00:00"Apakov, Yu. P."https://www.zbmath.org/authors/?q=ai:apakov.yusupjon-p|apakov.yusufjon-pSummary: We consider the second boundary-value problem for a third-order equation with multiple characteristics containing the second derivative with respect to time in a rectangular domain. Uniqueness of the solution is proved by the energy integral method. The Green's function is constructed for the second boundary-value problem, through which an explicit solution of the problem is written.Maxwell-Stokes system with \(L^2\) boundary data and div-curl system with potential.https://www.zbmath.org/1452.352042021-02-12T15:23:00+00:00"Pan, Xing-Bin"https://www.zbmath.org/authors/?q=ai:pan.xingbinSummary: This paper concerns the boundary value problems of two partial differential systems involving the operator curl and containing an unknown potential, and under boundary conditions with \(L^2\) boundary data. The first one is the \textit{Maxwell-Stokes system}. We study solvability of both linear and semilinear Maxwell-Stokes systems under either the Dirichlet boundary condition or the natural boundary condition, and examine regularity of the solutions. The second one is the \textit{div-curl system with potential}, and we derive solvability and regularity under the Dirichlet boundary condition.Formulation of a maximum principle satisfying a numerical scheme for traffic flow models.https://www.zbmath.org/1452.651562021-02-12T15:23:00+00:00"Farotimi, Oluwaseun"https://www.zbmath.org/authors/?q=ai:farotimi.oluwaseun"Vajravelu, Kuppalapalle"https://www.zbmath.org/authors/?q=ai:vajravelu.kuppalapalleSummary: We consider a non-local traffic flow model with Arrhenius look-ahead dynamics. In recent times, a maximum principle satisfying local conservation framework has been getting much attention, yet conventional numerical approximation scheme may lead to a breakdown of the maximum principle. In this paper, we construct a maximum principle satisfying a numerical scheme for a class of non-local conservation laws and present numerical simulations for the traffic flow models. The technique and the idea developed in this work are applicable to a large class of non-local conservation laws.A discontinuous Galerkin Trefftz type method for solving the two dimensional Maxwell equations.https://www.zbmath.org/1452.760882021-02-12T15:23:00+00:00"Fure, Håkon Sem"https://www.zbmath.org/authors/?q=ai:fure.hakon-sem"Pernet, Sébastien"https://www.zbmath.org/authors/?q=ai:pernet.sebastien"Sirdey, Margot"https://www.zbmath.org/authors/?q=ai:sirdey.margot"Tordeux, Sébastien"https://www.zbmath.org/authors/?q=ai:tordeux.sebastienSummary: Trefftz methods are known to be very efficient to reduce the numerical pollution when associated to plane wave basis. However, these local basis functions are not adapted to the computation of evanescent modes or corner singularities. In this article, we consider a two dimensional time-harmonic Maxwell system and we propose a formulation which allows to design an electromagnetic Trefftz formulation associated to local Galerkin basis computed thanks to an auxiliary Nédélec finite element method. The results are illustrated with numerous numerical examples. The considered test cases reveal that the short range and long range propagation phenomena are both well taken into account.Existence of solutions for a class of IBVP for nonlinear hyperbolic equations.https://www.zbmath.org/1452.350902021-02-12T15:23:00+00:00"Georgiev, Svetlin Georgiev"https://www.zbmath.org/authors/?q=ai:georgiev.svetlin-georgiev"Majdoub, Mohamed"https://www.zbmath.org/authors/?q=ai:majdoub.mohamedSummary: We study a class of initial boundary value problems of hyperbolic type. A~new topological approach is applied to prove the existence of non-negative classical solutions. The arguments are based upon a recent theoretical result.Time dependent center manifold in PDEs.https://www.zbmath.org/1452.350432021-02-12T15:23:00+00:00"Cheng, Hongyu"https://www.zbmath.org/authors/?q=ai:cheng.hongyu"de la Llave, Rafael"https://www.zbmath.org/authors/?q=ai:de-la-llave.rafaelSummary: We consider externally forced equations in an evolution form. Mathematically, these are skew systems driven by a finite dimensional dynamical system. Two very common cases included in our treatment are quasi-periodic forcing and forcing by a stochastic process. We allow that the evolution is a PDE and even that it is not well-posed and that it does not define a flow (not all initial conditions lead to a solution).
We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, smoothness, etc) assumptions, establishes the existence of a ``time-dependent invariant manifold'' (TDIM). These manifolds evolve with the forcing. They are such that the original equation is always tangent to a vector field in the manifold. Hence, for initial data in the TDIM, the original equation is equivalent to an ordinary differential equation. This allows us to define families of solutions of the full equation by studying the solutions of a finite dimensional system. Note that this strategy may apply even if the original equation is ill posed and does not admit solutions for arbitrary initial conditions (the TDIM selects initial conditions for which solutions exist). It also allows that the TDIM is infinite dimensional.
Secondly, we construct the center manifold for skew systems driven by the external forcing.
Thirdly, we present concrete applications of the abstract result to the differential equations whose linear operators are exponential trichotomy subject to quasi-periodic perturbations. The use of TDIM allows us to establish the existence of quasi-periodic solutions and to study the effect of resonances.Random time change and related evolution equations. Time asymptotic behavior.https://www.zbmath.org/1452.350382021-02-12T15:23:00+00:00"Kochubei, Anatoly N."https://www.zbmath.org/authors/?q=ai:kochubei.anatoly-n"Kondratiev, Yuri G."https://www.zbmath.org/authors/?q=ai:kondratiev.yuri-g"da Silva, José L."https://www.zbmath.org/authors/?q=ai:da-silva.jose-luisOn some rigorous aspects of fragmented condensation.https://www.zbmath.org/1452.811752021-02-12T15:23:00+00:00"Dimonte, Daniele"https://www.zbmath.org/authors/?q=ai:dimonte.daniele"Falconi, Marco"https://www.zbmath.org/authors/?q=ai:falconi.marco"Olgiati, Alessandro"https://www.zbmath.org/authors/?q=ai:olgiati.alessandroThe inviscid limit of Navier-Stokes with critical Navier-slip boundary conditions for analytic data.https://www.zbmath.org/1452.351342021-02-12T15:23:00+00:00"Nguyen, Trinh T."https://www.zbmath.org/authors/?q=ai:nguyen.trinh-tThe inviscid limit of the Navier-Stokes equations for incompressible fluids in the half space is studied. The (slip) boundary conditions are depending on \(\nu^{\beta}\), where \(\nu\) is the viscosity. Previous results exist for \(\beta \in[0, 1)\). The new element of the paper is the analysis for the case \(\beta=1\) with some hypothesis only on initial data. The point is to prove the stability of boundary layer expansions for the corresponding boundary conditions.
Some results given in [\textit{T. T. Nguyen} and \textit{T. T. Nguyen}, Arch. Ration. Mech. Anal. 230, No. 3, 1103--1129 (2018; Zbl 1432.35166)] are improved and used to this end.
The flow equations are written in terms of the vorticity, for which some analytic (time-dependent) boundary layer norms are introduced, based on a specific weight function \(\phi_P\) -- see formula (3.2).
The Fourier transforms of vorticity and tangential components of velocity are also used. Important tools are: a very interesting pointwise estimate for the Green function of the Stokes problem (given in Section 6), the Duhamel principle and Gronwall inequality.
Reviewer: Gelu Paşa (Bucureşti)On the existence of a solution of a class of non-stationary free boundary problems.https://www.zbmath.org/1452.351442021-02-12T15:23:00+00:00"Bousselsal, Mahmoud"https://www.zbmath.org/authors/?q=ai:bousselsal.mahmoud"Lyaghfouri, Abdeslem"https://www.zbmath.org/authors/?q=ai:lyaghfouri.abdeslem"Zaouche, Elmehdi"https://www.zbmath.org/authors/?q=ai:zaouche.elmehdiSummary: We consider a class of parabolic free boundary problems with heterogeneous coefficients including, from a physical point of view, the evolutionary dam problem. We establish existence of a solution for this problem. We use a regularized problem for which we prove existence of a solution by applying the Tychonoff fixed point theorem. Then we pass to the limit to get a solution of our problem. We also give a regularity result of the solutions.Large deviations for stochastic flows of diffeomorphisms.https://www.zbmath.org/1452.600212021-02-12T15:23:00+00:00"Budhiraja, Amarjit"https://www.zbmath.org/authors/?q=ai:budhiraja.amarjit-s"Dupuis, Paul"https://www.zbmath.org/authors/?q=ai:dupuis.paul-g"Maroulas, Vasileios"https://www.zbmath.org/authors/?q=ai:maroulas.vasileiosSummary: A large deviation principle is established for a general class of stochastic flows in the small noise limit. This result is then applied to a Bayesian formulation of an image matching problem, and an approximate maximum likelihood property is shown for the solution of an optimization problem involving the large deviations rate function.Current trends in the bifurcation methods of solutions of real world dynamical systems.https://www.zbmath.org/1452.651572021-02-12T15:23:00+00:00"Ferreira, Jocirei D."https://www.zbmath.org/authors/?q=ai:ferreira.jocirei-d"Rao, V. Sree Hari"https://www.zbmath.org/authors/?q=ai:sree-hari-rao.v|rao.vadrevu-sree-hariSummary: In this survey paper we discuss a class of dynamical systems giving rise to three types of bifurcations: Hopf bifurcation, Turing bifurcation and Zip bifurcation. We present methods from dynamical systems theory that help to classify the studied bifurcations. As application of the proposed methods, a class of predator-prey systems are analyzed. Special attention is bestowed to a class of models describing the interactions of two predator species competing for one prey. Under certain natural assumptions, it has been observed that the models admit a one dimensional continuum of equilibria leading to, what is described as a zip bifurcation phenomenon. The models presented in this survey research for these species exhibit rich dynamics for varying values of the vital parameters involved. Conditions for the existence and stability of equilibria of the model equations are established. The effort in the article is to create an awareness among the researchers that certain real world dynamical systems often give rise to the existence of non-isolated equilibria also known as a continuous of equilibria. We propose to present methodologies that would help one to analyze systems with a continuous of equilibria.
For the entire collection see [Zbl 1446.65004].A multi-dimensional, moment-accelerated deterministic particle method for time-dependent, multi-frequency thermal radiative transfer problems.https://www.zbmath.org/1452.652872021-02-12T15:23:00+00:00"Hammer, Hans"https://www.zbmath.org/authors/?q=ai:hammer.hans-werner"Park, HyeongKae"https://www.zbmath.org/authors/?q=ai:park.hyeongkae"Chacón, Luis"https://www.zbmath.org/authors/?q=ai:chacon.luisSummary: Thermal Radiative Transfer (TRT) is the dominant energy transfer mechanism in high-energy density physics with applications in inertial confinement fusion and astrophysics. The stiff interactions between the material and radiation fields make TRT problems challenging to model. In this study, we propose a multi-dimensional extension of the deterministic particle (DP) method. The DP method combines aspects from both particle and deterministic methods. If the emission source is known \textit{a priori}, and no physical scattering is present, the intensity of a particle can be integrated analytically. This introduces no statistical noise compared to Monte-Carlo methods, while maintaining the flexibility of particle methods. The method is closely related to the popular method of long characteristics. The combination of the DP-method with a discretely-consistent, nonlinear, gray low-order system enables an efficient solution algorithm for multi-frequency TRT problems. We demonstrate with numerical examples that the use of a linear-source approximation based on spatial moments improves the behavior of our method in the thick diffusion limit significantly.On a class of two-dimensional incomplete Riemann solvers.https://www.zbmath.org/1452.761142021-02-12T15:23:00+00:00"Gallardo, José M."https://www.zbmath.org/authors/?q=ai:gallardo.jose-m"Schneider, Kleiton A."https://www.zbmath.org/authors/?q=ai:schneider.kleiton-a"Castro, Manuel J."https://www.zbmath.org/authors/?q=ai:castro.manuel-jSummary: We propose a general class of genuinely two-dimensional incomplete Riemann solvers for systems of conservation laws. In particular, extensions of Balsara's multidimensional HLL scheme [\textit{D. S. Balsara}, J. Comput. Phys. 231, No. 22, 7476--7503 (2012; Zbl 1284.76261)] to two-dimensional PVM/RVM (Polynomial/Rational Viscosity Matrix) finite volume methods are considered. The numerical flux is constructed by assembling, at each edge of the computational mesh, a one-dimensional PVM/RVM flux with two purely two-dimensional PVM/RVM fluxes at vertices, which take into account transversal features of the flow. The proposed methods are applicable to general hyperbolic systems, although in this paper we focus on applications to magnetohydrodynamics. In particular, we propose an efficient technique for divergence cleaning of the magnetic field that provides good results when combined with our two-dimensional solvers. Several numerical tests including genuinely two-dimensional effects are presented to test the performances of the proposed schemes.Hessian recovery based finite element methods for the Cahn-Hilliard equation.https://www.zbmath.org/1452.652552021-02-12T15:23:00+00:00"Xu, Minqiang"https://www.zbmath.org/authors/?q=ai:xu.minqiang"Guo, Hailong"https://www.zbmath.org/authors/?q=ai:guo.hailong"Zou, Qingsong"https://www.zbmath.org/authors/?q=ai:zou.qingsongSummary: In this paper, we propose several novel recovery based finite element methods for the 2D Cahn-Hilliard equation. One distinguishing feature of those methods is that we discretize the fourth-order differential operator in a standard \(C^0\) linear finite elements space. Precisely, we first transform the fourth-order Cahn-Hilliard equation to its variational formulation in which only first-order and second-order derivatives are involved and then we compute the first and second-order derivatives of a linear finite element function by a least-squares fitting recovery procedure. When the underlying mesh is uniform meshes of regular pattern, our recovery scheme for the Laplacian operator coincides with the well-known five-point stencil. Another feature of the methods is some special treatments on Neumann type boundary conditions for reducing computational cost. The optimal-order convergence and energy stability are numerically proved through a series of benchmark tests. The proposed method can be regarded as a combination of the finite difference scheme and the finite element scheme.A Cahn-Hilliard model for cell motility.https://www.zbmath.org/1452.352162021-02-12T15:23:00+00:00"Cucchi, Alessandro"https://www.zbmath.org/authors/?q=ai:cucchi.alessandro"Mellet, Antoine"https://www.zbmath.org/authors/?q=ai:mellet.antoine"Meunier, Nicolas"https://www.zbmath.org/authors/?q=ai:meunier.nicolasThis work addresses mathematical features of the cell motility model, which involves a fourth-order degenerate Cahn-Hilliard equation, coupled to a second-order diffusion equation. Its solution is governed by the competing between a double-well potential and the surface tension parameter that results in the complex dynamics of the cell membrane. The main mathematical results list proving a weak existence for the one-dimensional problem and analysing the sharp interface limit in the case of arbitrary dimensions.
Reviewer: Eugene Postnikov (Kursk)Efficient gradient stencils for robust implicit finite-volume solver convergence on distorted grids.https://www.zbmath.org/1452.653062021-02-12T15:23:00+00:00"Nishikawa, Hiroaki"https://www.zbmath.org/authors/?q=ai:nishikawa.hiroakiSummary: Two gradient-stencil augmentation techniques are discussed: symmetric and F-decreasing augmentations, for efficient and robust implicit finite-volume-solver convergence on distorted unstructured grids. The former augments a face-neighbor stencil with extra cells to increase the symmetry of the stencil as much as possible, and the latter adds further extra cells to decrease the reciprocal of the Frobenius norm of a scaled least-squares matrix to minimize the lower bound of the magnitude of the gradient. These techniques are proposed as efficient ways of overcoming a known stability issue with a face-neighbor stencil. It is demonstrated that the F-decreasing augmentation in combination with the symmetric augmentation yields robust and efficient gradient stencils on highly distorted quadrilateral and triangular grids.Uncertainty quantification methodology for hyperbolic systems with application to blood flow in arteries.https://www.zbmath.org/1452.761302021-02-12T15:23:00+00:00"Petrella, M."https://www.zbmath.org/authors/?q=ai:petrella.m"Tokareva, S."https://www.zbmath.org/authors/?q=ai:tokareva.svetlana-a"Toro, E. F."https://www.zbmath.org/authors/?q=ai:toro.eleuterio-fSummary: We present a Stochastic Finite Volume - ADER (SFV-ADER) methodology for Uncertainty Quantification (UQ) in the general framework of systems of hyperbolic balance laws with uncertainty in parameters. The resulting method is weakly intrusive, meaning that deterministic solvers need minor modifications to include random contributions, and has no theoretical accuracy barrier. An illustration of the second-order version for the viscous Burgers equation with uncertain viscosity coefficient is first given in detail, under different deterministic initial conditions; the attainment of the theoretically expected convergence rate is demonstrated empirically; results and features of our scheme are discussed. We then extend the SFV-ADER method to a non-linear hyperbolic system with source terms that models one-dimensional blood flow in arteries, assuming uncertainties in a set of parameters of the problem. Results are then compared with measurements in a well-defined 1:1 experimental replica of the network of largest arteries in the human systemic circulation. For a 99\%-confidence level, severe influence of parameter fluctuations is seen in pressure profiles, while secondary effects on flow rate profiles become visible, particularly in terminal branches. Results suggest that the proposed methodology could successfully be applied to other physical problems involving hyperbolic balance laws.Data recovery in inverse scattering: from limited-aperture to full-aperture.https://www.zbmath.org/1452.352582021-02-12T15:23:00+00:00"Liu, Xiaodong"https://www.zbmath.org/authors/?q=ai:liu.xiaodong.1"Sun, Jiguang"https://www.zbmath.org/authors/?q=ai:sun.jiguangSummary: Inverse scattering has been an active research area for the past thirty years. While very successful in many cases, progress has lagged when only \textit{limited-aperture} measurement is available. In this paper, we perform some elementary study to recover data that can not be measured directly. In particular, we aim at recovering the \textit{full-aperture} far field data from \textit{limited-aperture} measurement. Due to the reciprocity relation, the multi-static response matrix (MSR) has a symmetric structure. Using the Green's formula and single layer potential, we propose two schemes to recover \textit{full-aperture} MSR. The recovered data is tested by a recently proposed direct sampling method and the factorization method. Numerical results show that it is possible to, at least, partially recover the missing data and consequently improve the reconstruction of the scatterer.A fifth-order shock capturing scheme with two-stage boundary variation diminishing algorithm.https://www.zbmath.org/1452.761132021-02-12T15:23:00+00:00"Deng, Xi"https://www.zbmath.org/authors/?q=ai:deng.xi"Shimizu, Yuya"https://www.zbmath.org/authors/?q=ai:shimizu.yuya"Xiao, Feng"https://www.zbmath.org/authors/?q=ai:xiao.fengSummary: A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called \(\operatorname{P}_4 \operatorname{T}_2 - \operatorname{BVD}\) (polynomial of 4-degree and THINC function of 2-level reconstruction based on BVD algorithm), is formulated as a two-stage spatial reconstruction scheme following the BVD (Boundary Variation Diminishing) principle that minimizes the jumps of the reconstructed values at cell boundaries. In the \(\operatorname{P}_4 \operatorname{T}_2 - \operatorname{BVD}\) scheme, polynomial of degree four and THINC (Tangent of Hyperbola for INterface Capturing) functions with two-level steepness are used as the candidate reconstruction functions. The final reconstruction function is selected through the two-stage BVD algorithm so as to effectively control both numerical oscillation and dissipation. Spectral analysis and numerical verifications show that the \(\operatorname{P}_4 \operatorname{T}_2 - \operatorname{BVD}\) scheme possesses the following desirable properties: 1) it effectively suppresses spurious numerical oscillation in the presence of strong shock or discontinuity; 2) it substantially reduces numerical dissipation errors; 3) it automatically retrieves the underlying linear 5th-order upwind scheme for smooth solution over all wave numbers; 4) it is able to resolve both smooth and discontinuous flow structures of all scales with substantially improved solution quality in comparison to other existing methods; and 5) it produces accurate solutions in long term computation. \(\operatorname{P}_4 \operatorname{T}_2 - \operatorname{BVD}\), as well as the underlying idea presented in this paper, provides an innovative and practical approach to design high-fidelity numerical schemes for compressible flows involving strong discontinuities and flow structures of wide range scales.Kinetics of swelling gels.https://www.zbmath.org/1452.762612021-02-12T15:23:00+00:00"Keener, James P."https://www.zbmath.org/authors/?q=ai:keener.james-p"Sircar, Sarthok"https://www.zbmath.org/authors/?q=ai:sircar.sarthok"Fogelson, Aaron L."https://www.zbmath.org/authors/?q=ai:fogelson.aaron-lReproducing kernel for elastic Herglotz functions.https://www.zbmath.org/1452.460202021-02-12T15:23:00+00:00"Luque, T."https://www.zbmath.org/authors/?q=ai:luque.teresa"Vilela, M. C."https://www.zbmath.org/authors/?q=ai:vilela.mari-cruzSummary: We study the elastic Herglotz wave functions, which are entire solutions of the spectral Navier equation appearing in linearized elasticity theory with \(L^2\)-far-field patterns. We characterize in three-dimensions the set of these functions \(\mathbf{\mathcal{W}}\), as a closed subspace of a Hilbert space \(\mathbf{\mathcal{H}}\) of vector-valued functions such that they and their spherical gradients belong to a certain weighted \(L^2\) space. This allows us to prove that \(\mathbf{\mathcal{W}}\) is a reproducing kernel Hilbert space and to calculate the reproducing kernel. Finally, we outline the proof for the two-dimensional case and give the corresponding reproducing kernel.Efficient simulation of thermally fluctuating biopolymers immersed in fluids on 1-micron, 1-second scales.https://www.zbmath.org/1452.740382021-02-12T15:23:00+00:00"Liu, Kai"https://www.zbmath.org/authors/?q=ai:liu.kai.2|liu.kai.4|liu.kai.3|liu.kai.5|liu.kai|liu.kai.1"Lowengrub, John"https://www.zbmath.org/authors/?q=ai:lowengrub.john-s"Allard, Jun"https://www.zbmath.org/authors/?q=ai:allard.junSummary: The combination of fluid-structure interactions with stochasticity, due to thermal fluctuations, remains a challenging problem in computational fluid dynamics. We develop an efficient scheme based on the stochastic immersed boundary method, Stokeslets, and multiple timestepping. We test our method for spherical particles and filaments under purely thermal and deterministic forces and find good agreement with theoretical predictions for Brownian Motion of a particle and equilibrium thermal undulations of a semi-flexible filament. As an initial application, we simulate bio-filaments with the properties of F-actin. We specifically study the average time for two nearby parallel filaments to bundle together. Interestingly, we find a two-fold acceleration in this time between simulations that account for long-range hydrodynamics compared to those that do not, suggesting that our method will reveal significant hydrodynamic effects in biological phenomena.Solvability theorem for a model of a unimolecular heterogeneous reaction with adsorbate diffusion.https://www.zbmath.org/1452.920462021-02-12T15:23:00+00:00"Ambrazevičius, A."https://www.zbmath.org/authors/?q=ai:ambrazevicius.algirdasSummary: A mathematical model of a unimolecular heterogeneous catalytic reaction is considered in the case where the adsorbate can diffuse along the surface of a catalyst and the desorption of the reaction product from the surface of the adsorbent is instantaneous. The model is described by a coupled parabolic system. The existence and uniqueness of a classical solution are established.A genuinely two-dimensional Riemann solver for compressible flows in curvilinear coordinates.https://www.zbmath.org/1452.761312021-02-12T15:23:00+00:00"Qu, Feng"https://www.zbmath.org/authors/?q=ai:qu.feng.1"Sun, Di"https://www.zbmath.org/authors/?q=ai:sun.di"Bai, Junqiang"https://www.zbmath.org/authors/?q=ai:bai.junqiang"Yan, Chao"https://www.zbmath.org/authors/?q=ai:yan.chaoSummary: A genuinely two-dimension Riemann solver for compressible flows in curvilinear coordinates is proposed. Following Balsara's idea, this two-dimension solver considers not only the waves orthogonal to the cell interfaces, but also those transverse to the cell interfaces. By adopting the Toro-Vasquez splitting procedure, this solver constructs the two-dimensional convective flux and the two-dimensional pressure flux separately. Systematic numerical test cases are conducted. One dimensional Sod shock tube case and moving contact discontinuity case indicate that such two-dimensional solver is capable of capturing one-dimensional shocks, contact discontinuities, and expansion waves accurately. Two-dimensional double Mach reflection of a strong shock case shows that this scheme is with a high resolution in Cartesian coordinates. Also, it is robust against the unphysical shock anomaly phenomenon. Hypersonic viscous flows over the blunt cone and the two-dimensional Double-ellipsoid cases show that the two-dimensional solver proposed in this manuscript is with a high resolution in curvilinear coordinates. It is promising to be widely used in engineering areas to simulate compressible flows.Breather solutions of the cubic Klein-Gordon equation.https://www.zbmath.org/1452.351072021-02-12T15:23:00+00:00"Scheider, Dominic"https://www.zbmath.org/authors/?q=ai:scheider.dominicA conservative, free energy dissipating, and positivity preserving finite difference scheme for multi-dimensional nonlocal Fokker-Planck equation.https://www.zbmath.org/1452.651742021-02-12T15:23:00+00:00"Qian, Yiran"https://www.zbmath.org/authors/?q=ai:qian.yiran"Wang, Zhongming"https://www.zbmath.org/authors/?q=ai:wang.zhongming"Zhou, Shenggao"https://www.zbmath.org/authors/?q=ai:zhou.shenggaoSummary: In this work, we design and analyze a conservative, positivity preserving, and free energy dissipating finite difference method for the multi-dimensional nonlocal Fokker-Planck (FP) equation. Based on a non-logarithmic Landau transformation, a central-differencing spatial discretization using harmonic-mean approximations is developed. Both forward and backward Euler discretizations in time are employed to derive an explicit scheme and a linearized semi-implicit scheme, respectively. Three desired properties that are possessed by analytical solutions: i) mass conservation, ii) free-energy dissipation, and iii) positivity, are proved to be maintained at discrete level. Remarkably, numerical analysis demonstrates that the semi-implicit time discretization ensures the property of positivity preserving unconditionally. Due to the advantages brought by the harmonic-mean approximations, an estimate on the upper bound of the condition number of the resulting coefficient matrix is further established for the semi-implicit scheme. Extensive numerical tests are performed to validate aforementioned properties numerically.A logarithmically improved regularity criterion for the Boussinesq equations in a bounded domain.https://www.zbmath.org/1452.351412021-02-12T15:23:00+00:00"Alghamdi, Ahmad M."https://www.zbmath.org/authors/?q=ai:alghamdi.ahmad-mohammad|alghamdi.ahmad-m-a"Gala, Sadek"https://www.zbmath.org/authors/?q=ai:gala.sadek"Ragusa, Maria Alessandra"https://www.zbmath.org/authors/?q=ai:ragusa.maria-alessandraSummary: The paper is concerned with the regularity of solutions of the Boussinesq equations for incompressible fluids without heat conductivity. The main goal is to prove a regularity criterion in terms of the vorticity for the initial boundary value problem in a bounded domain \(\Omega\) of \(\mathbb{R}^3\) with Navier-type boundary conditions and we prove that if
\[ \int_0^T\frac{\left\| \omega (\cdot,t)\right\|_{BMO(\Omega)}}{ \log \left( e+\left\| \omega (\cdot,t)\right\|_{BMO(\Omega)}\right) }dt<\infty, \]
where \(\omega :=\operatorname{curl} u\) is the vorticity, then the unique local in time smooth solution of the 3D Boussinesq equations can be prolonged up to any finite but arbitrary time.Algorithms for motion of networks by weighted mean curvature.https://www.zbmath.org/1452.652942021-02-12T15:23:00+00:00"Esedoğlu, Selim"https://www.zbmath.org/authors/?q=ai:esedoglu.selimTwo-dimensional vortex quantum droplets get thick.https://www.zbmath.org/1452.351902021-02-12T15:23:00+00:00"Lin, Zeda"https://www.zbmath.org/authors/?q=ai:lin.zeda"Xu, Xiaoxi"https://www.zbmath.org/authors/?q=ai:xu.xiaoxi"Chen, Zikang"https://www.zbmath.org/authors/?q=ai:chen.zikang"Yan, Ziteng"https://www.zbmath.org/authors/?q=ai:yan.ziteng"Mai, Zhijie"https://www.zbmath.org/authors/?q=ai:mai.zhijie"Liu, Bin"https://www.zbmath.org/authors/?q=ai:liu.bin.7|liu.bin.5|liu.bin.8|liu.bin.6|liu.bin.3|liu.bin|liu.bin.9|liu.bin.4|liu.bin.1|liu.bin.2Summary: We study two-dimensional (2D) vortex quantum droplets (QDs) trapped by a thicker transverse confinement with \(a_\bot>1\mu\text m\). Under this circumstance, the Lee-Huang-Yang (LHY) term should be described by its original form in the three-dimensional (3D) configuration. Previous studies have demonstrated that stable 2D vortex QDs can be supported by a thin transverse confinement with \(a_\bot\ll 1\mu\text m\). In this case, the LHY term is described by a logarithm. Hence, two kinds of confinement features result in different mechanisms of the vortex QDs. The stabilities and characteristics of the vortex QDs must be re-identified. In the current system, we find that stable 2D vortex QDs can be supported with topological charge number up to at least 4. We reformulated their density profile, chemical potential and threshold norm for supporting the stable vortex QDs according to the new condition. Unlike the QDs under thin confinement, the QDs in the current system strongly repel each other because the LHY term features a higher-order repulsion than that of the thin confinement system. Moreover, elastic and inelastic collisions between two moving vortex QDs are studied throughout the paper. Two kinds of collisions can be characterized by exerting different values of related speed. The dynamics of the stable nested vortex QD, which is constructed by embedding one vortex QD with a smaller topological number into another vortex QD with a larger number of topological charge, can be supported by the system.Quantization method and Schrödinger equation of fractional time and their weak effects on Hamiltonian: phase transitions of energy and wave functions.https://www.zbmath.org/1452.811102021-02-12T15:23:00+00:00"Zhang, Xiao"https://www.zbmath.org/authors/?q=ai:zhang.xiao"Yang, Bo"https://www.zbmath.org/authors/?q=ai:yang.bo.3|yang.bo|yang.bo.4|yang.bo.2|yang.bo.1|yang.bo.5"Wei, Chaozhen"https://www.zbmath.org/authors/?q=ai:wei.chaozhen"Luo, Maokang"https://www.zbmath.org/authors/?q=ai:luo.maokangSummary: Fractional time quantum mechanics (FTQM) is a method of describing the time evolution of quantum dynamics based on fractional derivatives. For any potential, we obtain the modeling method to describe quantum systems with fractional time consistent with fundamental quantum physics laws, which makes fractional time effects naturally enter quantum mechanics. The method is only using the start states, not based on usually directly replacing the integer derivatives by fractional ones or parallel introductions of standard models. In the process, we solve three open problems perplexing the studies on FTQM: What is the essential quantization method? How does one represent the fractional time Hamiltonian while retaining physical significance? How does one avoid the violations of current models of FTQM for many fundamental quantum physics laws? Then, a FTQM framework is built by amalgamating two quantization methods under a unified foundation of fractional time. The framework contains the quantization method, the Hamiltonian, the Hamilton operator, the Schrödinger equation, the energy correspondence relation, the Bohr correspondence principle and the time-energy uncertainty relation. And the effects of fractional time are revealed: containing historical information of particle's motions; representing weak actions of Hamilton operator. An example is provided, and analytic expressions of the energy and wave functions are obtained. These account for the distinct nonlinear phenomena: the phase transition of energy and wave functions, which can not be revealed in the previous methods. The phase transitions cause some classical physical effects and phenomena: (1) energy gaps filled by energy levels; (2) increase in particle orbits; (3) a famous bound states in continuum (BICs) firstly found in FTQM; (4) a new explanation of the discrete energy levels from the perspective of energy level filling.Gradient invariance of slow energy descent: spectral renormalization and energy landscape techniques.https://www.zbmath.org/1452.350312021-02-12T15:23:00+00:00"Cakir, Hayriye Guckir"https://www.zbmath.org/authors/?q=ai:cakir.hayriye-guckir"Promislow, Keith"https://www.zbmath.org/authors/?q=ai:promislow.keithThe direct scattering problem for the perturbed \(\mathrm{Gr}(1,2)_{\geq 0}\) Kadomtsev-Petviashvili II solitons.https://www.zbmath.org/1452.351762021-02-12T15:23:00+00:00"Wu, Derchyi"https://www.zbmath.org/authors/?q=ai:wu.derchyiSensitivity computations in higher order continuation methods.https://www.zbmath.org/1452.350872021-02-12T15:23:00+00:00"Charpentier, Isabelle"https://www.zbmath.org/authors/?q=ai:charpentier.isabelle"Lampoh, Komlanvi"https://www.zbmath.org/authors/?q=ai:lampoh.komlanviSummary: Sensitivity analysis is a key tool in the study of the relationships between the input parameters of a model and the output solution. Although sensitivity analysis is extensively addressed in the literature, little attention has been brought to the methodological aspects of the sensitivity of nonlinear parametric solutions computed through a continuation technique. This paper proposes four combinations of sensitivity analysis with continuation and homotopy methods, including sensitivity analysis along solution branches or at a particular point. Theoretical aspects are discussed in the higher order continuation framework Diamant. The sensitivity methods are applied to a thermal ignition problem and some free vibration problems. Remarkable eigenvalue maps are produced for the complex nonlinear eigenvalue problems.Decomposition methods for coupled 3D equations of applied mathematics and continuum mechanics: partial survey, classification, new results, and generalizations.https://www.zbmath.org/1452.351582021-02-12T15:23:00+00:00"Polyanin, Andrei D."https://www.zbmath.org/authors/?q=ai:polyanin.andrei-d"Lychev, Sergei A."https://www.zbmath.org/authors/?q=ai:lychev.sergei-aSummary: The present paper provides a systematic treatment of various decomposition methods for linear (and some model nonlinear) systems of coupled three-dimensional partial differential equations of a fairly general form. Special cases of the systems considered are commonly used in applied mathematics, continuum mechanics, and physics. The methods in question are based on the decomposition (splitting) of a system of equations into a few simpler subsystems or independent equations. We show that in the absence of mass forces the solution of the system of four three-dimensional stationary and nonstationary equations considered can be expressed via solutions of three independent equations (two of which having a similar form) in a number of ways. The notion of decomposition order is introduced. Various decomposition methods of the first, second, and higher orders are described. To illustrate the capabilities of the methods, more than fifteen distinct systems of coupled 3D equations are discussed which describe viscoelastic incompressible fluids, compressible barotropic fluids, thermoelasticity, thermoviscoelasticity, electromagnetic fields, etc. The results obtained may be useful when constructing exact and numerical solutions of linear problems in continuum mechanics and physics as well as when testing numerical and approximate methods for linear and some nonlinear problems.Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier-Stokes equations.https://www.zbmath.org/1452.351502021-02-12T15:23:00+00:00"Furukawa, Ken"https://www.zbmath.org/authors/?q=ai:furukawa.ken"Giga, Yoshikazu"https://www.zbmath.org/authors/?q=ai:giga.yoshikazu"Hieber, Matthias"https://www.zbmath.org/authors/?q=ai:hieber.matthias"Hussein, Amru"https://www.zbmath.org/authors/?q=ai:hussein.amru"Kashiwabara, Takahito"https://www.zbmath.org/authors/?q=ai:kashiwabara.takahito"Wrona, Marc"https://www.zbmath.org/authors/?q=ai:wrona.marcThe approximately synchronizable state for a kind of coupled system of wave equations.https://www.zbmath.org/1452.930042021-02-12T15:23:00+00:00"Lu, Xing"https://www.zbmath.org/authors/?q=ai:lu.xingSummary: This paper deals with the approximately synchronizable state by groups for a kind of coupled system of wave equations with Dirichlet boundary controls. So far, the approximate boundary synchronization for a kind of coupled system of wave equations has already been deeply studied; however, the study on the approximately synchronizable state still needs to be done in details. In this paper, we will give some results on the determination of approximately synchronizable state by groups and the attainable set of them.Realizability-preserving DG-IMEX method for the two-moment model of fermion transport.https://www.zbmath.org/1452.652292021-02-12T15:23:00+00:00"Chu, Ran"https://www.zbmath.org/authors/?q=ai:chu.ran"Endeve, Eirik"https://www.zbmath.org/authors/?q=ai:endeve.eirik"Hauck, Cory D."https://www.zbmath.org/authors/?q=ai:hauck.cory-d"Mezzacappa, Anthony"https://www.zbmath.org/authors/?q=ai:mezzacappa.anthonySummary: Building on the framework of \textit{X. Zhang} and \textit{C.-W. Shu} [ibid. 229, No. 9, 3091--3120 (2010; Zbl 1187.65096); ibid. 229, No. 23, 8918--8934 (2010; Zbl 1282.76128)], we develop a realizability-preserving method to simulate the transport of particles (fermions) through a background material using a two-moment model that evolves the angular moments of a phase space distribution function \(f\). The two-moment model is closed using algebraic moment closures; e.g., as proposed by \textit{J. Cernohorsky} and \textit{S. A. Bludman} [``Maximum entropy distribution and closure for Bose-Einstein and Fermi-Dirac radiation transport'', Astrophys. J. 433, No. 1, 250--255 (1994; \url{doi:10.1086/174640})] and \textit{Z. Banach} and \textit{W. Larecki} [Z. Angew. Math. Phys. 68, No. 4, Paper No. 100, 24 p. (2017; Zbl 1386.85003)]. Variations of this model have recently been used to simulate neutrino transport in nuclear astrophysics applications, including core-collapse supernovae and compact binary mergers. We employ the discontinuous Galerkin (DG) method for spatial discretization (in part to capture the asymptotic diffusion limit of the model) combined with implicit-explicit (IMEX) time integration to stably bypass short timescales induced by frequent interactions between particles and the background. Appropriate care is taken to ensure the method preserves strict algebraic bounds on the evolved moments (particle density and flux) as dictated by Pauli's exclusion principle, which demands a bounded distribution function (i.e., \(f \in [0, 1]\)). This realizability-preserving scheme combines a suitable CFL condition, a realizability-enforcing limiter, a closure procedure based on Fermi-Dirac statistics, and an IMEX scheme whose stages can be written as a convex combination of forward Euler steps combined with a backward Euler step. The IMEX scheme is formally only first-order accurate, but works well in the diffusion limit, and -- without interactions with the background -- reduces to the optimal second-order strong stability-preserving explicit Runge-Kutta scheme of \textit{C.-W. Shu} and \textit{S. Osher} [J. Comput. Phys. 77, No. 2, 439--471 (1988; Zbl 0653.65072)]. Numerical results demonstrate the realizability-preserving properties of the scheme. We also demonstrate that the use of algebraic moment closures not based on Fermi-Dirac statistics can lead to unphysical moments in the context of fermion transport.Dynamical analysis of age-structured pertussis model with covert infection.https://www.zbmath.org/1452.920422021-02-12T15:23:00+00:00"Tian, Xuan"https://www.zbmath.org/authors/?q=ai:tian.xuan"Wang, Wendi"https://www.zbmath.org/authors/?q=ai:wang.wendiSummary: An age-structured pertussis model with covert infection is proposed to understand the effect of covert infection on the recurrence of pertussis. It is found that vaccination only for young children does not have a decisive effect on whooping cough control. It is shown that although the vaccine coverage rate is relatively high, the model has a backward bifurcation for a larger covert infection rate. In addition, sufficient conditions for the disease-free steady state to be globally asymptotically stable are obtained.Assessing the origin and velocity of Ca\(^{2+}\) waves in three-dimensional tissue: insights from a mathematical model and confocal imaging in mouse pancreas tissue slices.https://www.zbmath.org/1452.352252021-02-12T15:23:00+00:00"Šterk, Marko"https://www.zbmath.org/authors/?q=ai:sterk.marko"Dolenšek, Jurij"https://www.zbmath.org/authors/?q=ai:dolensek.jurij"Bombek, Lidija Križančić"https://www.zbmath.org/authors/?q=ai:bombek.lidija-krizancic"Markovič, Rene"https://www.zbmath.org/authors/?q=ai:markovic.rene"Zakelšek, Darko"https://www.zbmath.org/authors/?q=ai:zakelsek.darko"Perc, Matjaž"https://www.zbmath.org/authors/?q=ai:perc.matjaz"Pohorec, Viljem"https://www.zbmath.org/authors/?q=ai:pohorec.viljem"Stožer, Andraž"https://www.zbmath.org/authors/?q=ai:stozer.andraz"Gosak, Marko"https://www.zbmath.org/authors/?q=ai:gosak.markoSummary: Many tissues are gap-junction-coupled syncytia that support cell-to-cell communication via propagating calcium waves. This also holds true for pancreatic islets of Langerhans, where several thousand beta cells work in synchrony to ensure proper insulin secretion. Two emerging functional parameters of islet function are the location of wave initiator regions and the velocity of spreading calcium waves. High-frequency confocal laser-scanning imaging in tissue slices is one of the best available methods to determine these markers, but it is limited to two-dimensional cross-sections of an otherwise three-dimensional islet. Here we show how mathematical modeling can significantly improve this limitation. Firstly, we analytically determine the shape of velocity profiles of spherical excitation waves in the focal plane of a homogeneous three-dimensional space. Secondly, we introduce a mathematical model consisting of coupled excitable cells that considers cellular heterogeneities to approach more realistic conditions by means of numerical simulations. We demonstrate the effectiveness of our approach on experimentally recorded waves from an islet that was stimulated with 9 mM glucose. Furthermore, we show that calcium waves were primarily triggered by a specific region located \(30\mu\)m bellow the focal plane at the periphery of the islet. Additionally, we show that the velocity of the calcium wave was around \(80\mu\)m/s. We discuss the importance of our approach for the correct determination of the origin and velocity of calcium waves from experimental data, as well as the pitfalls that are due to improper procedural simplifications.Global existence and blow-up of solutions to a nonlocal Kirchhoff diffusion problem.https://www.zbmath.org/1452.350732021-02-12T15:23:00+00:00"Ding, Hang"https://www.zbmath.org/authors/?q=ai:ding.hang"Zhou, Jun"https://www.zbmath.org/authors/?q=ai:zhou.jun.1Existence and uniqueness of mild solutions for fractional partial integro-differential equations.https://www.zbmath.org/1452.352482021-02-12T15:23:00+00:00"Zhu, Bo"https://www.zbmath.org/authors/?q=ai:zhu.bo"Han, Baoyan"https://www.zbmath.org/authors/?q=ai:han.baoyanAuthors' abstract: ``We study a class of nonlinear time fractional partial integro-differential equations. The main results for the problem are obtained by the measure of noncompactness, solution operator, convex-power condensing operator and general Banach contraction mapping principle.''
Fractional calculus plays an important role in modelling memory effect in several phenomena from science and engineering. Fractional differential equations are considered powerful tools in providing an interpretation to the physical and geometrical meanings of some certain systems (for more basic information about some fractional derivatives and its connection with other related fractional derivatives, we refer to the introduction section in [\textit{M. Kaabar}, J. New Theory 2020, No. 31, 56--85 (2020)]). In addition, various research works have investigated the existence of solutions for the fractional differential equations via many newly proposed approaches or developed numerical methods. For more information, we refer the reader to some interesting and important research studies related to this topic of research such as viscous Hamilton-Jacobi equations with Caputo time-fractional derivative [\textit{F. Camilli } and \textit{A. Goffi}, Nonlinear Differ. Equ. Appl. 27, No. 22, 1--37 (2020; Zbl 1452.35234)], fractional elliptic equations [\textit{N. Cusimano} et al., ESAIM, Math. Model. Numer. Anal. 54, No. 3, 751--774 (2020; Zbl 1452.35237)], fixed point results in set \(P_{h,e}\) [\textit{L. Zhang} et al., Topol. Methods Nonlinear Anal. 54, No. 2A, 537--566 (2019; Zbl 1445.47039)], the Calderon problem on the fractional Schrödinger equation [\textit{T. Ghosh} et al., Anal. PDE 13, No. 2, 455--475 (2020; Zbl 1439.35530)], singular fractional nonlinear differential equation [\textit{F. M. Gaafar}, J. Egypt. Math. Soc. 26, 469--482 (2018; Zbl 1441.34007)], analytic semigroup [\textit{K. Ryszewska}, J. Math. Anal. Appl. 483, No. 2, Article ID 123654, 17 p. (2020; Zbl 1436.35323)], fractional porous medium equation [\textit{L. C. F. Ferreira} et al., Bull. Sci. Math. 153, 86--117 (2019; Zbl 1433.35185)], nonlinear fractional parabolic inequalities [\textit{S. D. Taliaferro}, J. Math. Pures Appl. (9) 133, 287--328 (2020; Zbl 1437.35697)], fractional composite functions and the Cauchy problem for the nonlinear half wave equation [\textit{K. Hidano } and \textit{C. Wang}, Sel. Math., New Ser. 25, No. 1, 1--28 (2019; Zbl 1428.35662)], \(L_{p}\)-estimates for time fractional parabolic equations [\textit{H. Dong} and \textit{D. Kim}, J. Funct. Anal. 278, No. 3, Article ID 108338, 66 p. (2020; Zbl 1427.35316)], fractional
boundary value problems with p-Laplacian [\textit{A. Ghanmi} and \textit{Z. Zhang}, Bull. Korean Math. Soc. 56, No. 5, 1297--1314 (2019; Zbl 1432.34012)], and conformable derivatives and applications to differential impulsive systems [\textit{Y. Gholami} and \textit{K. Ghanbari}, S\(\vec{\text{e}}\)MA J. 75, No. 2, 305--333 (2018; Zbl 1400.26012)]. Therefore, researchers, mathematicians, and engineers are now studying the applicability of the fractional derivatives in various modelling scenarios (we refer the reader to this valuable book authored by Qing Du about the nonlocal modelling, analysis, and computation [\textit{Q. Du}, Nonlocal modeling, analysis, and computation. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2019; Zbl 1423.00007)]). Although fractional derivatives offer many advantages, there are also some disadvantages such as the difficulty of finding analytical or exact solutions or even impossible to find ones for some certain systems formulated in the senses of fractional derivatives. As a result, new approximate-analytical or numerical methods are needed to investigate the existence of solutions to some certain systems (see the approximate-analytical method, double Laplace transform, that was used in solving the nonlinear fractional Schrödinger equation with second-order spatio-temporal dispersion in [\textit{M. K. A. Kaabar} et al., ``New approximate-analytical solutions for the nonlinear fractional Schrödinger equation with second-order spatio-temporal dispersion via double Laplace transform method'', Preprint, \url{arXiv:2010.10977}]). In this research work, the initial boundary value problem (IBVP) of the nonlinear time-fractional partial integro-differential equations (TFPIDEs) which is very important in physics has been investigated to prove the existence and uniqueness of the mild solutions for this problem via the solution operator, noncompactness measure, convex-power condensing operator, and general Banach contraction mapping principle. The proposed problem in this paper can be expressed as follows:
Given continuous functions, \(h\) and \(f\), such that \(h:\Re \rightarrow \Re\) and \(f:[0,b] \times \Re^{2} \rightarrow \Re\); \(1< \beta < 2\); a Banach space denoted by \(L^{2}([0,\pi])\); a linear operator, denoted by \(\Omega\), is defined in this paper as: \(\Omega u(x,t)=\int_0^t \eta(t,s)u(x,s)ds\) where \(\eta \in C[D,\Re_{+}], \Re_{+}=[0,\infty)\) such that \(D={(t,s)\in \Re^{2}:0\leq s \leq t \leq b}\). Then, the IBVP of TFPIDEs can be written as follows:
\(\frac{\partial}{\partial t}(u(x,t)+h(u(x,t)))=\int_0^t \frac{(t-s)^{\beta-2}}{\Gamma(\beta-1)}\frac{\partial^{2}}{\partial x^{2}}(u(x,s)+h(u(x,s)))ds+f(t,u(x,t).\Omega u(x,t))\), where \(t\in[0,b];\) This problem is subject to \(u(0,t)=u(\pi,t)=0, t\in[0,b]\), and \(u(x,0)=\Lambda(x), x\in[0,\pi]\), where \(\Lambda \in L^{2}([0,\pi])\).
The degree of novelty in this paper is considered very good because as mentioned by authors that this paper has successfully improved some previous studies' results on this research topic where the challenge of satisfying the strong restrict inequality condition has been overcome using the new above approach in this research paper. All in all, this paper is well-organized, and it has a a good discussion of all results. Therefore, this research work is highly recommended for mathematicians and other interested researchers in this field of research. Further research studies can be conducted on finding other related approaches or open problems related to this research study can be also proposed in future research papers about this interesting topic of research.
Reviewer: Mohammed Kaabar (Gelugor)Spatiotemporal pattern formation in 2D prey-predator system with nonlocal intraspecific competition.https://www.zbmath.org/1452.352232021-02-12T15:23:00+00:00"Pal, Swadesh"https://www.zbmath.org/authors/?q=ai:pal.swadesh"Petrovskii, Sergei"https://www.zbmath.org/authors/?q=ai:petrovskii.sergei-v"Ghorai, S."https://www.zbmath.org/authors/?q=ai:ghorai.saktipada|ghorai.santu|ghorai.santanu"Banerjee, Malay"https://www.zbmath.org/authors/?q=ai:banerjee.malaySummary: There is growing evidence that ecological interactions are often nonlocal. Correspondingly, increasing attention is paid to mathematical models with nonlocal terms as such models may provide a more realistic description of ecological dynamics. Here we consider a nonlocal prey-predator model where the movement of both species is described by the standard Fickian diffusion, and hence is local, but the intra-specific competition of prey is nonlocal and is described by a convolution-type term with the `top-hat' (piecewise-constant) kernel. The prey growth rate also includes the strong Allee effect. The system is studied using a combination of analytical tools and extensive numerical simulations. We obtain that nonlocality makes possible the pattern formation due to the Turing instability, which is not possible in the corresponding local model. We also obtain that the nonlocality creates bistability: it depends on the initial conditions which of the two spatially heterogeneous distributions emerges. Finally, we show that the bifurcation structure of the system is less sensitive to the choice of parametrization than it is in the corresponding nonspatial case, suggesting that nonlocality may decrease the structural sensitivity of the system.Symmetry breaking in a two-component system with repulsive interactions and linear coupling.https://www.zbmath.org/1452.780242021-02-12T15:23:00+00:00"Sakaguchi, Hidetsugu"https://www.zbmath.org/authors/?q=ai:sakaguchi.hidetsugu"Malomed, Boris A."https://www.zbmath.org/authors/?q=ai:malomed.boris-aSummary: We extend the well-known theoretical treatment of the spontaneous symmetry breaking (SSB) in two-component systems combining linear coupling and self-attractive nonlinearity to a system in which the linear coupling competes with repulsive interactions. First, we address one- and two-dimensional (1D and 2D) ground-state (GS) solutions and 2D vortex states with topological charges \(S=1\) and 2, maintained by a confining harmonic-oscillator (HO) potential. The system can be implemented in BEC and optics. By means of the Thomas-Fermi approximation and numerical solution of the underlying coupled Gross-Pitaevskii equations, we demonstrate that SSB takes place, in the GSs and vortices alike, when the cross-component repulsion is stronger than the self-repulsion in each component. The SSB transition is categorized as a supercritical bifurcation, which gives rise to states featuring broken symmetry in an inner area, and intact symmetry in a surrounding layer. Unlike stable GSs and vortices with \(S=1\), the states with \(S=2\) are unstable against splitting. We also address SSB for 1D gap solitons in the system including a lattice potential. In this case, SSB takes place under the opposite condition, i.e., the cross-component repulsion must be weaker than the self-repulsion, and SSB is exhibited by antisymmetric solitons.Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model.https://www.zbmath.org/1452.350192021-02-12T15:23:00+00:00"Fischer, Julian"https://www.zbmath.org/authors/?q=ai:fischer.julian"Kniely, Michael"https://www.zbmath.org/authors/?q=ai:kniely.michaelA new global nonreflecting boundary condition with diagonal coefficient matrices for analysis of unbounded media.https://www.zbmath.org/1452.740132021-02-12T15:23:00+00:00"Mirzajani, M."https://www.zbmath.org/authors/?q=ai:mirzajani.m"Khaji, N."https://www.zbmath.org/authors/?q=ai:khaji.naser"Khodakarami, M. I."https://www.zbmath.org/authors/?q=ai:khodakarami.m-iSummary: In this paper, a new semi-analytical method is developed with introducing a new global nonreflecting boundary condition at mediumtructure interface, in which the coefficient matrices, as well as dynamic-stiffness matrix are diagonal. In this method, only the boundary of the problem's domain is discretized with higher-order sub-parametric elements, where special shape functions and higher-order Chebyshev mapping functions are employed. Implementing the weighted residual method and using Clenshaw-Curtis quadrature lead to diagonal Bessel's differential equations in the frequency domain. This method is then developed to calculate the dynamic-stiffness matrix throughout the unbounded medium. This method is a semi-analytical method which is based on substructure approach. Solving two first-order ordinary differential equations (i.e., interaction force-displacement relationship and governing differential equation in dynamic stiffness) allows the boundary condition of the mediumtructure interface and radiation condition at infinity to be satisfied, respectively. These two differential equations are then diagonalized by implementing the proposed semi-analytical method. The interaction force-displacement relationship may be regarded as a nonreflecting boundary condition for the substructure of bounded domain. Afterwards, this method is extended to calculate the asymptotic expansion of dynamic-stiffness matrix for high frequency and the unit-impulse response coefficient of the unbounded media. Finally, six benchmark problems are solved to illustrate excellent agreements between the results of the present method and analytical solutions and/or other numerical methods available in the literature.Solving engineering models using hyperbolic matrix functions.https://www.zbmath.org/1452.650832021-02-12T15:23:00+00:00"Defez, Emilio"https://www.zbmath.org/authors/?q=ai:defez.emilio"Sastre, Jorge"https://www.zbmath.org/authors/?q=ai:sastre.jorge"Ibáñez, Javier"https://www.zbmath.org/authors/?q=ai:ibanez.jacinto-javier"Peinado, Jesús"https://www.zbmath.org/authors/?q=ai:peinado.jesusSummary: In this paper a method for computing hyperbolic matrix functions based on Hermite matrix polynomial expansions is outlined. Hermite series truncation together with Paterson-Stockmeyer method allow to compute the hyperbolic matrix cosine efficiently. A theoretical estimate for the optimal value of its parameters is obtained. An efficient and highly-accurate Hermite algorithm and a MATLAB implementation have been developed. The MATLAB implementation has been compared with the MATLAB function on matrices of different dimensions, obtaining lower execution time and higher accuracy in most cases. To do this we used an NVIDIA Tesla K20 GPGPU card, the CUDA environment and MATLAB. With this implementation we get much better performance for large scale problems.A new treatment based on hybrid functions to the solution of telegraph equations of fractional order.https://www.zbmath.org/1452.352432021-02-12T15:23:00+00:00"Mollahasani, N."https://www.zbmath.org/authors/?q=ai:mollahasani.nasibeh"Moghadam, M. Mohseni"https://www.zbmath.org/authors/?q=ai:mohseni-moghadam.m"Afrooz, K."https://www.zbmath.org/authors/?q=ai:afrooz.kSummary: In this paper, a new operational method based on hybrid functions of Legendre polynomials and Block-Pulse-Functions will be presented. The operational matrix of fractional integration is derived and used to take an acceptable approximate for the solution of a telegraph equation of fractional order. An error estimation will be presented to give an image of the goodness of the solution. Some numerical examples demonstrate the efficiency of the proposed method.Stationary shock-like transition fronts in dispersive systems.https://www.zbmath.org/1452.351012021-02-12T15:23:00+00:00"Gavrilyuk, Sergey"https://www.zbmath.org/authors/?q=ai:gavrilyuk.sergey-l"Nkonga, Boniface"https://www.zbmath.org/authors/?q=ai:nkonga.boniface"Shyue, Keh-Ming"https://www.zbmath.org/authors/?q=ai:shyue.keh-ming"Truskinovsky, Lev"https://www.zbmath.org/authors/?q=ai:truskinovsky.levA cross-diffusive evolution system arising from biological transport networks.https://www.zbmath.org/1452.352212021-02-12T15:23:00+00:00"Li, Bin"https://www.zbmath.org/authors/?q=ai:li.bin.1|li.bin"Li, Xie"https://www.zbmath.org/authors/?q=ai:li.xieSummary: In this paper, we are concerned with a cross-diffusive evolution system arising from biological transport networks. We study the general regularity properties of solutions to the corresponding Dirichlet-Neumann initial-boundary value problem (DNibvp). By applying the properties of divergence equations and the Dirichlet heat semigroup, we find that DNibvp possesses a globally classical solution which is unique and uniformly bounded. Based on this uniform boundedness, we establish the existence of the steady states by means of dynamical methods. Our results demonstrate that the transient solution will stabilize to the stationary solution in infinite time with an exponential time-decay rate.Decays for Kelvin-Voigt damped wave equations. I: The black box perturbative method.https://www.zbmath.org/1452.350302021-02-12T15:23:00+00:00"Burq, Nicolas"https://www.zbmath.org/authors/?q=ai:burq.nicolasAuthor's abstract: We show in this article how perturbative approaches from \textit{N. Burq} and \textit{M. Hitrik} [Math. Res. Lett. 14, No. 1, 35--47 (2007; Zbl 1122.35015)] and the black box strategy from \textit{N. Burq} and \textit{M. Zworski} [J. Am. Math. Soc. 17, No. 2, 443--471 (2004; Zbl 1050.35058)] allow us to obtain decay rates for Kelvin-Voigt damped wave equations from quite standard resolvent estimates: Carleman estimates or geometric control estimates for the Helmholtz equation; Carleman or other resolvent estimates for the Helmholtz equation. Though in this context of Kelvin-Voigt damping, such an approach is unlikely to allow for the optimal results when additional geometric assumptions are considered, it turns out that using this method, we can obtain the usual logarithmic decay which is optimal in general cases. We also present some applications of this approach giving decay rates in some particular geometries (tori).
Reviewer: Kaïs Ammari (Monastir)Simultaneous identification of convection velocity and source strength in a convection-diffusion equation.https://www.zbmath.org/1452.652062021-02-12T15:23:00+00:00"Cheng, Ting"https://www.zbmath.org/authors/?q=ai:cheng.ting"Hu, Junjun"https://www.zbmath.org/authors/?q=ai:hu.junjun"Jiang, Daijun"https://www.zbmath.org/authors/?q=ai:jiang.daijunSummary: In this work we are concerned with the analysis on a simultaneous finite element reconstruction of the convection velocity and source strength in a time-dependent convection-diffusion equation. The ill-posed problem is formulated into an output least-squares nonlinear minimization by an appropriately selected Tikhonov regularization. The regularizing effect and mathematical properties of the regularized system are justified and demonstrated. The nonlinear optimization problem is approximated by a fully discrete finite element method, whose convergence is rigorously established.Uniform Sobolev estimates for Schrödinger operators with scaling-critical potentials and applications.https://www.zbmath.org/1452.351282021-02-12T15:23:00+00:00"Mizutani, Haruya"https://www.zbmath.org/authors/?q=ai:mizutani.haruyaSummary: We prove uniform Sobolev estimates for the resolvent of Schrödinger operators with large scaling-critical potentials without any repulsive condition. As applications, global-in-time Strichartz estimates including some nonadmissible retarded estimates, a Hörmander-type spectral multiplier theorem, and Keller-type eigenvalue bounds with complex-valued potentials are also obtained.Partial differential equations. An introduction.https://www.zbmath.org/1452.350012021-02-12T15:23:00+00:00"Shah, Nita H."https://www.zbmath.org/authors/?q=ai:shah.nita-h"Jani, Mrudul Y."https://www.zbmath.org/authors/?q=ai:jani.mrudul-yPublisher's description: Differential equations play a noticeable role in engineering, physics, economics and other disciplines. They permits us to model changing forms in both mathematical and physical problems. These equations are precisely used when a deterministic relation containing some continuously varying quantities and their rates of change in space and/or time is recognized or postulated.
This book is intended to provide a straightforward introduction to the concept of partial differential equations. It provides a diversity of numerical examples framed to nurture the intellectual level of scholars. It includes enough examples to provide students with a clear concept, and also offers short questions for reader comprehension. Construction of real-life problems are considered in the last chapter along with applications.
Research scholars and students working in the fields of engineering, physics, and different branches of mathematics need to learn the concepts of partial differential equations to solve their problems This book will serve their needs instead of having to use the more complex books that contain more concepts then needed.A logical account for linear partial differential equations.https://www.zbmath.org/1452.031352021-02-12T15:23:00+00:00"Kerjean, Marie"https://www.zbmath.org/authors/?q=ai:kerjean.marieAcoustic radiation analysis for a control domain based on Green's function.https://www.zbmath.org/1452.653792021-02-12T15:23:00+00:00"Chen, Luyun"https://www.zbmath.org/authors/?q=ai:chen.luyun"Liang, Xiaofeng"https://www.zbmath.org/authors/?q=ai:liang.xiaofeng"Yi, Hong"https://www.zbmath.org/authors/?q=ai:yi.hongSummary: The acoustic radiation problem in the control domain with complex boundaries is investigated. Acoustic Green's function for acoustic point sources in the control domain was obtained by using conformal transformation theory. The expression of the acoustic radiation function for the control domain was derived. The acoustic radiation characteristics of a pulsating sphere in a quarter-infinite domain and numerical simulations were compared to illustrate the effects of the boundary characteristic, location, and radiation frequency on acoustic radiation power and acoustic directivity. This study provides a new method to analyze the acoustic radiation problem with complex boundaries.Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network.https://www.zbmath.org/1452.912542021-02-12T15:23:00+00:00"Rasanan, Amir Hosein Hadian"https://www.zbmath.org/authors/?q=ai:rasanan.amir-hosein-hadian"Bajalan, Nastaran"https://www.zbmath.org/authors/?q=ai:bajalan.nastaran"Parand, Kourosh"https://www.zbmath.org/authors/?q=ai:parand.kourosh"Rad, Jamal Amani"https://www.zbmath.org/authors/?q=ai:rad.jamal-amaniSummary: By the rapid growth of available data, providing data-driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive-decision-making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg-Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.Extremal principal eigenvalue of the bi-Laplacian operator.https://www.zbmath.org/1452.653172021-02-12T15:23:00+00:00"Mohammadi, S. A."https://www.zbmath.org/authors/?q=ai:mohammadi.seyyed-abbas"Bahrami, F."https://www.zbmath.org/authors/?q=ai:bahrami.fariba|bahrami.faridSummary: In this paper we propose two numerical algorithms to derive the extremal principal eigenvalue of the bi-Laplacian operator under Navier boundary conditions or Dirichlet boundary conditions. Consider a non-homogeneous hinged or clamped plate \(\Omega\), the algorithms converge to the density functions on \(\Omega\) which yield the maximum or minimum basic frequency of the plate.Small scale creation in active scalars.https://www.zbmath.org/1452.351392021-02-12T15:23:00+00:00"Kiselev, Alexander A."https://www.zbmath.org/authors/?q=ai:kiselev.alexander-aSummary: The focus of the course is on small scale formation in solutions of the incompressible Euler equation of fluid dynamics and associated models. We first review the regularity results and examples of small scale growth in two dimensions. Then we discuss a specific singular scenario for the three-dimensional Euler equation discovered by \textit{T. Y. Hou} and \textit{P. Liu} [Res. Math. Sci. 2, Paper No. 5, 26 p. (2015; Zbl 1320.35269)] and \textit{G. Luo} and \textit{T. Y. Hou} [Multiscale Model. Simul. 12, No. 4, 1722--1776 (2014; Zbl 1316.35235)], and analyze some associated models. Finally, we will also talk about the surface quasi-geostrophic (SQG) equation, and construct an example of singularity formation in the modified SQG patch solutions as well as examples of unbounded growth of derivatives for the smooth solutions.
For the entire collection see [Zbl 1448.76006].Localized method of approximate particular solutions for solving unsteady Navier-Stokes problem.https://www.zbmath.org/1452.652922021-02-12T15:23:00+00:00"Zhang, Xueying"https://www.zbmath.org/authors/?q=ai:zhang.xueying"Chen, Muyuan"https://www.zbmath.org/authors/?q=ai:chen.muyuan"Chen, C. S."https://www.zbmath.org/authors/?q=ai:chen.caisheng|chen.changsong|chen.chun-shu|chen.chian-shu|chen.ching-shung|chen.chiou-shiun|chen.chuin-shan|chen.chao-shi|chen.chunsheng|chen.changsheng|chen.chen-san|chen.caisen|chen.chii-sheng|chen.ching-shun|chen.chun-shuo|chen.chuansheng|chen.chung-shue|chen.chzhun-syan|chen.chang-shuai|chen.chaio-shiung|chen.chin-sheng|chen.chishu|chen.chongshuang|chen.chun-syan|chen.chun-shwu|chen.chun-sian|chen.ching-shyang|chen.chang-shing|chen.chu-song|chen.chengsen|chen.chin-sung"Li, Zhiyong"https://www.zbmath.org/authors/?q=ai:li.zhiyongSummary: The localized method of approximate particular solutions (LMAPS) is proposed to solve two-dimensional transient incompressible Navier-Stokes systems of equations in primitive variables. The equations contain the Laplacian operator. In avoiding ill-conditioning problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear system within local supporting domain in which five nearest neighboring points and multiquadrics are used for interpolation. Then local matrices are reformulated in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The method is assessed on driven cavity problem and flow around cylinder. The numerical experiments show that the newly developed LMAPS is suitable for solving incompressible Navier-Stokes equations with high accuracy and efficiency.Coexistence states of a periodic cooperative reaction-diffusion system with nonlinear functional response.https://www.zbmath.org/1452.350752021-02-12T15:23:00+00:00"Wang, Ying"https://www.zbmath.org/authors/?q=ai:wang.ying.2|wang.ying.6|wang.ying.3|wang.ying.1|wang.ying|wang.ying.8"Jia, Yunfeng"https://www.zbmath.org/authors/?q=ai:jia.yunfengSummary: In this paper, by using the existence and comparison conclusions for the \(T\)-periodic quasi-monotone nondecreasing system, we investigate a cooperative system with fractional functional response and the homogeneous Dirichlet boundary conditions. We mainly discuss the existence, uniqueness and nonexistence problems of positive solutions of the system by the method of upper and lower solutions. The numerical examples show that the relevant results hold with certain parameter values.High-order lump-type solutions and their interaction solutions to a \((3+1)\)-dimensional nonlinear evolution equation.https://www.zbmath.org/1452.350612021-02-12T15:23:00+00:00"Fang, Tao"https://www.zbmath.org/authors/?q=ai:fang.tao"Wang, Hui"https://www.zbmath.org/authors/?q=ai:wang.hui"Wang, Yun-Hu"https://www.zbmath.org/authors/?q=ai:wang.yunhu"Ma, Wen-Xiu"https://www.zbmath.org/authors/?q=ai:ma.wen-xiuDarboux transformation for a negative order AKNS equation.https://www.zbmath.org/1452.350652021-02-12T15:23:00+00:00"Wajahat, H."https://www.zbmath.org/authors/?q=ai:wajahat.h"Riaz, A."https://www.zbmath.org/authors/?q=ai:riaz.anam|riaz.arshad|riaz.amirSoliton solutions to coupled nonlinear evolution equations modelling a third harmonic resonance in the theory of capillary-gravity waves.https://www.zbmath.org/1452.760362021-02-12T15:23:00+00:00"Jones, Mark C. W."https://www.zbmath.org/authors/?q=ai:jones.m-christopher-wSummary: Soliton solutions are sought to a pair of coupled nonlinear partial differential equations which model the interface of two stratified ideal fluids and which occur when a fundamental mode and its third harmonic component induce a resonant interaction. These equations bear a resemblance to the standard coupled nonlinear Schrodinger equations, but they contain additional terms which makes their analysis quite different. There are two parameters in the problem: the ratio of the velocity of the fluids and of their densities. A large number of solutions is found and some important special cases are studied in detail.Numerical analysis of a dual-phase-lag model involving two temperatures.https://www.zbmath.org/1452.652232021-02-12T15:23:00+00:00"Bazarra, Noelia"https://www.zbmath.org/authors/?q=ai:bazarra.noelia"Fernández, José R."https://www.zbmath.org/authors/?q=ai:fernandez.jose-ramon"Magaña, Antonio"https://www.zbmath.org/authors/?q=ai:magana.antonio"Quintanilla, Ramón"https://www.zbmath.org/authors/?q=ai:quintanilla.ramonSummary: In this paper, we numerically analyse a phase-lag model with two temperatures which arises in the heat conduction theory. The model is written as a linear partial differential equation of third order in time. The variational formulation, written in terms of the thermal acceleration, leads to a linear variational equation, for which we recall an existence and uniqueness result and an energy decay property. Then, using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives, fully discrete approximations are introduced. A discrete stability property is proved, and a priori error estimates are obtained, from which the linear convergence of the approximation is derived. Finally, some one-dimensional numerical simulations are described to demonstrate the accuracy of the approximation and the behaviour of the solution.Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity.https://www.zbmath.org/1452.350392021-02-12T15:23:00+00:00"Kukavica, Igor"https://www.zbmath.org/authors/?q=ai:kukavica.igor"Wang, Weinan"https://www.zbmath.org/authors/?q=ai:wang.weinanSummary: We address long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity in the cases of the torus, \(\mathbb{R}^2\), and on a bounded domain with Lions or Dirichlet boundary conditions. In all the cases, we obtain bounds on the long time behavior for the norms of the velocity and the vorticity. In particular, we obtain that the norm \(\Vert(u,\rho)\Vert_{H^2\times H^1}\) is bounded by a single exponential, improving earlier bounds.Existence of attractors for a nonlinear Timoshenko system with delay.https://www.zbmath.org/1452.350422021-02-12T15:23:00+00:00"Ramos, Anderson J. A."https://www.zbmath.org/authors/?q=ai:ramos.anderson-j-a"Santos, Manoel J. Dos"https://www.zbmath.org/authors/?q=ai:santos.manoel-j-dos"Freitas, Mirelson M."https://www.zbmath.org/authors/?q=ai:freitas.mirelson-m"Almeida Júnior, Dilberto S."https://www.zbmath.org/authors/?q=ai:almeida.dilberto-s-junSummary: This paper deals with Timoshenko's classic model for beams vibrations. Regarding the linear model of Timoshenko, there are several known results on exponential decay, controllability and numerical approximation, but there are few results that deal with the nonlinear case or even the linear case with delay type damping. In this paper, we will establish the existence of global and exponential attractors for a semilinear Timoshenko system with delay in the rotation angle equation and a friction-type damping in the transverse displacement equation. Since the damping acts on the two equations of the system, we should not assume the well-known velocity equality.Effective slow dynamics models for a class of dispersive systems.https://www.zbmath.org/1452.350142021-02-12T15:23:00+00:00"Baumstark, Simon"https://www.zbmath.org/authors/?q=ai:baumstark.simon"Schneider, Guido"https://www.zbmath.org/authors/?q=ai:schneider.guido"Schratz, Katharina"https://www.zbmath.org/authors/?q=ai:schratz.katharina"Zimmermann, Dominik"https://www.zbmath.org/authors/?q=ai:zimmermann.dominikSummary: We consider dispersive systems of the form
\[
\begin{aligned}\partial_tU=\Lambda_UU+B_U(U,V),\qquad\varepsilon\partial_tV=\Lambda_VV+B_V(U,U)\end{aligned}
\]
in the singular limit \(\varepsilon\rightarrow 0\), where \(\Lambda_U,\Lambda_V\) are linear and \(B_U,B_V\) bilinear mappings. We are interested in deriving error estimates for the approximation obtained through the regular limit system
\[
\begin{aligned}\partial_t\psi_U=\Lambda_U\psi_U-B_U(\psi_U,\Lambda_V^{-1}B_V(\psi_U,\psi_U))\end{aligned}
\]
from a more general point of view. Our abstract approximation theorem applies to a number of semilinear systems, such as the Dirac-Klein-Gordon system, the Klein-Gordon-Zakharov system, and a mean field polaron model. It extracts the common features of scattered results in the literature, but also gains an approximation result for the Dirac-Klein-Gordon system which has not been documented in the literature before. We explain that our abstract approximation theorem is sharp in the sense that there exists a quasilinear system of the same structure where the regular limit system makes wrong predictions.Halfspaces minimise nonlocal perimeter: a proof via calibrations.https://www.zbmath.org/1452.490252021-02-12T15:23:00+00:00"Pagliari, Valerio"https://www.zbmath.org/authors/?q=ai:pagliari.valerioThe paper deals with a nonlocal version of Plateau's problem. The author considers a nonlocal extension of the total variation functional, featuring a positive kernel which may be not summable at the origin. This includes fractional kernels and extends previous literature to the case where the kernel is not monotone or not radial. The main result of the paper is that half-spaces are the unique minimizers of the nonlocal perimeter in a ball.
The minimality is proven by showing that halfspaces admit calibrations, where the classical notion of calibration is adapted to the nonlocal setting. In this context, a calibration \(\zeta\) of a given set \(E\) is a scalar function, defined on pairs of points \((x,y)\) in the space, with values in \([-1,1]\), and satisfying two technical properties: The first property is a nonlocal extension of the vanishing divergence condition. The second property implies that \(\zeta(x,y)=-1\) if \(x\in E\) and \(y\in E^{c}\), while \(\zeta(x,y)=1\) if \(x\in E^{c}\) and \(y\in E\); thus (under regularity assumptions) \(\zeta(x,y)\) gives the sign of the scalar product between the vector \(y-x\) and the inner normal to \(E\) at a point close to the segment between \(x\) and \(y\). It is proven that a set with a calibration is a local minimizer; moreover, any local minimizer is associated to a calibration.
The final section concerns an application to a \(\Gamma\)-convergence result for nonlocal functionals with a certain scaling of the kernel. It is shown that the functionals converge to (an extension of) De Giorgi's perimeter, featuring an anisotropic norm. The main tool to prove this result is a characterization of the anisotropic norm, obtained by using the minimality of half-spaces shown in the first part of the paper.
Reviewer: Giuliano Lazzaroni (Firenze)Lump solutions for two mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko equations.https://www.zbmath.org/1452.350632021-02-12T15:23:00+00:00"Ren, Bo"https://www.zbmath.org/authors/?q=ai:ren.bo"Ma, Wen-Xiu"https://www.zbmath.org/authors/?q=ai:ma.wen-xiu"Yu, Jun"https://www.zbmath.org/authors/?q=ai:yu.jun.3|yu.jun|yu.jun.2|yu.jun.1Quantum Bernoulli noises approach to stochastic Schrödinger equation of exclusion type.https://www.zbmath.org/1452.811032021-02-12T15:23:00+00:00"Ren, Suling"https://www.zbmath.org/authors/?q=ai:ren.suling"Wang, Caishi"https://www.zbmath.org/authors/?q=ai:wang.caishi"Tang, Yuling"https://www.zbmath.org/authors/?q=ai:tang.yulingSummary: Stochastic Schrödinger equations are a special type of stochastic evolution equations in complex Hilbert spaces, which arise in the study of open quantum systems. Quantum Bernoulli noises refer to annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. In this paper, we investigate a linear stochastic Schrödinger equation of exclusion type in terms of quantum Bernoulli noises. Among others, we prove the well-posedness of the equation, illustrate the results with examples, and discuss the consequences. Our main work extends that of \textit{J. Chen} and \textit{C. Wang}, [ibid. 58, No. 5, 053510, 12 p. (2017; Zbl 1364.81164)].
{\copyright 2020 American Institute of Physics}Exact solutions of an Alice-Bob KP equation.https://www.zbmath.org/1452.351772021-02-12T15:23:00+00:00"Wu, Wen-Biao"https://www.zbmath.org/authors/?q=ai:wu.wen-biao"Lou, Sen-Yue"https://www.zbmath.org/authors/?q=ai:lou.senyueDecay structures for the equations of porous elasticity in one-dimensional whole space.https://www.zbmath.org/1452.350402021-02-12T15:23:00+00:00"Quintanilla, Ramón"https://www.zbmath.org/authors/?q=ai:quintanilla.ramon"Ueda, Yoshihiro"https://www.zbmath.org/authors/?q=ai:ueda.yoshihiroSummary: This paper investigates the solutions of the porous-elastic materials with dissipation in the case of the whole real line. We consider three different cases. First we consider the case when there are dissipation mechanisms at the elastic structure and the porous structure and we prove the decay structure is standard type. Second we consider the cases when the dissipation is only on the elastic structure or on the porous structure. In these cases we show that the decay structure is regularity-loss type. Furthermore, we will show the optimality for the decay estimates for all cases.New families of non-parity-time-symmetric complex potentials with all-real spectra.https://www.zbmath.org/1452.811122021-02-12T15:23:00+00:00"Bagchi, Bijan"https://www.zbmath.org/authors/?q=ai:bagchi.bijan-kumar"Yang, Jianke"https://www.zbmath.org/authors/?q=ai:yang.jiankeSummary: New families of non-parity-time-symmetric complex potentials with all-real spectra are derived by the supersymmetry method and the pseudo-Hermiticity method. With the supersymmetry method, we find families of non-parity-time-symmetric complex partner potentials, which share the same spectrum as base potentials with known real spectra, such as the (complex) Wadati potentials. Different from previous supersymmetry derivations of potentials with real spectra, our derivation does not utilize discrete eigenmodes of base potentials. As a result, our partner potentials feature explicit analytical expressions, which contain free functions. With the pseudo-Hermiticity method, we derive a new class of non-parity-time-symmetric complex potentials with free functions and constants, whose eigenvalues appear as conjugate pairs. This eigenvalue symmetry forces the spectrum to be all-real for a wide range of choices of these functions and constants in the potential. Tuning these free functions and constants, phase transition can also be induced, where conjugate pairs of complex eigenvalues emerge in the spectrum.
{\copyright 2020 American Institute of Physics}Standing waves of fixed period for \(n+1\) vortex filaments.https://www.zbmath.org/1452.350272021-02-12T15:23:00+00:00"Craig, Walter"https://www.zbmath.org/authors/?q=ai:craig.walter"García-Azpeitia, Carlos"https://www.zbmath.org/authors/?q=ai:garcia-azpeitia.carlosSummary: The \(n+1\) vortex filament problem has explicit solutions consisting of \(n\) parallel filaments of equal circulation in the form of nested polygons uniformly rotating around a central filament which has circulation of opposite sign. We show that when the relation between temporal and spatial periods is fixed at certain rational numbers, these configurations have an infinite number of homographic time dependent standing wave patterns that bifurcate from these uniformly rotating central configurations.Interaction of Dirac \(\delta\)-waves in the nonlinear Klein-Gordon equation.https://www.zbmath.org/1452.460312021-02-12T15:23:00+00:00"Paiva, A."https://www.zbmath.org/authors/?q=ai:paiva.adelino|paiva.alberto|paiva.antonio-r-c|paiva.ana|paiva.a-p|paiva.anselmo-cardoso|paiva.afonso|paiva.alfonsoThe Cauchy problem for the nonlinear Klein-Gordon equation is considered. Necessary and sufficient conditions for the propagation of \(\delta\)-waves in the nonlinear Klein-Gordon equation are established. Special attention is paid to the so-called \(\alpha\)-solution concept, which is shown to be a convenient tool in the study of singular solutions of nonlinear PDEs.
Reviewer: Denis Sidorov (Irkutsk)The unique global solvability and optimal time decay rates for a multi-dimensional compressible generic two-fluid model with capillarity effect.https://www.zbmath.org/1452.762512021-02-12T15:23:00+00:00"Xu, Fuyi"https://www.zbmath.org/authors/?q=ai:xu.fuyi"Chi, Meiling"https://www.zbmath.org/authors/?q=ai:chi.meilingGeneral higher-order breather and hybrid solutions of the Fokas system.https://www.zbmath.org/1452.351822021-02-12T15:23:00+00:00"Chen, Ting-Ting"https://www.zbmath.org/authors/?q=ai:chen.tingting"Hu, Peng-Yan"https://www.zbmath.org/authors/?q=ai:hu.pengyan"He, Jing-Song"https://www.zbmath.org/authors/?q=ai:he.jingsongResidual symmetry of the Alice-Bob modified Korteweg-de Vries equation.https://www.zbmath.org/1452.351702021-02-12T15:23:00+00:00"Hu, Ya-Hong"https://www.zbmath.org/authors/?q=ai:hu.yahong"Ma, Zheng-Yi"https://www.zbmath.org/authors/?q=ai:ma.zhengyi"Chen, Li"https://www.zbmath.org/authors/?q=ai:chen.li.6|chen.li.3|chen.li.1|chen.li.5|chen.li.4|chen.li.7|chen.li.2Long-time asymptotics for the nonlocal mKdV equation.https://www.zbmath.org/1452.351692021-02-12T15:23:00+00:00"He, Feng-Jing"https://www.zbmath.org/authors/?q=ai:he.feng-jing"Fan, En-Gui"https://www.zbmath.org/authors/?q=ai:fan.engui"Xu, Jian"https://www.zbmath.org/authors/?q=ai:xu.jianSmooth positons of the second-type derivative nonlinear Schrödinger equation.https://www.zbmath.org/1452.351912021-02-12T15:23:00+00:00"Liu, Shu-Zhi"https://www.zbmath.org/authors/?q=ai:liu.shuzhi"Zhang, Yong-Shuai"https://www.zbmath.org/authors/?q=ai:zhang.yongshuai"He, Jing-Song"https://www.zbmath.org/authors/?q=ai:he.jingsongThe magnetic Scott correction for relativistic matter at criticality.https://www.zbmath.org/1452.811712021-02-12T15:23:00+00:00"Bley, Gonzalo A."https://www.zbmath.org/authors/?q=ai:bley.gonzalo-a"Fournais, Søren"https://www.zbmath.org/authors/?q=ai:fournais.sorenSummary: We provide a proof of the first correction to the leading asymptotics of the minimal energy of pseudo-relativistic molecules in the presence of magnetic fields, the so-called ``relativistic Scott correction,'' when max \(Z_k \alpha \leq 2/ \pi \), where \(Z_k\) is the charge of the \(k\) th nucleus and \(\alpha\) is the fine structure constant. Our theorem extends a previous result by Erdős, Fournais, and Solovej to the critical constant \(2/ \pi\) in the relativistic Hardy inequality \(| p | - \frac{2}{\pi | x |} \geq 0\).
{\copyright 2020 American Institute of Physics}The Crank-Nicolson finite spectral element method and numerical simulations for 2D non-stationary Navier-Stokes equations.https://www.zbmath.org/1452.652782021-02-12T15:23:00+00:00"Luo, Zhendong"https://www.zbmath.org/authors/?q=ai:luo.zhendong"Jiang, Wenrui"https://www.zbmath.org/authors/?q=ai:jiang.wenruiSummary: In this paper, we first build a semi-discretized Crank-Nicolson (CN) model about time for the two-dimensional (2D) non-stationary Navier-Stokes equations about vorticity-stream functions and discuss the existence, stability, and convergence of the time semi-discretized CN solutions. And then, we build a fully discretized finite spectral element CN (FSECN) model based on the bilinear trigonometric basic functions on quadrilateral elements for the 2D non-stationary Navier-Stokes equations about the vorticity-stream functions and discuss the existence, stability, and convergence of the FSECN solutions. Finally, we utilize two sets of numerical experiments to check out the correctness of theoretical consequences.Combined mean field limit and non-relativistic limit of Vlasov-Maxwell particle system to Vlasov-Poisson system.https://www.zbmath.org/1452.820232021-02-12T15:23:00+00:00"Chen, Li"https://www.zbmath.org/authors/?q=ai:chen.li.6|chen.li.4|chen.li.5|chen.li.2|chen.li.1|chen.li.3|chen.li.7"Li, Xin"https://www.zbmath.org/authors/?q=ai:li.xin.7|li.xin.14|li.xin.10|li.xin.15|li.xin.6|li.xin.3|li.xin.5|li.xin.11|li.xin|li.xin.13|li.xin.4|li.xin.2|li.xin.1|li.xin.9|li.xin.12"Pickl, Peter"https://www.zbmath.org/authors/?q=ai:pickl.peter"Yin, Qitao"https://www.zbmath.org/authors/?q=ai:yin.qitaoSummary: In this paper, we consider the mean field limit and non-relativistic limit of the relativistic Vlasov-Maxwell particle system to the Vlasov-Poisson equation. With the relativistic Vlasov-Maxwell particle system being a starting point, we carry out the estimates (with respect to \(N\) and \(c)\) between the characteristic equation of both the Vlasov-Maxwell particle model and the Vlasov-Poisson equation, where the probabilistic method is exploited. In the last step, we take both a large \(N\) limit and a non-relativistic limit (meaning \(c\) tending to infinity) to close the argument. Deriving such a mean-field limit for interactions with Coulomb singularity is a difficult task; therefore, we introduce an \(N\)-dependent mollifier to smoothen the interactions. Note that the length scale of our mollifier is \(N^{-\alpha}\) for an \(\alpha\) that can be chosen arbitrarily close to but smaller than 1/3, which improves previous results in the literature on deriving Vlasov-Maxwell in the non-relativistic setting.
{\copyright 2020 American Institute of Physics}Pattern formation of a diffusive predator-prey model with herd behavior and nonlocal prey competition.https://www.zbmath.org/1452.920332021-02-12T15:23:00+00:00"Djilali, Salih"https://www.zbmath.org/authors/?q=ai:djilali.salihSummary: In this paper, we study the influence of the nonlocal interspecific competition of the prey population on the dynamics of the diffusive predator-prey model with prey social behavior. Using the linear stability analysis, the conditions for the positive constant steady state at which undergoes Hopf bifurcation, T-H bifurcation (Turing-Hopf bifurcation) are investigated. The Turing patterns occur in the presence of the nonlocal competition and cannot be found in the original system. For determining the dynamical behavior near T-H bifurcation point, the normal form of the T-H bifurcation has been used. Some graphical representations are provided to illustrate the theoretical results.Stochastic discontinuous Galerkin methods (SDGM) based on fluctuation-dissipation balance.https://www.zbmath.org/1452.652902021-02-12T15:23:00+00:00"Pazner, W."https://www.zbmath.org/authors/?q=ai:pazner.will"Trask, N."https://www.zbmath.org/authors/?q=ai:trask.nat"Atzberger, P. J."https://www.zbmath.org/authors/?q=ai:atzberger.paul-jAn approach to discretize stochastic partial differential equations based on fluctuation-dissipation balance (instead of a direct approach based on random fluxes) is proposed.
The authors develop robust discontinuous Galerkin methods and apply them to various initial-boundary value problems.
Reviewer: Piotr Biler (Wrocław)The large-time behavior of solutions in the critical \(L^p\) framework for compressible viscous and heat-conductive gas flows.https://www.zbmath.org/1452.762032021-02-12T15:23:00+00:00"Shi, Weixuan"https://www.zbmath.org/authors/?q=ai:shi.weixuan"Xu, Jiang"https://www.zbmath.org/authors/?q=ai:xu.jiangSummary: In this paper, we are concerned with non-isentropic Navier-Stokes equations governing compressible viscous and heat-conductive gases. We formulate an additional regularity assumption of low frequencies in \(\mathbb{R}^d\) \((d \geq 3)\) and then establish the sharp time-weighted inequality, which allows us to get the time-decay estimates of global strong solutions in the \(L^p\) critical Besov spaces. Precisely, it is shown that if the initial data belong to some Besov space \(\dot{B}_{2, \infty}^{- s_1}\) with \(s_1 \in(1 - \frac{d}{2}, s_0]\) \((s_0 \triangleq \frac{2 d}{p} - \frac{d}{2})\), then the \(L^p\) norm of the critical global solutions admits the time decay \(t^{- \frac{s_1}{2} - \frac{d}{2}(\frac{1}{2} - \frac{1}{p})} \) (in particular, \(t^{- \frac{d}{2 p}}\) if \(s_1 = s_0)\), which coincides with that of the heat kernel in the \(L^p\) framework. In comparison with the work of \textit{R. Danchin} and \textit{J. Xu} [J. Math. Fluid Mech. 20, No. 4, 1641--1665 (2018; Zbl 1404.76224)], the low-frequency assumption is improved such that the regularity \(s_1\) belongs to the whole range \((1 - \frac{d}{2}, s_0]\).
{\copyright 2020 American Institute of Physics}On the weak solutions for the rotation-two-component Camassa-Holm equation.https://www.zbmath.org/1452.762622021-02-12T15:23:00+00:00"Yang, Li"https://www.zbmath.org/authors/?q=ai:yang.li.1|yang.li.3|yang.li.2|yang.li"Mu, Chunlai"https://www.zbmath.org/authors/?q=ai:mu.chunlai"Zhou, Shouming"https://www.zbmath.org/authors/?q=ai:zhou.shouming"Tu, Xinyu"https://www.zbmath.org/authors/?q=ai:tu.xinyuSummary: This paper deals with a model the equatorial water waves with the Coriolis effect in the rotating fluid, which is called rotation-two-component Camassa-Holm system. The purpose of this work is to utilize the pseudo-parabolic regularization to establish the existence and uniqueness of weak solutions in a lower order Sobolev space \(H^s(\mathbb{R}) \times H^{s - 1}(\mathbb{R})\) with \(1 < s \leq \frac{3}{2} \).
{\copyright 2020 American Institute of Physics}A note on the transformation of variables of KP equation, cylindrical KP equation and spherical KP equation.https://www.zbmath.org/1452.351722021-02-12T15:23:00+00:00"Liu, Chun-Ping"https://www.zbmath.org/authors/?q=ai:liu.chunpingNew bilinear Bäcklund transformation and higher order rogue waves with controllable center of a generalized \((3+1)\)-dimensional nonlinear wave equation.https://www.zbmath.org/1452.350912021-02-12T15:23:00+00:00"Shen, Ya-Li"https://www.zbmath.org/authors/?q=ai:shen.yali"Yao, Ruo-Xia"https://www.zbmath.org/authors/?q=ai:yao.ruoxia"Li, Yan"https://www.zbmath.org/authors/?q=ai:li.yan.9|li.yan.8|li.yan.1|li.yan.4|li.yan.10|li.yan|li.yan.5|li.yan.6|li.yan.7|li.yan.2|li.yan.3Characteristics of rogue waves on a soliton background in the general coupled nonlinear Schrödinger equation.https://www.zbmath.org/1452.351972021-02-12T15:23:00+00:00"Wang, Xiu-Bin"https://www.zbmath.org/authors/?q=ai:wang.xiubin"Han, Bo"https://www.zbmath.org/authors/?q=ai:han.boRadially symmetric solutions for Navier-Stokes-Smoluchowski system: global existence in unbounded annular domain and center singularity.https://www.zbmath.org/1452.762582021-02-12T15:23:00+00:00"Zhu, Limei"https://www.zbmath.org/authors/?q=ai:zhu.limei"Huang, Bingyuan"https://www.zbmath.org/authors/?q=ai:huang.bingyuan"Huang, Jinrui"https://www.zbmath.org/authors/?q=ai:huang.jinruiSummary: In this paper, we establish the global existence of radially symmetric strong solutions of a fluid-particle interaction system in an unbounded annular domain. Furthermore, the description for possible breakdown of regularity for the 3D problem is studied: the concentration of mass on the center.
{\copyright 2020 American Institute of Physics}An integrable matrix Camassa-Holm equation.https://www.zbmath.org/1452.370672021-02-12T15:23:00+00:00"Chan, Li-Feng"https://www.zbmath.org/authors/?q=ai:chan.li-feng"Xia, Bao-Qiang"https://www.zbmath.org/authors/?q=ai:xia.baoqiang"Zhou, Ru-Guang"https://www.zbmath.org/authors/?q=ai:zhou.ruguangBilinear forms and dark-dark solitons for the coupled cubic-quintic nonlinear Schrödinger equations with variable coefficients in a twin-core optical fiber or non-Kerr medium.https://www.zbmath.org/1452.351832021-02-12T15:23:00+00:00"Chu, Mei-Xia"https://www.zbmath.org/authors/?q=ai:chu.mei-xia"Tian, Bo"https://www.zbmath.org/authors/?q=ai:tian.bo"Yuan, Yu-Qiang"https://www.zbmath.org/authors/?q=ai:yuan.yu-qiang"Zhang, Ze"https://www.zbmath.org/authors/?q=ai:zhang.ze"Tian, He-Yuan"https://www.zbmath.org/authors/?q=ai:tian.he-yuanGlobal solution and blow-up of the stochastic nonlinear Schrödinger system.https://www.zbmath.org/1452.811092021-02-12T15:23:00+00:00"Zhang, Qi"https://www.zbmath.org/authors/?q=ai:zhang.qi.2|zhang.qi-shuhuason|zhang.qi.1|zhang.qi|zhang.qi.4"Duan, Jinqiao"https://www.zbmath.org/authors/?q=ai:duan.jinqiao"Chen, Yong"https://www.zbmath.org/authors/?q=ai:chen.yong.7|chen.yong.4|chen.yong.2|chen.yong|chen.yong.6|chen.yong.5|chen.yong.1|chen.yong.3Summary: We study a stochastic nonlinear Schrödinger system with multiplicative white noise in energy space \(H^1\). Based on deterministic and stochastic Strichartz estimates, we prove the local well-posedness of a mild solution. Then, we prove the global well-posedness in the mass subcritical case and the defocusing case. For the mass subcritical case, we also investigate the global existence when the \(L^2\) norm of the initial value is small enough. We also study the blow-up phenomenon and give a sharp criterion via a general virial identity.
{\copyright 2020 American Institute of Physics}Potential well theory for the focusing fractional Choquard equation.https://www.zbmath.org/1452.811052021-02-12T15:23:00+00:00"Saanouni, Tarek"https://www.zbmath.org/authors/?q=ai:saanouni.tarekSummary: This note studies the non-linear fractional Schrödinger equation \(i \dot{u} -(- \operatorname{\Delta})^s u +(I_\alpha * | u |^p) | u |^{p - 2} u = 0\). In the mass super-critical and energy sub-critical regimes, the local solutions exist globally and scatter in the energy space or blow-up in finite time if the data belong to some stable sets.
{\copyright 2020 American Institute of Physics}Darboux transformations, higher-order rational solitons and rogue wave solutions for a \((2+1)\)-dimensional nonlinear Schrödinger equation.https://www.zbmath.org/1452.351812021-02-12T15:23:00+00:00"Chen, Mi"https://www.zbmath.org/authors/?q=ai:chen.mi"Li, Biao"https://www.zbmath.org/authors/?q=ai:li.biao"Yu, Ya-Xuan"https://www.zbmath.org/authors/?q=ai:yu.yaxuanTime and space fractional Schrödinger equation with fractional factor.https://www.zbmath.org/1452.352462021-02-12T15:23:00+00:00"Xiang, Pei"https://www.zbmath.org/authors/?q=ai:xiang.pei"Guo, Yong-Xin"https://www.zbmath.org/authors/?q=ai:guo.yongxin"Fu, Jing-Li"https://www.zbmath.org/authors/?q=ai:fu.jingliOn nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems.https://www.zbmath.org/1452.351202021-02-12T15:23:00+00:00"Cao, Xinlin"https://www.zbmath.org/authors/?q=ai:cao.xinlin"Diao, Huaian"https://www.zbmath.org/authors/?q=ai:diao.huaian"Liu, Hongyu"https://www.zbmath.org/authors/?q=ai:liu.hongyu|liu.hongyu.1"Zou, Jun"https://www.zbmath.org/authors/?q=ai:zou.jun.1|zou.junThe authors deal with geometric structures of Laplacian eigenfunctions and their behaviours in two dimensions. The article has 8 sections. In the first section, the history of the Laplacian eigenvalue problem, the motivation and basic definitions for the authors' main results are given. In the second part, the authors consider a relatively simple case which is two nodal lines or two singular lines intersect at in irrational angle.The problem that is considered
by the authors below is the following: $\Delta u=\lambda u$ (0.1), where $u\in L^2(\Omega)$ and $\lambda\in \mathbb R$, such that $\Omega$ is an open set in $\mathbb R$. In Section 3, the authors consider the general case that two line segments intersect at a rational age. Section 3 is devoted to the presentation and discussion of the main results. In Section 4, the authors show the case that the intersecting angle is irrational and the vanishing order is infinity. In Sections 5 and 6, the authors study the case that the intersecting angle is rational and the vanishing order is finite. In Section 7 is the discussion of a generic condition. And finally, in the last section, the authors establish unique recovery results for the inverse problem and the inverse diffraction grating problem by at most two incident waves.
Reviewer: Khanlar R. Mamedov (Mersin)Degenerate solutions of the nonlinear self-dual network equation.https://www.zbmath.org/1452.351752021-02-12T15:23:00+00:00"Qiu, Ying-Yang"https://www.zbmath.org/authors/?q=ai:qiu.ying-yang"He, Jing-Song"https://www.zbmath.org/authors/?q=ai:he.jingsong"Li, Mao-Hua"https://www.zbmath.org/authors/?q=ai:li.maohuaNumerical analysis of backward subdiffusion problems.https://www.zbmath.org/1452.652162021-02-12T15:23:00+00:00"Zhang, Zhengqi"https://www.zbmath.org/authors/?q=ai:zhang.zhengqi"Zhou, Zhi"https://www.zbmath.org/authors/?q=ai:zhou.zhiA parareal finite volume method for variable-order time-fractional diffusion equations.https://www.zbmath.org/1452.651912021-02-12T15:23:00+00:00"Liu, Huan"https://www.zbmath.org/authors/?q=ai:liu.huan"Cheng, Aijie"https://www.zbmath.org/authors/?q=ai:cheng.aijie"Wang, Hong"https://www.zbmath.org/authors/?q=ai:wang.hong.1Summary: In this paper, we investigate the well-posedness and solution regularity of a variable-order time-fractional diffusion equation, which is often used to model the solute transport in complex porous media where the micro-structure of the porous media may changes over time. We show that the variable-order time-fractional diffusion equations have flexible abilities to eliminate the nonphysical singularity of the solutions to their constant-order analogues. We also present a finite volume approximation and provide its stability and convergence analysis in a weighted discrete norm. Furthermore, we develop an efficient parallel-in-time procedure to improve the computational efficiency of the variable-order time-fractional diffusion equations. Numerical experiments are performed to confirm the theoretical results and to demonstrate the effectiveness and efficiency of the parallel-in-time method.New optical soliton solutions of the perturbed Fokas-Lenells equation.https://www.zbmath.org/1452.350642021-02-12T15:23:00+00:00"Shehata, Maha S. M."https://www.zbmath.org/authors/?q=ai:shehata.maha-s-m"Rezazadeh, Hadi"https://www.zbmath.org/authors/?q=ai:rezazadeh.hadi"Zahran, Emad H. M."https://www.zbmath.org/authors/?q=ai:zahran.emad-h-m"Tala-Tebue, Eric"https://www.zbmath.org/authors/?q=ai:tala-tebue.eric"Bekir, Ahmet"https://www.zbmath.org/authors/?q=ai:bekir.ahmetAnalytical and approximate solutions for complex nonlinear Schrödinger equation via generalized auxiliary equation and numerical schemes.https://www.zbmath.org/1452.351892021-02-12T15:23:00+00:00"Khater, Mostafa M. A."https://www.zbmath.org/authors/?q=ai:khater.mostafa-m-a"Lu, Dian-Chen"https://www.zbmath.org/authors/?q=ai:lu.dianchen"Attia, Raghda A. M."https://www.zbmath.org/authors/?q=ai:attia.raghda-a-m"Inç, Mustafa"https://www.zbmath.org/authors/?q=ai:inc.mustafaSemi-implicit Hermite-Galerkin spectral method for distributed-order fractional-in-space nonlinear reaction-diffusion equations in multidimensional unbounded domains.https://www.zbmath.org/1452.652742021-02-12T15:23:00+00:00"Guo, Shimin"https://www.zbmath.org/authors/?q=ai:guo.shimin"Mei, Liquan"https://www.zbmath.org/authors/?q=ai:mei.liquan"Li, Can"https://www.zbmath.org/authors/?q=ai:li.can.1|li.can"Zhang, Zhengqiang"https://www.zbmath.org/authors/?q=ai:zhang.zhengqiang"Li, Ying"https://www.zbmath.org/authors/?q=ai:li.ying.3|li.ying|li.ying.1|li.ying.2Summary: In this paper, we construct an efficient Hermite-Galerkin spectral method for the nonlinear reaction-diffusion equations with distributed-order fractional Laplacian in multidimensional unbounded domains. By applying Gauss-Legendre quadrature rule for the distributed integral term, we first approximate the original distributed-order fractional problem by the multi-term fractional-in-space differential equation. Applying Hermite-Galerkin spectral method in space and backward difference method in time, we establish semi-implicit fully discrete scheme. For two- and three-dimensional cases of the original fractional problem, the linear systems are solved by the preconditioned conjugate gradients method. The main advantage of our method is that the original fractional problem is directly solved in the unbounded domains, thus avoiding the errors introduced by the domain truncations. The stability analysis is rigourously established, which shows that our scheme is unconditionally stable under suitable assumption on the nonlinear term. Several numerical examples are presented to validate both stability and accuracy of the numerical method. The numerical results of the fractional Allen-Cahn, Gray-Scott, and Belousov-Zhabotinskii models show that our semi-implicit methods produce good numerical solutions.Wiener-Hopf approach for the coaxial waveguide with an impedance-coated groove on the inner wall.https://www.zbmath.org/1452.352022021-02-12T15:23:00+00:00"Öztürk, Hülya"https://www.zbmath.org/authors/?q=ai:ozturk.hulyaSummary: In the present work, the band-pass filter characteristics of a coaxial waveguide with an impedance coated groove on the inner wall and perfectly conducting outer wall is analyzed rigorously through the Wiener-Hopf technique. By using the direct Fourier transform, the related boundary value problem is reduced to the Wiener-Hopf equation whose solution contains infinitely many constants satisfying an infinite system of linear algebraic equations. These equations are solved numerically and the field terms, which depends on the solution obtained numerically, are derived explicitly. The problem is also analyzed by applying a mode-matching technique and the results are compared numerically. Besides, some computational results illustrating the effects of parameters such as depth of the groove, surface impedances and radii of the walls are also presented.Asymptotic of the solution of the contact problem for a thin elastic plate and a viscoelastic layer.https://www.zbmath.org/1452.740852021-02-12T15:23:00+00:00"Panasenko, Grigory P."https://www.zbmath.org/authors/?q=ai:panasenko.grigory-p"Elbert, A. E."https://www.zbmath.org/authors/?q=ai:elbert.aleksandr-evgenevichSummary: The contact problem for a thin elastic rigid plate described by the elasticity equations and a viscoelastic layer is solved. The ratio of the thicknesses of the plate and the layer is a small parameter, while the ratio of the Young's moduli of the layer and the plate is proportional to the cube of this parameter. The asymptotic expansion of the solution is constructed. A theorem on the estimate of the error of asymptotic approximation is formulated. Such problem appears in geophysics, in modeling of the Earth crust-magma interaction.Stochastic Navier-Stokes-Fourier equations.https://www.zbmath.org/1452.352652021-02-12T15:23:00+00:00"Breit, Dominic"https://www.zbmath.org/authors/?q=ai:breit.dominic"Feireisl, Eduard"https://www.zbmath.org/authors/?q=ai:feireisl.eduardThe paper under review studies the complete Navier-Stokes-Fourier system for viscous, heat-conducting compressible fluids subject to random effects. Randomness in the system is enforced by way of the initial data, a heat source, or by way of a multiplicative noise in the momentum equation.
The authors construct weak martingale solutions to the underlying system. These solutions are weak in the analytical sense (so that the equations making up the full system are satisfied weakly in the sense of distributions) and also weak in the probabilistic sense (so that the underlying probability space and the driving noise are also unknowns of the problem). Furthermore, crucially, the total energy is balanced.
The proof follows the now-standard multi-layer approximation scheme in the construction of weak solutions to compressible fluid systems. In particular, the authors combine the deterministic techniques in [\textit{E. Feireisl} and \textit{A. Novotný}, Singular limits in thermodynamics of viscous fluids. 2nd edition. Cham: Birkhäuser (2017; Zbl 1432.76002)] for the construction of weak solutions for the full deterministic Navier-Stokes-Fourier system with the stochastic arguments in [\textit{M. Hofmanová} and \textit{D. Breit}, Indiana Univ. Math. J. 65, No. 4, 1183--1250 (2016; Zbl 1358.60072)] for the construction of finite energy weak martingale solutions for the stochastic isothermal compressible Navier-Stokes system of equations.
As a by-product of their argument, one can expect to obtain analogous relative energy estimates for stochastic fluid systems which are very useful for stability arguments. In particular, one can expect to prove weak-strong uniqueness of solutions for the stochastic analogue of \textit{E. Feireisl} and \textit{A. Novotný} [Arch. Ration. Mech. Anal. 204, No. 2, 683--706 (2012; Zbl 1285.76034)].
Reviewer: Prince Romeo Mensah (London)Consistency of finite volume approximations to nonlinear hyperbolic balance laws.https://www.zbmath.org/1452.651842021-02-12T15:23:00+00:00"Ben-Artzi, Matania"https://www.zbmath.org/authors/?q=ai:ben-artzi.matania"Li, Jiequan"https://www.zbmath.org/authors/?q=ai:li.jiequanSummary: This paper addresses the three concepts of \textit{consistency, stability and convergence} in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of ``balance laws''. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here allow the computation of several quantities per mesh cell (e.g., slopes) and the notion of consistency must be extended to this framework. Then a suitable convergence theorem is established, generalizing the classical convergence theorem of Lax and Wendroff. Finally, the limit functions are shown to be entropy solutions by using a notion of ``Godunov compatibility'', which serves as a substitute to the entropy condition.A second-order numerical method for the aggregation equations.https://www.zbmath.org/1452.651862021-02-12T15:23:00+00:00"Carrillo, José A."https://www.zbmath.org/authors/?q=ai:carrillo.jose-antonio"Fjordholm, Ulrik S."https://www.zbmath.org/authors/?q=ai:fjordholm.ulrik-skre"Solem, Susanne"https://www.zbmath.org/authors/?q=ai:solem.susanneAuthors' abstract: Inspired by so-called TVD limiter-based second-order schemes for hyperbolic conservation laws, the authors develop a formally second-order accurate numerical method for multi-dimensional aggregation equations. The method allows for simulations to be continued after the first blow-up time of the solution. In the case of symmetric, \(\lambda\)-convex potentials with a possible Lipschitz singularity at the origin, the authors prove that the method converges in the Monge-Kantorovich distance towards the unique gradient flow solution. Several numerical experiments are presented to validate the second-order convergence rate and to explore the performance of the scheme.
Reviewer: Victor Michel-Dansac (Strasbourg)New analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media.https://www.zbmath.org/1452.653592021-02-12T15:23:00+00:00"Sun, Weiwei"https://www.zbmath.org/authors/?q=ai:sun.weiwei"Wu, Chengda"https://www.zbmath.org/authors/?q=ai:wu.chengdaSummary: Analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media has been investigated extensively in the last several decades. Of particular interest in practical applications is the lowest-order Galerkin-mixed method, in which a linear Lagrange FE approximation is used for the concentration and the lowest-order Raviart-Thomas FE approximation is used for the velocity/pressure. The previous works only showed the first-order accuracy of the method in \(L^2\)-norm in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial mesh. In this paper, we provide new and optimal \(L^2\)-norm error estimates of Galerkin-mixed FEMs for all three components in a general case. In particular, for the lowest-order Galerkin-mixed FEM, we show unconditionally the second-order accuracy in \(L^2\)-norm for the concentration. Numerical results for both two- and three-dimensional models are presented to confirm our theoretical analysis. More important is that our approach can be extended to the analysis of mixed FEMs for many strongly coupled systems to obtain optimal error estimates for all components.Convergence of the optimality criteria method for multiple state optimal design problems.https://www.zbmath.org/1452.490022021-02-12T15:23:00+00:00"Burazin, Krešimir"https://www.zbmath.org/authors/?q=ai:burazin.kresimir"Crnjac, Ivana"https://www.zbmath.org/authors/?q=ai:crnjac.ivanaSummary: We consider multiple state optimal design problems with two isotropic materials from the conductivity point of view. Since the classical solutions of these problems usually do not exist, a proper relaxation of the original problem is obtained, using the homogenization method. In [\textit{K. Burazin} et al., Commun. Math. Sci. 16, No. 6, 1597--1614 (2019; Zbl 1408.49001)] we derive necessary conditions of optimality of the relaxed problem, which enables us to implement a new variant of the optimality criteria method. It appears that this variant gives converging sequence of designs for the energy minimization problems. In this work we prove convergence of the method for energy minimization problems in the spherically symmetric case and in a case when the number of states is less than the space dimension.Error analysis of an \(L2\)-type method on graded meshes for a fractional-order parabolic problem.https://www.zbmath.org/1452.652372021-02-12T15:23:00+00:00"Kopteva, Natalia"https://www.zbmath.org/authors/?q=ai:kopteva.natalia.1|kopteva.nataliaSummary: An initial-boundary value problem with a Caputo time derivative of fractional order \(\alpha \in (0,1)\) is considered, solutions of which typically exhibit a singular behaviour at an initial time. An \(L2\)-type discrete fractional-derivative operator of order \(3-\alpha\) is considered on nonuniform temporal meshes. Sufficient conditions for the inverse-monotonicity of this operator are established, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. In particular, those results imply that milder (compared to the optimal) grading yields optimal convergence rates in positive time. Semi-discretizations in time and full discretizations are addressed. The theoretical findings are illustrated by numerical experiments.Solvability conditions for the nonlocal boundary-value problem for a differential-operator equation with weak nonlinearity in the refined Sobolev scale of spaces of functions of many real variables.https://www.zbmath.org/1452.351292021-02-12T15:23:00+00:00"Il'kiv, V. S."https://www.zbmath.org/authors/?q=ai:ilkiv.volodymyr-stepanovich"Strap, N. I."https://www.zbmath.org/authors/?q=ai:strap.nataliya-igorivna"Volyanska, I. I."https://www.zbmath.org/authors/?q=ai:volyanska.i-iSummary: We study the solvability of the nonlocal boundary-value problem for a differential equation with weak nonlinearity. By using the Nash-Mozer iterative scheme, we establish the solvability conditions for the posed problem in the Hilbert Hörmander spaces of functions of several real variables, which form a refined Sobolev scale.Identification and reconstruction of body forces in a Stokes system using shear waves.https://www.zbmath.org/1452.652132021-02-12T15:23:00+00:00"Martins, Nuno F. M."https://www.zbmath.org/authors/?q=ai:martins.nuno-f-mSummary: In this paper we consider an inverse source problem for the Brinkman system of equations (or unsteady Stokes equations). The uniqueness problem of recovering the source term from traction boundary data is studied and established from density properties of shear waves with complex frequency. This result is then applied to the reconstruction of the body force term as a superposition of shear waves. Some numerical results will be presented in order to illustrate the accuracy of the proposed method.
For the entire collection see [Zbl 1445.65001].Two-grid IPDG discretization scheme for nonlinear elliptic PDEs.https://www.zbmath.org/1452.653642021-02-12T15:23:00+00:00"Zhong, Liuqiang"https://www.zbmath.org/authors/?q=ai:zhong.liuqiang"Zhou, Liangliang"https://www.zbmath.org/authors/?q=ai:zhou.liangliang"Liu, Chunmei"https://www.zbmath.org/authors/?q=ai:liu.chunmei"Peng, Jie"https://www.zbmath.org/authors/?q=ai:peng.jieSummary: Two-grid interior penalty discontinuous Galerkin (IPDG) method for the mildly nonlinear second-order elliptic partial differential equations is studied in this paper. The IPDG finite element discretizations are developed and the corresponding well-posedness is established by introducing the equivalent weak formulation of IPDG method and combining Brouwer's fixed point theorem. Some priori error estimates for discrete solution in the broken \(H^1\)-norm, \(L^2\)-norm and \(L^\infty\)-norm are derived, respectively. Two-grid method is designed for solving IPDG discretization scheme and the corresponding error estimate is provided. Numerical experiments are also shown to confirm the efficiency of the proposed approach.Neumann type problems for the polyharmonic equation in ball.https://www.zbmath.org/1452.310132021-02-12T15:23:00+00:00"Karachik, V. V."https://www.zbmath.org/authors/?q=ai:karachik.valery-vSummary: For Neumann type problems for the homogeneous polyharmonic equation in the unit ball we obtain necessary solvability conditions in the form of orthogonality of homogeneous harmonic polynomials to linear combinations of boundary functions with coefficients taken from the integer Neumann triangle.Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains.https://www.zbmath.org/1452.652072021-02-12T15:23:00+00:00"Imperiale, S."https://www.zbmath.org/authors/?q=ai:imperiale.sebastien"Moireau, P."https://www.zbmath.org/authors/?q=ai:moireau.philippe"Tonnoir, A."https://www.zbmath.org/authors/?q=ai:tonnoir.antoineSummary: We are interested in reconstructing the initial condition of a wave equation in an unbounded domain configuration from measurements available in time on a subdomain. To solve this problem, we adopt an iterative strategy of reconstruction based on observers and time reversal adjoint formulations. We prove the convergence of our reconstruction algorithm with perfect measurements and its robustness to noise. Moreover, we develop a complete strategy to practically solve this problem on a bounded domain using artificial transparent boundary conditions to account for the exterior domain. Our work then demonstrates that the consistency error introduced by the use of approximate transparent boundary conditions is compensated by the stabilization properties obtained from the use of the available measurements, hence allowing to still be able to reconstruct the unknown initial condition.Restricting Riesz-logarithmic-Besov potentials.https://www.zbmath.org/1452.310092021-02-12T15:23:00+00:00"Liu, Liguang"https://www.zbmath.org/authors/?q=ai:liu.liguang"Yue, Chengjun"https://www.zbmath.org/authors/?q=ai:yue.chengjun"Zhang, Lunchuan"https://www.zbmath.org/authors/?q=ai:zhang.lunchuanSummary: This paper studies the restriction properties of the Riesz-logarithmic-Besov potential. Moreover, applications to the regularity of the solutions to the fractional Laplace equation with measure data are given.Regularity theory for time-fractional advection-diffusion-reaction equations.https://www.zbmath.org/1452.652032021-02-12T15:23:00+00:00"McLean, William"https://www.zbmath.org/authors/?q=ai:mclean.william"Mustapha, Kassem"https://www.zbmath.org/authors/?q=ai:mustapha.kassem"Ali, Raed"https://www.zbmath.org/authors/?q=ai:ali.raed"Knio, Omar M."https://www.zbmath.org/authors/?q=ai:knio.omar-mSummary: We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of fractional integrals.Point vortex approximation for 2D Navier-Stokes equations driven by space-time white noise.https://www.zbmath.org/1452.760542021-02-12T15:23:00+00:00"Flandoli, Franco"https://www.zbmath.org/authors/?q=ai:flandoli.franco"Luo, Dejun"https://www.zbmath.org/authors/?q=ai:luo.dejunSummary: We show that the system of point vortices, perturbed by a certain transport type noise, converges weakly to the vorticity form of 2D Navier-Stokes equations driven by the space-time white noise.Computation of interior elastic transmission eigenvalues using a conforming finite element and the secant method.https://www.zbmath.org/1452.351272021-02-12T15:23:00+00:00"Ji, Xia"https://www.zbmath.org/authors/?q=ai:ji.xia"Li, Peijun"https://www.zbmath.org/authors/?q=ai:li.peijun.2|li.peijun.1"Sun, Jiguang"https://www.zbmath.org/authors/?q=ai:sun.jiguangSummary: The interior elastic transmission eigenvalue problem, arising from the inverse scattering theory of non-homogeneous elastic media, is nonlinear, non-self-adjoint and of fourth order. This paper proposes a numerical method to compute real elastic transmission eigenvalues. To avoid treating the non-self-adjoint operator, an auxiliary nonlinear function is constructed. The values of the function are generalized eigenvalues of a series of self-adjoint fourth order problems. The roots of the function are the transmission eigenvalues. The self-adjoint fourth order problems are computed using the \(H^2\)-conforming Argyris element. The secant method is employed to search the roots of the nonlinear function. The convergence of the proposed method is proved.An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes.https://www.zbmath.org/1452.651882021-02-12T15:23:00+00:00"Escalante, C."https://www.zbmath.org/authors/?q=ai:escalante.c"Dumbser, M."https://www.zbmath.org/authors/?q=ai:dumbser.michael"Castro, M. J."https://www.zbmath.org/authors/?q=ai:castro.manuel-jSummary: In this paper we propose a novel set of first-order hyperbolic equations that can model dispersive non-hydrostatic free surface flows. The governing PDE system is obtained via a hyperbolic approximation of the family of non-hydrostatic free-surface flow models recently derived by \textit{M.-O. Bristeau} et al. [Discrete Contin. Dyn. Syst., Ser. B 20, No. 4, 961--988 (2015; Zbl 1307.35162)]. Our new hyperbolic reformulation is based on an augmented system in which the divergence constraint of the velocity is coupled with the other conservation laws via an evolution equation for the depth-averaged non-hydrostatic pressure, similar to the hyperbolic divergence cleaning applied in generalized Lagrangian multiplier methods (GLM) for magnetohydrodynamics (MHD). We suggest a formulation in which the divergence errors of the velocity field are transported with a large but finite wave speed that is directly related to the maximal eigenvalue of the governing PDE.
We then use arbitrary high order accurate (ADER) discontinuous Galerkin (DG) finite element schemes with an \textit{a posteriori} subcell finite volume limiter in order to solve the proposed PDE system numerically. The final scheme is highly accurate in smooth regions of the flow and very robust and positive preserving for emerging topographies and wet-dry fronts. It is well-balanced making use of a path-conservative formulation of HLL-type Riemann solvers based on the straight line segment path. Furthermore, the proposed ADER-DG scheme with \textit{a posteriori} subcell finite volume limiter adapts very well to modern GPU architectures, resulting in a very accurate, robust and computationally efficient computational method for non-hydrostatic free surface flows. The new model proposed in this paper has been applied to idealized academic benchmarks such as the propagation of solitary waves, as well as to more challenging physical situations that involve wave runup on a shore including wave breaking in both one and two space dimensions. In all cases the achieved agreement with analytical solutions or experimental data is very good, thus showing the validity of both, the proposed mathematical model and the numerical solution algorithm.New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations.https://www.zbmath.org/1452.351682021-02-12T15:23:00+00:00"Goldman, Michael"https://www.zbmath.org/authors/?q=ai:goldman.michael"Josien, Marc"https://www.zbmath.org/authors/?q=ai:josien.marc"Otto, Felix"https://www.zbmath.org/authors/?q=ai:otto.felixSummary: We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from a previous paper of the third author [J. Funct. Anal. 257, No. 7, 2188--2245 (2009; Zbl 1194.35082)], at the same time slightly improving the result. That result relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving a modified Kármán-Howarth-Monin identity for solutions of the inhomogeneous inviscid Burgers equation. We show that this provides a new interpretation of recent results obtained by \textit{F. Golse} and \textit{B. Perthame} [Rev. Mat. Iberoam. 29, No. 4, 1477--1504 (2013; Zbl 1288.35343)].A uniqueness theorem for the two-dimensional sigma function.https://www.zbmath.org/1452.300152021-02-12T15:23:00+00:00"Domrin, A. V."https://www.zbmath.org/authors/?q=ai:domrin.andrei-victorovichSummary: We prove that the sigma functions of Weierstrass \((g = 1)\) and Klein \((g = 2)\) are the unique solutions (up to multiplication by a complex constant) of the corresponding systems of \(2g\) linear differential heat equations in a nonholonomic frame (for a function of \(3g\) variables) that are holomorphic in a neighborhood of at least one point where all modular variables vanish. We also show that all local holomorphic solutions of these systems can be extended analytically to entire functions of angular variables. For \(g =1\), we give a complete description of the envelopes of holomorphy of such solutions.Sine-Gordon solitons and breathers in rod-like magnetic liquid crystals under external magnetic field.https://www.zbmath.org/1452.760202021-02-12T15:23:00+00:00"Li, Yan"https://www.zbmath.org/authors/?q=ai:li.yan.5|li.yan.7|li.yan.3|li.yan|li.yan.2|li.yan.1|li.yan.9|li.yan.8|li.yan.6|li.yan.10|li.yan.4"Lu, Xiao-Bo"https://www.zbmath.org/authors/?q=ai:lu.xiaobo"Hou, Chun-Feng"https://www.zbmath.org/authors/?q=ai:hou.chunfengExistence and dynamics of bounded traveling wave solutions to Getmanou equation.https://www.zbmath.org/1452.350592021-02-12T15:23:00+00:00"Wen, Zhen-Shu"https://www.zbmath.org/authors/?q=ai:wen.zhenshuMixed local-nonlocal vector Schrödinger equations and their breather solutions.https://www.zbmath.org/1452.351842021-02-12T15:23:00+00:00"Fan, Rui"https://www.zbmath.org/authors/?q=ai:fan.rui"Yu, Fa-Jun"https://www.zbmath.org/authors/?q=ai:yu.fajunApplications of kinetic tools to inverse transport problems.https://www.zbmath.org/1452.652112021-02-12T15:23:00+00:00"Li, Qin"https://www.zbmath.org/authors/?q=ai:li.qin"Sun, Weiran"https://www.zbmath.org/authors/?q=ai:sun.weiranA dynamic viscoelastic problem with friction and rate-depending contact interactions.https://www.zbmath.org/1452.352072021-02-12T15:23:00+00:00"Cocou, Marius"https://www.zbmath.org/authors/?q=ai:cocou.mariusSummary: The aim of this work is to study a dynamic problem that constitutes a unified approach to describe some rate-depending interactions between the boundaries of two viscoelastic bodies, including relaxed unilateral contact, pointwise friction or adhesion conditions. The classical formulation of the problem is presented and two variational formulations are given as three and four-field evolution implicit equations. Based on some approximation results and an equivalent fixed point problem for a multivalued function, we prove the existence of solutions to these variational evolution problems.Breathers and rogue waves derived from an extended multi-dimensional \(N\)-coupled higher-order nonlinear Schrödinger equation in optical communication systems.https://www.zbmath.org/1452.780222021-02-12T15:23:00+00:00"Bai, Cheng-Lin"https://www.zbmath.org/authors/?q=ai:bai.chenglin"Cai, Yue-Jin"https://www.zbmath.org/authors/?q=ai:cai.yuejin"Luo, Qing-Long"https://www.zbmath.org/authors/?q=ai:luo.qinglongDouble Wronskian solutions for a generalized nonautonomous nonlinear equation in a nonlinear inhomogeneous fiber.https://www.zbmath.org/1452.351632021-02-12T15:23:00+00:00"Xie, Xi-Yang"https://www.zbmath.org/authors/?q=ai:xie.xi-yang"Meng, Gao-Qing"https://www.zbmath.org/authors/?q=ai:meng.gao-qingHyperbolic function solutions for positive Gardner-KP equation.https://www.zbmath.org/1452.350602021-02-12T15:23:00+00:00"Demiray, Seyma Tuluce"https://www.zbmath.org/authors/?q=ai:demiray.seyma-tuluce"Bulut, Hasan"https://www.zbmath.org/authors/?q=ai:bulut.hasanSummary: In this paper, modified \(\exp(-\Omega(\xi))\)-expansion function method has been handled for finding exact solutions of positive Gardner-KP equation. Hyperbolic function solutions and dark soliton solution of positive Gardner-KP equation have been obtained by means of this method. Moreover, by the help of Mathematica 9, certain graphical notations were given to clarify the action of these solutions.A conservative coupling algorithm between a compressible flow and a rigid body using an embedded boundary method.https://www.zbmath.org/1452.762062021-02-12T15:23:00+00:00"Monasse, L."https://www.zbmath.org/authors/?q=ai:monasse.laurent"Daru, V."https://www.zbmath.org/authors/?q=ai:daru.virginie"Mariotti, C."https://www.zbmath.org/authors/?q=ai:mariotti.christian"Piperno, S."https://www.zbmath.org/authors/?q=ai:piperno.serge"Tenaud, C."https://www.zbmath.org/authors/?q=ai:tenaud.christianSummary: This paper deals with a new solid-fluid coupling algorithm between a rigid body and an unsteady compressible fluid flow, using an Embedded Boundary method. The coupling with a rigid body is a first step towards the coupling with a Discrete Element method. The flow is computed using a finite volume approach on a Cartesian grid. The expression of numerical fluxes does not affect the general coupling algorithm and we use a one-step high-order scheme proposed by the second and the last author [ibid. 193, No. 2, 563--594 (2004; Zbl 1109.76338)]. The Embedded Boundary method is used to integrate the presence of a solid boundary in the fluid. The coupling algorithm is totally explicit and ensures exact mass conservation and a balance of momentum and energy between the fluid and the solid. It is shown that the scheme preserves uniform movement of both fluid and solid and introduces no numerical boundary roughness. The efficiency of the method is demonstrated on challenging one- and two-dimensional benchmarks.A reduced-order extrapolated approach to solution coefficient vectors in the Crank-Nicolson finite element method for the uniform transmission line equation.https://www.zbmath.org/1452.652502021-02-12T15:23:00+00:00"Teng, Fei"https://www.zbmath.org/authors/?q=ai:teng.fei"Luo, Zhendong"https://www.zbmath.org/authors/?q=ai:luo.zhendongSummary: This work is concerned mainly with developing and testing the reduced-order extrapolated approach to the unknown coefficient vectors in the Crank-Nicolson finite element (CNFE) solutions for the uniform transmission line equation. For this objective, the CNFE functional and matrix models and the existence, stability, and errors of the CNFE solutions of the uniform transmission line equation are first derived. Then a reduced-order extrapolated CNFE (ROECNFE) matrix model is established by means of a proper orthogonal decomposition technique, and the existence, stability, and error estimates of the ROECNFE solutions are demonstrated by matrix analysis, leading to an elegant theoretical development. Especially, our work shows that the basis functions and accuracy of the ROECNFE matrix model are the same as those of the CNFE matrix model. Finally, some numerical tests are illustrated to computationally experimentally confirm the validity and sharpness of the ROECNFE method.Gaspard Monge (1746--1818): application of geometry to analysis.https://www.zbmath.org/1452.010182021-02-12T15:23:00+00:00"Borgato, Maria Teresa"https://www.zbmath.org/authors/?q=ai:borgato.maria-teresa"Pepe, Luigi"https://www.zbmath.org/authors/?q=ai:pepe.luigiOn the occasion of the bicentennial anniversary of the death of Gaspard Monge (1746--1818), the authors give a summary of some of his early work, mostly from the 1770s (though not all was published then) when he was at Mézières. The authors focus on three parts of Monge's varied output: contributions to first order partial differential equations; minimal surfaces, and first order optimal transport (the Monge-Ampère equation).
After the initial wave of development of the theory of PDEs at the hands of d'Alembert and Euler, and the success of applications to vibrating strings, the search for an analytical general theory of PDEs ran into difficulties. The authors show both the value, and later constraints, of Monge concentrating on connections with geometry and narrowing the scope of the equations under consideration. In contrast to Monge's own title of \textit{Application of analysis to geometry} for his later publications in 1807 and 1809, the authors term his early work as the application of geometry to analysis. Monge began with geometric questions on space curves, curvature of surfaces, etc. and then worked on the first order partial differential equations associated to them. The authors contrast this with Lagrange's contemporary work and philosophy that analysis absorbed geometry.
With the encouragement and support of Condorcet, Monge took up the study of minimal surfaces, where his student Jean-Baptiste Meusnier de la Place made substantial contributions including giving the first examples of surfaces of minimum area, the catenoid, and helicoid. The reliance on geometrical intuition in Monge's approach was criticized by Legendre and Cauchy.
Later in the 1770s, Monge began studying the question of optimizing transport of material, specifically excavation and embankment of ground for fortifications. Leaving aside the practical nature of the problem, Monge developed a nonlinear second order partial differential equation, although he did not publish his results at the time or give a solution to the equation. The problem was later studied by Ampère and Dupin and is now known as the Monge-Ampère equation; it continues to be an active area of research.
Reviewer: Duncan J. Melville (Canton)Vanishing diffusion in a dynamic boundary condition for the Cahn-Hilliard equation.https://www.zbmath.org/1452.350802021-02-12T15:23:00+00:00"Colli, Pierluigi"https://www.zbmath.org/authors/?q=ai:colli.pierluigi"Fukao, Takeshi"https://www.zbmath.org/authors/?q=ai:fukao.takeshiSummary: The initial boundary value problem for a Cahn-Hilliard system subject to a dynamic boundary condition of Allen-Cahn type is treated. The vanishing of the surface diffusion on the dynamic boundary condition is the point of emphasis. By the asymptotic analysis as the diffusion coefficient tends to 0, one can expect that the solutions of the surface diffusion problem converge to the solution of the problem without the surface diffusion. This is actually the case, but the solution of the limiting problem naturally looses some regularity. Indeed, the system we investigate is rather complicate due to the presence of nonlinear terms including general maximal monotone graphs both in the bulk and on the boundary. The two graphs are related each to the other by a growth condition, with the boundary graph that dominates the other one. In general, at the asymptotic limit a weaker form of the boundary condition is obtained, but in the case when the two graphs exhibit the same growth the boundary condition still holds almost everywhere.Entropy solutions for nonlinear parabolic equations in Musilak-Orlicz spaces without \(\Delta_2\)-condition.https://www.zbmath.org/1452.350792021-02-12T15:23:00+00:00"El Haji, Badr"https://www.zbmath.org/authors/?q=ai:el-haji.badr"El Moumni, Mostafa"https://www.zbmath.org/authors/?q=ai:el-moumni.mostafa"Talha, Abdeslam"https://www.zbmath.org/authors/?q=ai:talha.abdeslamSummary: In this paper, we are interested in results concerning entropy solutions for nonlinear parabolic equations in Musielak-Orlicz spaces without \(\Delta_2\)-condition.Strong backward uniqueness for sublinear parabolic equations.https://www.zbmath.org/1452.350042021-02-12T15:23:00+00:00"Arya, Vedansh"https://www.zbmath.org/authors/?q=ai:arya.vedansh"Banerjee, Agnid"https://www.zbmath.org/authors/?q=ai:banerjee.agnidSummary: In this paper, we establish strong backward uniqueness for solutions to sublinear parabolic equations of the type (1.1). The proof of our main result Theorem 1.3 is achieved by means of a new Carleman estimate and a Weiss type monotonicity formula that are tailored for such parabolic sublinear operators.Estimates of a single problem of electrodynamics arising in magnetic hydrodynamics in space \(W_p^{2,1}(Q_T)\), \(p>1\).https://www.zbmath.org/1452.350512021-02-12T15:23:00+00:00"Sakhaev, Sharipkhan"https://www.zbmath.org/authors/?q=ai:sakhaev.sharipkhanSummary: In this paper, unique solvability is obtained and estimates of solutions to the problem of magnetic hydrodynamics are obtained.Optimal regularity for all time for entropy solutions of conservation laws in \(BV^s\).https://www.zbmath.org/1452.350972021-02-12T15:23:00+00:00"Ghoshal, Shyam Sundar"https://www.zbmath.org/authors/?q=ai:ghoshal.shyam-sundar"Guelmame, Billel"https://www.zbmath.org/authors/?q=ai:guelmame.billel"Jana, Animesh"https://www.zbmath.org/authors/?q=ai:jana.animesh"Junca, Stéphane"https://www.zbmath.org/authors/?q=ai:junca.stephaneSummary: This paper deals with the optimal regularity for entropy solutions of conservation laws. For this purpose, we use two key ingredients: (a) fine structure of entropy solutions and (b) fractional \textit{BV} spaces. We show that optimality of the regularizing effect for the initial value problem from \(L^\infty\) to fractional Sobolev space and fractional \textit{BV} spaces is valid for all time. Previously, such optimality was proven only for a finite time, before the nonlinear interaction of waves. Here for some well-chosen examples, the sharp regularity is obtained after the interaction of waves. Moreover, we prove sharp smoothing in \(BV^s\) for a convex scalar conservation law with a linear source term. Next, we provide an upper bound of the maximal smoothing effect for nonlinear scalar multi-dimensional conservation laws and some hyperbolic systems in one or multi-dimension.Some representations of solutions to Blokhintsev equation.https://www.zbmath.org/1452.351162021-02-12T15:23:00+00:00"Anikonov, Yuriĭ Evgenievich"https://www.zbmath.org/authors/?q=ai:anikonov.yurii-evgenievich"Ayupova, Natalia Borisovna"https://www.zbmath.org/authors/?q=ai:ayupova.natalia-borisovna"Neshchadim, Mikhaĭl Vladimirovich"https://www.zbmath.org/authors/?q=ai:neshchadim.mikhail-vladimirovichSummary: In the paper, we obtain some representations for solutions and coefficients of Blokhintsev equation under condition the solutions satisfy to supplementary quasi-linear equation. These results may be used in the problems of identification of solutions and coefficients given supplementary initial-boundary information.A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension.https://www.zbmath.org/1452.760192021-02-12T15:23:00+00:00"Ding, Shijin"https://www.zbmath.org/authors/?q=ai:ding.shijin"Huang, Jinrui"https://www.zbmath.org/authors/?q=ai:huang.jinrui"Xia, Fengguang"https://www.zbmath.org/authors/?q=ai:xia.fengguangSummary: In this paper, we consider the free boundary problem for a simplified version of Ericksen-Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one. We obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum.Formulas in the theory of identification.https://www.zbmath.org/1452.352512021-02-12T15:23:00+00:00"Anikonov, Yuriĭ Evgenievich"https://www.zbmath.org/authors/?q=ai:anikonov.yurii-evgenievichSummary: The paper is devoted to some identication problems for evolution and other partial dieretial equation. Explicit formulas are obtained and discussed.Dimension drop for harmonic measure on Ahlfors regular boundaries.https://www.zbmath.org/1452.310062021-02-12T15:23:00+00:00"Azzam, Jonas"https://www.zbmath.org/authors/?q=ai:azzam.jonasThis article discusses the dimension of harmonic measure for a connected domain in \(\mathbb{R}^{d+1}\). The main result of the article establishes that for any domain \(\Omega\subset\mathbb{R}^{d+1}\), with uniformly non-flat Ahlfors \(s\)-regular boundary with \(s \geq d\), the dimension of its harmonic measure is strictly less than \(s\).
Reviewer: Marius Ghergu (Dublin)Unique determination of fractional order and source term in a fractional diffusion equation from sparse boundary data.https://www.zbmath.org/1452.352592021-02-12T15:23:00+00:00"Li, Zhiyuan"https://www.zbmath.org/authors/?q=ai:li.zhiyuan"Zhang, Zhidong"https://www.zbmath.org/authors/?q=ai:zhang.zhidongNonlinear Cauchy problem and identification in contact mechanics: a solving method based on Bregman-gap.https://www.zbmath.org/1452.352502021-02-12T15:23:00+00:00"Andrieux, Stephane"https://www.zbmath.org/authors/?q=ai:andrieux.stephane"Baranger, Thouraya N."https://www.zbmath.org/authors/?q=ai:baranger.thouraya-nouriRecovering a potential in damped wave equation from Neumann-to-Dirichlet operator.https://www.zbmath.org/1452.352602021-02-12T15:23:00+00:00"Romanov, Vladimir"https://www.zbmath.org/authors/?q=ai:romanov.vladimir-g"Hasanov, Alemdar"https://www.zbmath.org/authors/?q=ai:hasanoglu.alemdarNumerical solution of inverse problems by weak adversarial networks.https://www.zbmath.org/1452.653082021-02-12T15:23:00+00:00"Bao, Gang"https://www.zbmath.org/authors/?q=ai:bao.gang"Ye, Xiaojing"https://www.zbmath.org/authors/?q=ai:ye.xiaojing"Zang, Yaohua"https://www.zbmath.org/authors/?q=ai:zang.yaohua"Zhou, Haomin"https://www.zbmath.org/authors/?q=ai:zhou.hao-minThe stability for an inverse problem of bottom recovering in water-waves.https://www.zbmath.org/1452.352572021-02-12T15:23:00+00:00"Lecaros, Rodrigo"https://www.zbmath.org/authors/?q=ai:lecaros.rodrigo"López-Ríos, J."https://www.zbmath.org/authors/?q=ai:lopez-rios.j-c"Ortega, J. H."https://www.zbmath.org/authors/?q=ai:ortega.jaime-h"Zamorano, Sebastian"https://www.zbmath.org/authors/?q=ai:zamorano.sebastianA partial data inverse problem for the electro-magnetic wave equation and application to the related Borg-Levinson theorem.https://www.zbmath.org/1452.352522021-02-12T15:23:00+00:00"Bellassoued, Mourad"https://www.zbmath.org/authors/?q=ai:bellassoued.mourad"Mannoubi, Yosra"https://www.zbmath.org/authors/?q=ai:mannoubi.yosraModified forward and inverse Born series for the Calderon and diffuse-wave problems.https://www.zbmath.org/1452.352492021-02-12T15:23:00+00:00"Abhishek, Anuj"https://www.zbmath.org/authors/?q=ai:abhishek.anuj"Bonnet, Marc"https://www.zbmath.org/authors/?q=ai:bonnet.marc"Moskow, Shari"https://www.zbmath.org/authors/?q=ai:moskow.shariThe Monge-Ampère equation for non-integrable almost complex structures.https://www.zbmath.org/1452.320352021-02-12T15:23:00+00:00"Chu, Jianchun"https://www.zbmath.org/authors/?q=ai:chu.jianchun"Tosatti, Valentino"https://www.zbmath.org/authors/?q=ai:tosatti.valentino"Weinkove, Ben"https://www.zbmath.org/authors/?q=ai:weinkove.benLet \((M^{2n}, J, g)\) be a compact almost Hermitian \(2n\)-manifold and denote by \(\omega = g(J \cdot, \cdot)\) its associated Kähler form. By Yau's Theorem, if \(J\) is integrable and \((M, J, g)\) is Kähler, for any smooth volume form \(\text{vol}^{(F)} = e^F \omega^n\) satisfying the normalisation condition \(\int_M \text{vol}^{(F)} = \int_M \omega^n\), there exists a unique smooth function \(\varphi\) satisfying the conditions
\[ (\omega + \sqrt{-1} \partial \bar \partial \varphi)^n = \text{vol}^{(F)},\qquad \omega + \sqrt{-1} \partial \bar \partial \varphi > 0,\qquad \sup_M \varphi = 0\tag{\(\ast\)}.\]
The result has been generalised by the second and third author in [J. Amer. Math. Soc. 23, 19--40 (2010; Zbl 1208.53075)] to arbitrary Hermitian manifolds up to addition to \(F\) of a (uniquely determined) constant \(b\). In this paper this result is proved to be true on arbitrary compact almost Hermitian manifolds, provided that the operator \(\varphi \mapsto \sqrt{-1} \partial \bar \partial \varphi\) is replaced by the operator \(\varphi \mapsto \frac{1}{2}(d (J d \varphi))^{(1,1)}\), where
\((\cdot)^{(1,1)}\) denotes the natural pointwise projection onto the \(2\)-forms of bidegree \((1,1)\) with respect to the complex structure \(J_x\), \(x \in M\). Note that if \(J\) is integrable, this reduces to the classical definition of the \(\partial \bar \partial\) operator.
The result is crucially based on the following theorem, whose proof represents the hardest part of the whole paper: ``Given a compact almost Hermitian manifold \((M, J, g)\), for any pair of smooth real functions \(F\) and \(\varphi\) satisfying \((\ast)\), there exist a priori \(\mathcal C^\infty\) estimates on \(\varphi\) depending only on \((M, J, g)\) and bounds for \(F\).''
The main result of this paper provides a positive solution to a problem posed by Gromov, provided that \(\partial \bar \partial \varphi\) is understood as \((\frac{1}{2} d (J d \varphi))^{(1,1)}\). The original statement of the problem, where \(\partial \bar \partial \varphi\) is understood as \(\frac{1}{2} d (J d \varphi)\), is known to have a negative answer by the results of \textit{P. Delanoë} [Osaka J. Math. 33, No. 4, 829--846 (1996; Zbl 0878.53030)] and \textit{M. Warren} and \textit{Y. Yuan} [Commun. Pure Appl. Math. 62, No. 3, 305--321 (2009; Zbl 1173.35388)].
Reviewer: Andrea Spiro (Camerino)Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary.https://www.zbmath.org/1452.352052021-02-12T15:23:00+00:00"Akil, Mohammad"https://www.zbmath.org/authors/?q=ai:akil.mohammad"Chitour, Yacine"https://www.zbmath.org/authors/?q=ai:chitour.yacine"Ghader, Mouhammad"https://www.zbmath.org/authors/?q=ai:ghader.mouhammad"Wehbe, Ali"https://www.zbmath.org/authors/?q=ai:wehbe.aliSummary: In this paper, we study the indirect boundary stability and exact controllability of a one-dimensional Timoshenko system. In the first part of the paper, we consider the Timoshenko system with only one boundary fractional damping. We first show that the system is strongly stable but not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using frequency domain arguments combined with the multiplier method, we prove that the energy decay rate depends on coefficients appearing in the PDE and on the order of the fractional damping. Moreover, under the equal speed propagation condition, we obtain the optimal polynomial energy decay rate. In the second part of this paper, we study the indirect boundary exact controllability of the Timoshenko system with mixed Dirichlet-Neumann boundary conditions and boundary control. Using non-harmonic analysis, we first establish a weak observability inequality, which depends on the ratio of the waves propagation speeds. Next, using the HUM method, we prove that the system is exactly controllable in appropriate spaces and that the control time can be small.Fractional diffusion limit for a kinetic equation with an interface.https://www.zbmath.org/1452.352122021-02-12T15:23:00+00:00"Komorowski, Tomasz"https://www.zbmath.org/authors/?q=ai:komorowski.tomasz"Olla, Stefano"https://www.zbmath.org/authors/?q=ai:olla.stefano"Ryzhik, Lenya"https://www.zbmath.org/authors/?q=ai:ryzhik.lenyaSummary: We consider the limit of a linear kinetic equation with reflection-transmission-absorption at an interface and with a degenerate scattering kernel. The equation arises from a microscopic chain of oscillators in contact with a heat bath. In the absence of the interface, the solutions exhibit a superdiffusive behavior in the long time limit. With the interface, the long time limit is the unique solution of a version of the fractional in space heat equation with reflection-transmission-absorption at the interface. The limit problem corresponds to a certain stable process that is either absorbed, reflected or transmitted upon crossing the interface.Generalized spectrum of second order differential operators.https://www.zbmath.org/1452.650662021-02-12T15:23:00+00:00"Gergelits, Tomáš"https://www.zbmath.org/authors/?q=ai:gergelits.tomas"Nielsen, Bjørn Fredrik"https://www.zbmath.org/authors/?q=ai:nielsen.bjorn-fredrik"Strakoš, Zdeněk"https://www.zbmath.org/authors/?q=ai:strakos.zdenekTraffic flow models with nonlocal looking ahead-behind dynamics.https://www.zbmath.org/1452.351002021-02-12T15:23:00+00:00"Lee, Yongki"https://www.zbmath.org/authors/?q=ai:lee.yongkiThis paper deals with the traffic flow model with look ahead relaxation and look behind intensification:
\[
u_t+(u(1-u)e^{-\overline{u}+\widetilde{u}})_x=0,
\]
where
\[
\overline{u}=K_a* u,\qquad \widetilde{u}=K_b* u,
\]
\(K_a\) and \(K_b\) are constant and linear interaction potentials. The author identifies threshold conditions for the finite time shock formation of the traffic flow model and investigates the performance of the proposed model via numerical examples in comparison with the Arrhenius look-ahead model
\[
u_t+(u(1-u)e^{-\overline{u}})_x=0,
\]
and the Lighthill-Whitham-Richards one
\[
u_t+(u(1-u))_x=0.
\]
Reviewer: Giuseppe Maria Coclite (Bari)Representation of solutions to wave equations with profile functions.https://www.zbmath.org/1452.350892021-02-12T15:23:00+00:00"Lamacz, Agnes"https://www.zbmath.org/authors/?q=ai:lamacz.agnes"Schweizer, Ben"https://www.zbmath.org/authors/?q=ai:schweizer.benBoundary controllability of the Korteweg-de Vries equation on a tree-shaped network.https://www.zbmath.org/1452.351472021-02-12T15:23:00+00:00"Cerpa, Eduardo"https://www.zbmath.org/authors/?q=ai:cerpa.eduardo"Crépeau, Emmanuelle"https://www.zbmath.org/authors/?q=ai:crepeau.emmanuelle"Valein, Julie"https://www.zbmath.org/authors/?q=ai:valein.julieSummary: Controllability of coupled systems is a complex issue depending on the coupling conditions and the equations themselves. Roughly speaking, the main challenge is controlling a system with less inputs than equations. In this paper this is successfully done for a system of Korteweg-de Vries equations posed on an oriented tree shaped network. The couplings and the controls appear only on boundary conditions.On the scattering of electromagnetic waves by cylindrical bodies with non-coordinate boundaries.https://www.zbmath.org/1452.780132021-02-12T15:23:00+00:00"Pleshchinskaya, I. E."https://www.zbmath.org/authors/?q=ai:pleshchinskaya.irina-e"Pleshchinskii, N. B."https://www.zbmath.org/authors/?q=ai:pleshchinskii.nikolaij-borisovichThe authors consider the classical problem of diffraction of plane electromagnetic waves by cylindrical bodies. Different known methods are combined in order to solve numerically that kind of problem (in which is included the possibility of having metal bands on the surface).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Deep ReLU networks and high-order finite element methods.https://www.zbmath.org/1452.653542021-02-12T15:23:00+00:00"Opschoor, Joost A. A."https://www.zbmath.org/authors/?q=ai:opschoor.joost-a-a"Petersen, Philipp C."https://www.zbmath.org/authors/?q=ai:petersen.philipp-c"Schwab, Christoph"https://www.zbmath.org/authors/?q=ai:schwab.christophBlow-up and global solubility in the classical sense of the Cauchy problem for a formally hyperbolic equation with a non-coercive source.https://www.zbmath.org/1452.350482021-02-12T15:23:00+00:00"Korpusov, Maxim O."https://www.zbmath.org/authors/?q=ai:korpusov.maksim-olegovichAsymptotical analysis of a nonlinear Sturm-Liouville problem: linearisable and non-linearisable solutions.https://www.zbmath.org/1452.352012021-02-12T15:23:00+00:00"Kurseeva, Valeria"https://www.zbmath.org/authors/?q=ai:kurseeva.valeria-yu"Moskaleva, Marina"https://www.zbmath.org/authors/?q=ai:moskaleva.marina"Valovik, Dmitry"https://www.zbmath.org/authors/?q=ai:valovik.dmitry-vSummary: The paper focuses on a nonlinear eigenvalue problem of Sturm-Liouville type with real spectral parameter under first type boundary conditions and additional local condition. The nonlinear term is an arbitrary monotonically increasing function. It is shown that for small nonlinearity the negative eigenvalues can be considered as perturbations of solutions to the corresponding linear eigenvalue problem, whereas big positive eigenvalues cannot be considered in this way. Solvability results are found, asymptotics of negative as well as positive eigenvalues are derived, distribution of zeros of the eigenfunctions is presented. As a by-product, a comparison theorem between eigenvalues of two problems with different data is derived. Applications of the found results in electromagnetic theory are given.\(L^p\) polyharmonic Robin problems on Lipschitz domains.https://www.zbmath.org/1452.310152021-02-12T15:23:00+00:00"Li, Weifeng"https://www.zbmath.org/authors/?q=ai:li.weifeng"Dang, Pei"https://www.zbmath.org/authors/?q=ai:dang.pei"Du, Zhihua"https://www.zbmath.org/authors/?q=ai:du.zhihua"Guo, Guoan"https://www.zbmath.org/authors/?q=ai:guo.guoan"Li, Yumei"https://www.zbmath.org/authors/?q=ai:li.yumeiSummary: In this paper, we study a class of boundary value problems (BVPs) with Robin conditions in some \(L^p\) spaces for polyharmonic equation on Lipschitz domains. Utilizing polyharmonic fundamental solutions, these Robin BVPs are solved by the method of layer potentials. The crucial ingredients of our approach are the classical single layer potential and its higher order analog (which are called multi-layer \(S\)-potentials), and the main results generalize ones of second order (Laplacian) case to higher order (polyharmonic) case.Asymptotic behavior of reaction-advection-diffusion population models with Allee effect.https://www.zbmath.org/1452.350372021-02-12T15:23:00+00:00"Jerez, Silvia"https://www.zbmath.org/authors/?q=ai:jerez.silvia"Verdugo, Jonathan"https://www.zbmath.org/authors/?q=ai:verdugo.jonathanSummary: In this work, a qualitative analysis is carried out for reaction-advection-diffusion (RAD) systems modeling the interactions between two species with Allee effect. In particular, we study different scenarios: mutualism, competition, and a predator-prey relationship in order to investigate the survival or extinction of both populations. Global existence and uniqueness of positive solutions of the proposed RAD problems are demonstrated. Equilibrium states and asymptotic behavior of solutions are obtained using the monotone method and the upper and lower solutions technique. Numerical simulations by a Crank-Nicolson monotone iterative method of the different asymptotic solution dynamics are shown to illustrate our theoretical results.Integral transform solution of random coupled parabolic partial differential models.https://www.zbmath.org/1452.352662021-02-12T15:23:00+00:00"Casabán, María Consuelo"https://www.zbmath.org/authors/?q=ai:casaban.maria-consuelo"Company, Rafael"https://www.zbmath.org/authors/?q=ai:company.rafael"Egorova, Vera N."https://www.zbmath.org/authors/?q=ai:egorova.vera-n"Jódar, Lucas"https://www.zbmath.org/authors/?q=ai:jodar-sanchez.lucas-aSummary: Random coupled parabolic partial differential models are solved numerically using random cosine Fourier transform together with non-Gaussian random numerical integration that captures the highly oscillatory behaviour of the involved integrands. Sufficient condition of spectral type imposed on the random matrices of the system is given so that the approximated stochastic process solution and its statistical moments are numerically convergent. Numerical experiments illustrate the results.Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation.https://www.zbmath.org/1452.351792021-02-12T15:23:00+00:00"Campos, Luccas"https://www.zbmath.org/authors/?q=ai:campos.luccasSummary: We prove scattering below the mass-energy threshold for the focusing inhomogeneous nonlinear Schrödinger equation
\[
iu_t + \Delta u + |x|^{-b} |u|^{p - 1} u = 0,
\]
when \(b \geq 0\) and \(N > 2\) in the intercritical case \(0 < s_c < 1\). This work generalizes the results of \textit{L. G. Farah} and \textit{C. M. Guzmán} [Bull. Braz. Math. Soc. (N.S.) 51, No. 2, 449--512 (2020; Zbl 1437.35623)], allowing a broader range of values for the parameters \(p\) and \(b\). We use a modified version of Dodson-Murphy's approach [\textit{B. Dodson} and \textit{J. Murphy}, Proc. Am. Math. Soc. 145, No. 11, 4859--4867 (2017; Zbl 1373.35287)] allowing us to deal with the inhomogeneity. The proof is also valid for the classical nonlinear Schrödinger equation \((b = 0)\), extending the work in [Dodson and Murphy, loc. cit.] for radial solutions in all intercritical cases.Bounding the spectral gap for an elliptic eigenvalue problem with uniformly bounded stochastic coefficients.https://www.zbmath.org/1452.653152021-02-12T15:23:00+00:00"Gilbert, Alexander D."https://www.zbmath.org/authors/?q=ai:gilbert.alexander-d"Graham, Ivan G."https://www.zbmath.org/authors/?q=ai:graham.ivan-g"Scheichl, Robert"https://www.zbmath.org/authors/?q=ai:scheichl.robert"Sloan, Ian H."https://www.zbmath.org/authors/?q=ai:sloan.ian-hSummary: A key quantity that occurs in the error analysis of several numerical methods for eigenvalue problems is the distance between the eigenvalue of interest and the next nearest eigenvalue. When we are interested in the smallest or fundamental eigenvalue, we call this the \textit{spectral} or \textit{fundamental gap}. In a recent manuscript [\textit{A. D. Gilbert} et al., Numer. Math. 142, No. 4, 863--915 (2019; Zbl 1416.65018)], the current authors, together with Frances Kuo, studied an elliptic eigenvalue problem with homogeneous Dirichlet boundary conditions, and with coefficients that depend on an infinite number of uniformly distributed stochastic parameters. In this setting, the eigen-values, and in turn the eigenvalue gap, also depend on the stochastic parameters. Hence, for a robust error analysis one needs to be able to bound the gap over all possible realisations of the parameters, and because the gap depends on infinitely-many random parameters, this is not trivial. This short note presents, in a simplified setting, an important result that was shown in the paper above. Namely, that, under certain decay assumptions on the coefficient, the spectral gap of such a random elliptic eigenvalue problem can be bounded away from 0, uniformly over the entire infinite-dimensional parameter space.
For the entire collection see [Zbl 1445.37004].Existence and regularity results for viscous Hamilton-Jacobi equations with Caputo time-fractional derivative.https://www.zbmath.org/1452.352342021-02-12T15:23:00+00:00"Camilli, Fabio"https://www.zbmath.org/authors/?q=ai:camilli.fabio"Goffi, Alessandro"https://www.zbmath.org/authors/?q=ai:goffi.alessandroAuthors' abstract: ``We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton-Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional Hamilton-Jacobi equation obtained via nonlinear adjoint method and sharp estimates in Sobolev and Hölder spaces for the corresponding linear problem.''
Recently, the research topic of fractional differential equations has been applied by mathematicians and scientists to model various scenarios from science and engineering due to the main advantages of fractional derivatives in modelling memory effect and in providing a better explanation for the physical and geometrical meanings of the studied system than the studied systems using integer-order derivatives. Some interesting research studies have been recently conducted on finding analytical, approximate-analytical, or numerical solutions to the fractional differential equations formulated in the senses of different fractional derivatives. For example, \textit{M. K. A. Kaabar} et al. in [``New approximate-analytical solutions for the nonlinear fractional Schrödinger equation with second-order spatio-temporal dispersion via double Laplace transform method'', Preprint, \url{arXiv:2010.10977}] have investigated the approximate-analytical solutions for the proposed nonlinear fractional Schrödinger equation with second-order spatio-temporal dispersion in the senses of both Caputo fractional derivative and conformable derivative using a new approach called double Laplace transform method coupled with Adomian decomposition method. In addition, the two-dimensional wave equation formulated in the sense of conformable derivative has been solved in [\textit{M. Kaabar}, ``Novel methods for solving the conformable wave equations'', J. New Theory 2020, No. 31, 56--85 (2020)]. Moreover, many research studies have been dedicated to the investigation of the mathematical analysis of fractional differential equations in [\textit{Q. Du}, Nonlocal modeling, analysis, and computation. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2019; Zbl 1423.00007); \textit{N. Cusimano} et al., ESAIM, Math. Model. Numer. Anal. 54, No. 3, 751--774 (2020; Zbl 1452.35237); \textit{L. Zhang} et al., Topol. Methods Nonlinear Anal. 54, No. 2A, 537--566 (2019; Zbl 1445.47039); \textit{S. D. Taliaferro}, J. Math. Pures Appl. (9) 133, 287--328 (2020; Zbl 1437.35697); \textit{A. Ghanmi} and \textit{Z. Zhang}, Bull. Korean Math. Soc. 56, No. 5, 1297--1314 (2019; Zbl 1432.34012); \textit{T. Ghosh} et al., Anal. PDE 13, No. 2, 455--475 (2020; Zbl 1439.35530); \textit{K. Ryszewska}, J. Math. Anal. Appl. 483, No. 2, Article ID 123654, 17 p. (2020; Zbl 1436.35323); \textit{H. Dong} and \textit{D. Kim}, J. Funct. Anal. 278, No. 3, Article ID 108338, 66 p. (2020; Zbl 1427.35316); \textit{L. C. F. Ferreira} et al., Bull. Sci. Math. 153, 86--117 (2019; Zbl 1433.35185); \textit{Y. Gholami} and \textit{K. Ghanbari}, S\(\vec{\text{e}}\)MA J. 75, No. 2, 305--333 (2018; Zbl 1400.26012); \textit{F. M. Gaafar}, J. Egypt. Math. Soc. 26, 469--482 (2018; Zbl 1441.34007)]. In addition, some research works have discussed some important topics in fractional calculus such as complex integral [\textit{F. Martínez} et al., New results on complex conformable integral. AIMS Mathematics. 5, No. 6, 7695--7710 (2020)], fractional derivatives and integrals of complex-valued functions of a real variable [\textit{F. Martínez} et al., Note on the Conformable Fractional Derivatives and Integrals of Complex-valued Functions of a Real Variable. IAENG International Journal of Applied Mathematics. 50, No. 3, 609--615 (2020)], Sturm's theorems and Green's function [\textit{F. Martínez} et al., ``Note on the conformable boundary value problems: Sturm's theorems and Green's function'', Preprint (2020; \url{doi: 10.20944/preprints202009.0440.v1})], and fractional power series [\textit{F. Martínez} et al., ``Some new results on conformable fractional power series'', Asia Pac. J. Math. 7, No. 31, 1--14 (2020)]. In this research paper, the authors provide a well-written investigation of the classical solutions' existence, uniqueness, and properties for the proposed time-fractional Hamilton-Jacobi equation in the context of Caputo fractional derivative which is based on the combination of the gradient bound solution using two approaches: Sharp a priori bounds in fractional parabolic Sobolev and Hölder spaces for the linear problem and the nonlinear adjoint method that uses the integration by parts investigate the adjoint of the linearized Hamilton-Jacobi equation. The proposed equation can be expressed as follows: Given a unit torus, denoted by \(\mathbb{T}^d\); convex is denoted by \(H\); coercive Hamiltonian in \(Du\); and for \(0<\beta<1\), the time-fractional Caputo derivative is denoted by \(\partial ^{\beta}_{(0,t]} u\), then the time-fractional Hamilton-Jacobi equation formulated in the sense of Caputo fractional derivative can be written as: \(\partial ^{\beta}_{(0,t]} u(x,t)-\Delta u+H(x, Du)=V(x,t)(x,t) \in Q_{T}=\mathbb{T}^d \times (0,T)\) subject to \(u(x,0)=u_{0}(x)\) where \(\partial ^{\beta}_{(0,t]} u(x,t)=\frac{1}{\Gamma (1-\beta)} \int_0^t \frac{\partial_{\xi}u(x,\xi)}{(t-\xi)^{\beta}}d\xi\). This research study is considered novel and worthy due to the importance of investigating the Hamilton-Jacobi equation with its connection to classical mechanics. In this research work, since this study is dependent on the combination of gradient bound of the solution to the proposed time-fractional Hamilton-Jacobi equation using a priori bounds and adjoint method, then the proposed problem (time-fractional Hamilton-Jacobi) can be studied as a perturbation of the time-fractional heat equation as mentioned in the paper; therefore, the authors have discussed the Schauder estimates for the time-fractional heat equation to provide the necessary and sufficient condition for the maximal Hölder's regularity in the abstract linear differential equation (see problem 3 in the paper) where the maximal regularity is extended in Lebesgue and Hölder spaces. In conclusion, this paper has many new results that have an excellent level of novelty. This research paper deserves to be referenced by all interested researchers in this field of research, and many further research works can be developed based on all discussed results in this paper.
Reviewer: Mohammed Kaabar (Gelugor)Convergence of a class of nonlinear time delays reaction-diffusion equations.https://www.zbmath.org/1452.350172021-02-12T15:23:00+00:00"Anza Hafsa, Omar"https://www.zbmath.org/authors/?q=ai:anza-hafsa.omar"Mandallena, Jean Philippe"https://www.zbmath.org/authors/?q=ai:mandallena.jean-philippe"Michaille, Gérard"https://www.zbmath.org/authors/?q=ai:michaille.gerardThe paper is devoted to the convergence and the stochastic homogenization of sequences \((P_n)_{n\in \mathbb N}\) of reaction-diffusion equations of the type
\[
\begin{cases}
\frac{du_n}{dt}(t)+\partial \Phi _n(u_n(t))\ni F_n(t,u_n(t))&\text{for a.e. } t\in(0,T)\\
u_n(t)=\eta_n(t) &\text{for all }t\in(-\infty,0).
\end{cases}
\tag{\(P_n\)}
\]
The results obtained in the paper can be applied to various models of population dynamics or diseases in heterogeneous environments where delays terms may depend on the space variable.
Reviewer: Leonid Berezanski (Beer-Sheva)The semi-analytical method for time-dependent wave problems with uncertainties.https://www.zbmath.org/1452.352642021-02-12T15:23:00+00:00"Bartual, Maria Consuelo Casabán"https://www.zbmath.org/authors/?q=ai:bartual.maria-consuelo-casaban"López, Juan Carlos Cortés"https://www.zbmath.org/authors/?q=ai:cortes.juan-carlos"Sánchez, Lucas Jódar"https://www.zbmath.org/authors/?q=ai:jodar-sanchez.lucas-aSummary: This paper provides a constructive procedure for the computation of approximate solutions of random time-dependent hyperbolic mean square partial differential problems. Based on the theoretical representation of the solution as an infinite random improper integral, obtained via the random Fourier transform method, a double approximation process is implemented. Firstly, a random Gauss-Hermite quadrature is applied, and then, the evaluations at the nodes of the integrand are approximated by using a random Störmer numerical method. Numerical results are illustrated with examples.A Euclidean signature semi-classical program.https://www.zbmath.org/1452.830062021-02-12T15:23:00+00:00"Marini, Antonella"https://www.zbmath.org/authors/?q=ai:marini.antonella"Maitra, Rachel"https://www.zbmath.org/authors/?q=ai:maitra.rachel-lash"Moncrief, Vincent"https://www.zbmath.org/authors/?q=ai:moncrief.vincent-eSummary: In this article we discuss our ongoing program to extend the scope of certain, well-developed microlocal methods for the asymptotic solution of Schrödinger's equation (for suitable `nonlinear oscillatory' quantum mechanical systems) to the treatment of several physically significant, interacting quantum field theories. Our main focus is on applying these `Euclidean-signature semi-classical' methods to self-interacting (real) scalar fields of renormalizable type in \(2, 3\) and \(4\) spacetime dimensions and to Yang-Mills fields in 3 and 4 spacetime dimensions. A central argument in favor of our program is that the asymptotic methods for Schrödinger operators developed in the microlocal literature are far superior, for the quantum mechanical systems to which they naturally apply, to the conventional WKB methods of the physics literature and that these methods can be modified, by techniques drawn from the calculus of variations and the analysis of elliptic boundary value problems, to apply to certain (bosonic) quantum field theories. Unlike conventional (Rayleigh/ Schrödinger) perturbation theory these methods avoid the artificial decomposition of an interacting system into an approximating `unperturbed' system and its perturbation and instead keep the nonlinearities (and, if present, gauge invariances) of an interacting system intact at every level of the analysis.Solving two-phase freezing Stefan problems: stability and monotonicity.https://www.zbmath.org/1452.352622021-02-12T15:23:00+00:00"Piqueras, Miguel A."https://www.zbmath.org/authors/?q=ai:piqueras.miguel-a"Company, Rafael"https://www.zbmath.org/authors/?q=ai:company.rafael"Jódar, Lucas"https://www.zbmath.org/authors/?q=ai:jodar-sanchez.lucas-aSummary: The two-phase Stefan problems with phase formation and depletion are special cases of moving boundary problems with interest in science and industry. In this work, we study a solidification problem, introducing a front-fixing transformation. The resulting non-linear partial differential system involves singularities, both at the beginning of the freezing process and when the depletion is complete, that are treated with special attention in the numerical modelling. The problem is decomposed in three stages, in which implicit and explicit finite difference schemes are used. Numerical analysis reveals qualitative properties of the numerical solution spatial monotonicity of both solid and liquid temperatures and the evolution of the solidification front. Numerical experiments illustrate the behaviour of the temperatures profiles with time, as well as the dynamics of the solidification front.Padé schemes with Richardson extrapolation for the sine-Gordon equation.https://www.zbmath.org/1452.651702021-02-12T15:23:00+00:00"Martin-Vergara, Francisca"https://www.zbmath.org/authors/?q=ai:martin-vergara.francisca"Rus, Francisco"https://www.zbmath.org/authors/?q=ai:rus.francisco"Villatoro, Francisco R."https://www.zbmath.org/authors/?q=ai:villatoro.francisco-rSummary: Four novel implicit finite difference methods with \((q+s)\)-th order in space based on \((q,s)\)-Padé approximations have been analyzed and developed for the sine-Gordon equation. Specifically, \((4,0)\)-, \((2,2)\)-, \((4,2)\)-, and \((4,4)\)-Padé methods. All of them share the treatment for the nonlinearity and integration in time, specifically, the one that results in an energy-conserving \((2,0)\)-Padé scheme. The five methods have been developed with and without Richardson extrapolation in time. All the methods are linearly, unconditionally stable. A comparison among them for both the kink-antikink and breather solutions in terms of global error, computational cost and energy conservation is presented. Our results indicate that the \((4,0)\)- and \((4,4)\)-Padé methods without Richardson extrapolation are the most cost-effective ones for small and large global error, respectively; and the \((4,4)\)-Padé methods in all the cases when Richardson extrapolation is used.Breather and solitons waves in optical fibers via exponential time differencing method.https://www.zbmath.org/1452.651462021-02-12T15:23:00+00:00"Ashi, Hala A."https://www.zbmath.org/authors/?q=ai:ashi.hala-a"Aljahdaly, Noufe H."https://www.zbmath.org/authors/?q=ai:aljahdaly.noufe-hSummary: This work proposes the numerical solutions of the cubic nonlinear Schrödinger equation subjected to rational initial value for three cases, (i) Akhmediev breather (ii) Peregrine and (iii) Kuznetsov-Ma solitons by operating exponential time differencing method (ETDM). The work shows that ETDM is a reliable and accurate method to find the solutions comparing to Adomian decomposition method (ADM) and Laplace-Adomian decomposition method (LADM).Stability of traveling waves in a driven Frenkel-Kontorova model.https://www.zbmath.org/1452.370792021-02-12T15:23:00+00:00"Vainchtein, Anna"https://www.zbmath.org/authors/?q=ai:vainchtein.anna"Cuevas-Maraver, Jesús"https://www.zbmath.org/authors/?q=ai:cuevas-maraver.jesus"Kevrekidis, Panayotis G."https://www.zbmath.org/authors/?q=ai:kevrekidis.panayotis-g"Xu, Haitao"https://www.zbmath.org/authors/?q=ai:xu.haitaoSummary: In this work we revisit a classical problem of traveling waves in a damped Frenkel-Kontorova lattice driven by a constant external force. We compute these solutions as fixed points of a nonlinear map and obtain the corresponding kinetic relation between the driving force and the velocity of the wave for different values of the damping coefficient. We show that the kinetic curve can become \textit{non-monotone} at small velocities, due to resonances with linear modes, and also at large velocities where the kinetic relation becomes \textit{multivalued}. Exploring the spectral stability of the obtained waveforms, we identify, at the level of numerical accuracy of our computations, a precise criterion for instability of the traveling wave solutions: monotonically decreasing portions of the kinetic curve always bear an unstable eigendirection. We discuss why the validity of this criterion in the \textit{dissipative} setting is a rather remarkable feature offering connections to the Hamiltonian variant of the model and of lattice traveling waves more generally. Our stability results are corroborated by direct numerical simulations which also reveal the possible outcomes of dynamical instabilities.\(H(\operatorname{div})\) conforming methods for the rotation form of the incompressible fluid equations.https://www.zbmath.org/1452.653272021-02-12T15:23:00+00:00"Chen, Xi"https://www.zbmath.org/authors/?q=ai:chen.xi.2|chen.xi.1|chen.xi|chen.xi.4|chen.xi.5"Drapaca, Corina"https://www.zbmath.org/authors/?q=ai:drapaca.corina-stefaniaThe authors present new \(H(\operatorname{div})\) conforming methods for incompressible flows represented using the rotation form of the equation of motion and the Bernoulli function. They provide error estimates for the associated semidiscrete method and, through numerical simulations, show better convergence performance than theoretically predicted. For the incompressible Navier-Stokes equation, they use the full version of the stress tensor to enforce the divergence free constraint with consistency. In this formulation, the grad-div stabilization term arises naturally with \(H1\) conforming, and the symmetric gradient formulation of the viscous term is recovered with \(H(\operatorname{div})\) conforming. Numerical results are used to confirm the accuracy of their formulations. With \(H1\) conforming and Taylor-Hood elements, the authors find that the use of the full stress tensor helps to reduce errors both in the velocity and the Bernoulli function. They also found that the \(H(\operatorname{div})\) conforming method does a better job in the long time structure preservation compared with classical mixed method even with the grad-div stabilization.
Reviewer: Murli Gupta (Washington, D. C.)Discretization of multipole sources in a finite difference setting for wave propagation problems.https://www.zbmath.org/1452.651942021-02-12T15:23:00+00:00"Bencomo, Mario J."https://www.zbmath.org/authors/?q=ai:bencomo.mario-j"Symes, William W."https://www.zbmath.org/authors/?q=ai:symes.william-wSummary: Seismic sources are commonly idealized as point-sources due to their small spatial extent relative to seismic wavelengths. The acoustic isotropic point-radiator is inadequate as a model of seismic wave generation for seismic sources that are known to exhibit directivity. Therefore, accurate modeling of seismic wavefields must include source representations generating anisotropic radiation patterns. Such seismic sources can be modeled as \textit{multipoles}, that is, a time-dependent linear combination of spatial derivatives of the spatial delta function. Since the solutions of linear hyperbolic systems with point-source right hand sides are necessarily singular, standard results on convergence of grid-based numerical methods (finite difference or finite element) do not imply convergence of numerical solutions. We present a method for discretizing multipole sources in a finite difference setting, an extension of the moment matching conditions developed for the Dirac delta function in other applications, along with numerical evidence demonstrating the accuracy of these approximations. Using this analysis, we develop a weak convergence theory for the discretization of a family of symmetric hyperbolic systems of first-order partial differential equations, with singular source terms, solved via staggered-grid finite difference methods: we show that grid-independent space-time averages of the numerical solutions converge to the same averages of the continuum solution, and provide an estimate for the error in terms of moment matching and truncation error conditions. Numerical experiments confirm this result, but also suggest a stronger one: optimal convergence rates appear to be achieved point-wise in space away from the source.Cauchy problem for the equation of torsional vibrations of a rod in a viscoelastic medium.https://www.zbmath.org/1452.740522021-02-12T15:23:00+00:00"Umarov, Kh. G."https://www.zbmath.org/authors/?q=ai:umarov.khasan-galsanovichSummary: We study the Cauchy problem in the space of continuous functions for a nonlinear Sobolev type differential equation generalizing the equation of torsional vibrations of an infinite rod in a viscoelastic medium. Conditions for the existence of a global solution and for the blow-up of the solution of the Cauchy problem in finite time are considered.Estimates of the exponential decay of perturbations superimposed on the longitudinal harmonic vibrations of a viscous layer.https://www.zbmath.org/1452.760662021-02-12T15:23:00+00:00"Georgievskii, D. V."https://www.zbmath.org/authors/?q=ai:georgievskii.dmitrii-vladimirovichSummary: We study how the pattern of perturbations superimposed on a plane-parallel time-periodic flow of a Newtonian viscous fluid evolves in a layer in which one of the boundaries performs longitudinal harmonic vibrations along itself, with the zero-friction slip of material allowed on the other boundary. We pose a generalized Orr-Sommerfeld problem as a linearized problem of hydrodynamic stability of unsteady-state viscous incompressible flows. Using the integral relation method, based on variational inequalities for quadratic functionals and developed as applied to unsteady-state flows, we derive integral estimates sufficient for the exponential decay of the initial perturbations. For each wave number, these estimates are inequalities relating three constant dimensionless quantities, viz., period-average depth-maximum shear velocity in the layer, boundary vibration amplitude, and the Reynolds number. We compare the established stability estimates for the planar and three-dimensional perturbation patterns.Interplay of the pseudo-Raman term and trapping potentials in the nonlinear Schrödinger equation.https://www.zbmath.org/1452.351852021-02-12T15:23:00+00:00"Gromov, E. M."https://www.zbmath.org/authors/?q=ai:gromov.eugenie-m"Malomed, B. A."https://www.zbmath.org/authors/?q=ai:malomed.boris-aSummary: We introduce a nonlinear Schrödinger equation (NLSE) which combines the \textit{pseudo-stimulated-Raman-scattering} (pseudo-SRS) term, i.e., a non-conservative cubic one with the first spatial derivative, and an external potential, which helps to stabilize solitons against the pseudo-SRS effect. Dynamics of solitons is addressed by means of analytical and numerical methods. The quasi-particle approximation (QPA) for the solitons demonstrates that the SRS-induced downshift of the soliton's wavenumber may be compensated by a potential force, producing a stable stationary soliton. Three physically relevant potentials are considered: a harmonic-oscillator (HO) trap, a spatially periodic cosinusoidal potential, and the HO trap subjected to periodic temporal modulation. Both equilibrium positions of trapped pulses (solitons) and their regimes of motion with trapped and free trajectories are accurately predicted by the QPA and corroborated by direct simulations of the underlying NLSE. In the case of the time-modulated HO trap, a parametric resonance is demonstrated, in the form of the motion of the driven soliton with an exponentially growing amplitudes of oscillations.Weak solvability of equations modeling steady-state flows of second-grade fluids.https://www.zbmath.org/1452.351422021-02-12T15:23:00+00:00"Baranovskii, E. S."https://www.zbmath.org/authors/?q=ai:baranovskii.evgenii-sergeevichSummary: We prove the existence of continuous weak solutions of the nonlinear equations describing steady-state flows of second-grade fluids in a bounded three-dimensional domain under the no-slip boundary condition. A weak solution is found using the Galerkin method with special basis functions constructed with the help of a perturbed Stokes operator. An energy inequality for the resulting solution is derived.Asymptotic and numerical results for a model of solvent-dependent drug diffusion through polymeric spheres.https://www.zbmath.org/1452.761952021-02-12T15:23:00+00:00"McCue, Scott W."https://www.zbmath.org/authors/?q=ai:mccue.scott-william"Hsieh, Mike"https://www.zbmath.org/authors/?q=ai:hsieh.mike"Moroney, Timothy J."https://www.zbmath.org/authors/?q=ai:moroney.timothy-j"Nelson, Mark I."https://www.zbmath.org/authors/?q=ai:nelson.mark-ianAsymptotic behavior of a solution of the Cauchy problem for the generalized damped multidimensional Boussinesq equation.https://www.zbmath.org/1452.351572021-02-12T15:23:00+00:00"Polat, Necat"https://www.zbmath.org/authors/?q=ai:polat.necat"Pişkin, Erhan"https://www.zbmath.org/authors/?q=ai:piskin.erhanSummary: This work studies the Cauchy problem for the generalized damped multidimensional Boussinesq equation. By using a multiplier method, it is proven that the global solution of the problem decays to zero exponentially as the time approaches infinity, under a very simple and mild assumption regarding the nonlinear term.An inverse spectral problem for a fourth-order Sturm-Liouville operator based on trace formulae.https://www.zbmath.org/1452.652082021-02-12T15:23:00+00:00"Jiang, Xiaoying"https://www.zbmath.org/authors/?q=ai:jiang.xiaoying"Xu, Xiang"https://www.zbmath.org/authors/?q=ai:xu.xiangSummary: In this paper, an efficient algorithm for recovering a density of a fourth-order Sturm-Liouville operator from two given spectra is investigated. Based on Lidskii's theorem and Mercer's theorem, we build a sequence of trace formulae which bridge explicitly the density and eigenvalues in terms of nonlinear Fredholm integral equations. Due to intrinsic difficulties on ill-posedness of an inverse spectral problem, a truncated Fourier series regularization method is utilized for reconstructing the unknown density. Moreover, shifted Legendre polynomials are carried to balance the different order of trace formulae. Numerical results are presented to illustrate the effectiveness of the proposed reconstruction algorithm.A uniformly convergent weak Galerkin finite element method on Shishkin mesh for 1d convection-diffusion problem.https://www.zbmath.org/1452.653652021-02-12T15:23:00+00:00"Zhu, Peng"https://www.zbmath.org/authors/?q=ai:zhu.peng"Xie, Shenglan"https://www.zbmath.org/authors/?q=ai:xie.shenglanSummary: In this paper, a weak Galerkin finite element method is proposed and analyzed for one-dimensional singularly perturbed convection-diffusion problems. This finite element scheme features piecewise polynomials of degree \(k\geq 1\) on interior of each element plus piecewise constant on the node of each element. Our WG scheme is parameter-free and has competitive number of unknowns since the interior unknowns can be eliminated efficiently from the discrete linear system. An \(\varepsilon\)-uniform error bound of \(\mathcal{O}((N^{-1}\ln N)^k)\) in the energy-like norm is established on Shishkin mesh, where \(N\) is the number of elements. Finally, the numerical experiments are carried out to confirm the theoretical results. Moreover, the numerical results show that the present method has the optimal convergence rate of \(\mathcal{O}(N^{-(k+1)})\) in the \(L^2\)-norm and the superconvergence rates of \(\mathcal{O}((N^{-1}\ln N)^{2k})\) in the discrete \(L^\infty\)-norm.A variational finite volume scheme for Wasserstein gradient flows.https://www.zbmath.org/1452.490192021-02-12T15:23:00+00:00"Cancès, Clément"https://www.zbmath.org/authors/?q=ai:cances.clement"Gallouët, Thomas O."https://www.zbmath.org/authors/?q=ai:gallouet.thomas-o"Todeschi, Gabriele"https://www.zbmath.org/authors/?q=ai:todeschi.gabrieleA novel variational finite volume scheme based on a first discretize then optimize approach to approximate the solutions to Wasserstein gradient flows is proposed, where the time discretization is based on an implicit linearization of the Wasserstein distance expressed by means of the Benamou-Brenier formula, while the space discretization relies on upstream mobility two-point flux approximation finite volumes. The uniqueness of the solution of such a scheme is proved when the continuous problem involves a convex energy, and the corresponding convergence in the case of the linear Fokker-Planck equation with positive initial density is also derived. As possible applications of this new scheme, the authors mention various energies as well as the non-negativity of the discrete solutions and decay of the energy. Numerical illustrations of the theoretical results are provided, too, involving the Fokker-Planck equation, the porous medium equation, the thin film equation and the salinity intrusion problem.
Reviewer: Sorin-Mihai Grad (Wien)Fast high order difference schemes for the time fractional telegraph equation.https://www.zbmath.org/1452.651642021-02-12T15:23:00+00:00"Liang, Yuxiang"https://www.zbmath.org/authors/?q=ai:liang.yuxiang"Yao, Zhongsheng"https://www.zbmath.org/authors/?q=ai:yao.zhongsheng"Wang, Zhibo"https://www.zbmath.org/authors/?q=ai:wang.zhiboSummary: In this paper, a fast high order difference scheme is first proposed to solve the time fractional telegraph equation based on the \(\mathcal{FL}2-1_\sigma\) formula for the Caputo fractional derivative, which reduces the storage and computational cost for calculation. A compact scheme is then presented to improve the convergence order in space. The unconditional stability and convergence in maximum norm are proved for both schemes, with the accuracy order \(\mathcal{O}(\tau^2+h^2)\) and \(\mathcal{O}(\tau^2+h^4)\), respectively. Difficulty arising from the two Caputo fractional derivatives is overcome by some detailed analysis. Finally, we carry out numerical experiments to show the efficiency and accuracy, by comparing with the \(\mathcal{L}2-1_\sigma\) method.A residual a posteriori error estimate for the time-domain boundary element method.https://www.zbmath.org/1452.653822021-02-12T15:23:00+00:00"Gimperlein, Heiko"https://www.zbmath.org/authors/?q=ai:gimperlein.heiko"Özdemir, Ceyhun"https://www.zbmath.org/authors/?q=ai:ozdemir.ceyhun"Stark, David"https://www.zbmath.org/authors/?q=ai:stark.david-r|stark.david-b"Stephan, Ernst P."https://www.zbmath.org/authors/?q=ai:stephan.ernst-peterSummary: This article investigates residual a posteriori error estimates and adaptive mesh refinements for time-dependent boundary element methods for the wave equation. We obtain reliable estimates for Dirichlet and acoustic boundary conditions which hold for a large class of discretizations. Efficiency of the error estimate is shown for a natural discretization of low order. Numerical examples confirm the theoretical results. The resulting adaptive mesh refinement procedures in \(3d\) recover the adaptive convergence rates known for elliptic problems.Small-amplitude discontinuities of solutions to equations of continuum mechanics.https://www.zbmath.org/1452.352082021-02-12T15:23:00+00:00"Golubyatnikov, A. N."https://www.zbmath.org/authors/?q=ai:golubyatnikov.aleksandr-nikolaevichSummary: A general approach is developed for problems of propagation of weak discontinuities against a known background for systems of hyperbolic equations that can be represented in a variational form. A weak shock wave is considered as an approximation to a solution containing a weak discontinuity. This method is applicable to the description of various adiabatic processes in continuum mechanics in the presence of variable force fields.Transformed snapshot interpolation with high resolution transforms.https://www.zbmath.org/1452.652972021-02-12T15:23:00+00:00"Welper, G."https://www.zbmath.org/authors/?q=ai:welper.gerritUnified a posteriori error estimator for finite element methods for the Stokes equations.https://www.zbmath.org/1452.760592021-02-12T15:23:00+00:00"Wang, Junping"https://www.zbmath.org/authors/?q=ai:wang.junping"Wang, Yanqiu"https://www.zbmath.org/authors/?q=ai:wang.yanqiu"Ye, Xiu"https://www.zbmath.org/authors/?q=ai:ye.xiuSummary: This paper is concerned with residual type a posteriori error estimators for finite element methods for the Stokes equations. In particular, the authors established a unified approach for deriving and analyzing a posteriori error estimators for velocity-pressure based finite element formulations for the Stokes equations. A general a posteriori error estimator was presented with a unified mathematical analysis for the general finite element formulation that covers conforming, non-conforming, and discontinuous Galerkin methods as examples. The key behind the mathematical analysis is the use of a lifting operator from discontinuous finite element spaces to continuous ones for which all the terms involving jumps at interior edges disappear.Symmetries of fundamental solutions and their application in continuum mechanics.https://www.zbmath.org/1452.352062021-02-12T15:23:00+00:00"Aksenov, A. V."https://www.zbmath.org/authors/?q=ai:aksenov.a-vSummary: An application of the symmetries of fundamental solutions in continuum mechanics is presented. It is shown that the Riemann function of a second-order linear hyperbolic equation in two independent variables is invariant with respect to the symmetries of fundamental solutions, and a method is proposed for constructing such a function. A fourth-order linear elliptic partial differential equation is considered that describes the displacements of a transversely isotropic linear elastic medium. The symmetries of this equation and the symmetries of the fundamental solutions are found. The symmetries of the fundamental solutions are used to construct an invariant fundamental solution in terms of elementary functions.Generalized regularized least-squares approximation of noisy data with application to stochastic PDEs.https://www.zbmath.org/1452.653752021-02-12T15:23:00+00:00"Shirzadi, Mohammad"https://www.zbmath.org/authors/?q=ai:shirzadi.mohammad"Dehghan, Mehdi"https://www.zbmath.org/authors/?q=ai:dehghan.mehdiSummary: The regularized least-squares radial basis approximation is a kernel-based method to approximate a set of scattered data by a least-squares fit based on an optimization procedure that balances a tradeoff between smoothness of approximation and closeness to the data via a smoothing parameter. This paper suggests the generalized regularized least-squares radial basis approximation for noisy data and its application to the numerical solution of stochastic elliptic PDEs. Numerical observations show that the proposed method is more stable than the typical kernel-based method.Low-memory, discrete ordinates, discontinuous Galerkin methods for radiative transport.https://www.zbmath.org/1452.653602021-02-12T15:23:00+00:00"Sun, Zheng"https://www.zbmath.org/authors/?q=ai:sun.zheng"Hauck, Cory D."https://www.zbmath.org/authors/?q=ai:hauck.cory-dThe authors are concerned with low-memory variation of the upwind discrete ordinates discontinuous Galerkin method used to solve 2D and 3D scaled, steady-state, linear transport equation. The equation involves a positive scaling parameter $\varepsilon$ which characterizes the relative strength of scattering. Thus, instead of using a tensor product finite element space for the discrete ordinates discontinuous Galerkin system, the authors seek the solution in a proper subspace, in which all the elements have isotropic slopes. This alternative produces a significant reduction in memory per spatial cell and still preserves the asymptotic diffusion limit as well as maintains the characteristic structure needed for sweeps. It recovers second-order accuracy for arbitrary and fixed $\varepsilon$. Some numerical examples are carried out.
Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)Convexification for an inverse problem for a 1D wave equation with experimental data.https://www.zbmath.org/1452.652142021-02-12T15:23:00+00:00"Smirnov, A. V."https://www.zbmath.org/authors/?q=ai:smirnov.alexey-v"Klibanov, M. V."https://www.zbmath.org/authors/?q=ai:klibanov.michael-victor"Sullivan, Anders J."https://www.zbmath.org/authors/?q=ai:sullivan.anders-j"Nguyen, L. H."https://www.zbmath.org/authors/?q=ai:nguyen.loc-hoang|nguyen.lam-hCauchy problem for fractional non-autonomous evolution equations.https://www.zbmath.org/1452.352362021-02-12T15:23:00+00:00"Chen, Pengyu"https://www.zbmath.org/authors/?q=ai:chen.pengyu"Zhang, Xuping"https://www.zbmath.org/authors/?q=ai:zhang.xuping"Li, Yongxiang"https://www.zbmath.org/authors/?q=ai:li.yongxiangThe paper considers the Cauchy problem for a semilinear fractional nonautonomous integro-differential equation in a Banach space \(E\). Two existence results for mild solutions to the Cauchy problem are given by the fixed point method with assumptions that for each \(t\in [0, \ a]\), the related operator \(A(t)\) is a closed linear operator from a dense domain \(D(A)\subseteq E\) to \(E\) such that \(D(A)\) is independent of \(t\), and \(-A(t)\) generates an analytic semigroup; the related nonlinear function is Carathéodory continuous and satisfies some growth conditions and a noncompactness measure condition, and others.
Reviewer: Ti-Jun Xiao (Fudan)A \(TV-L^2-H^{-1}\) PDE model for effective denoising.https://www.zbmath.org/1452.350722021-02-12T15:23:00+00:00"Halim, Abdul"https://www.zbmath.org/authors/?q=ai:halim.abdul-hakim"Kumar, B. V. Rathish"https://www.zbmath.org/authors/?q=ai:kumar.b-v-rathish|kumar.bayya-venkatesulu-rathishSummary: In this paper, we have proposed a higher order non-linear PDE model containing diffusion terms of \(TV-L^2\) and \(TV-H^{-1}\) model for effective image denoising. Fourier spectral method for space and convexity splitting method for time have been used to solve the proposed PDE model. Stability analysis for the time discretized equation has been derived. Numerical experiment has been done on some test images and the results have been compared with the results of some existing models.Finite element methods for fractional-order diffusion problems with optimal convergence order.https://www.zbmath.org/1452.653512021-02-12T15:23:00+00:00"Maros, Gábor"https://www.zbmath.org/authors/?q=ai:maros.gabor"Izsák, Ferenc"https://www.zbmath.org/authors/?q=ai:izsak.ferencSummary: A convergence result is stated for the numerical solution of space-fractional diffusion problems. For the spatial discretization, an arbitrary family of finite elements can be used combined with the matrix transformation technique. The analysis covers the application of the implicit Euler method for time integration to ensure unconditional stability. The spatial convergence rate does not depend on the fractional power of the Laplacian operator. An efficient numerical implementation is developed avoiding the direct computation of matrix powers.Spectrally accurate approximate solutions and convergence analysis of fractional Burgers' equation.https://www.zbmath.org/1452.652792021-02-12T15:23:00+00:00"Mittal, A. K."https://www.zbmath.org/authors/?q=ai:mittal.ashok-kumarSummary: In this paper, a new numerical technique implements on the time-space pseudospectral method to approximate the numerical solutions of nonlinear time- and space-fractional coupled Burgers' equation. This technique is based on orthogonal Chebyshev polynomial function and discretizes using Chebyshev-Gauss-Lobbato (CGL) points. Caputo-Riemann-Liouville fractional derivative formula is used to illustrate the fractional derivatives matrix at CGL points. Using the derivatives matrices, the given problem is reduced to a system of nonlinear algebraic equations. These equations can be solved using Newton-Raphson method. Two model examples of time- and space-fractional coupled Burgers' equation are tested for a set of fractional space and time derivative order. The figures and tables show the significant features, effectiveness, and good accuracy of the proposed method.Identification of point-like objects with multifrequency sparse data.https://www.zbmath.org/1452.352562021-02-12T15:23:00+00:00"Ji, Xia"https://www.zbmath.org/authors/?q=ai:ji.xia"Liu, Xiaodong"https://www.zbmath.org/authors/?q=ai:liu.xiaodong.1The \(p\)- and \(hp\)-versions of the virtual element method for elliptic eigenvalue problems.https://www.zbmath.org/1452.653262021-02-12T15:23:00+00:00"Čertík, O."https://www.zbmath.org/authors/?q=ai:certik.ondrej"Gardini, Francesca"https://www.zbmath.org/authors/?q=ai:gardini.francesca"Manzini, G."https://www.zbmath.org/authors/?q=ai:manzini.gianmarco"Mascotto, Lorenzo"https://www.zbmath.org/authors/?q=ai:mascotto.lorenzo"Vacca, Giuseppe"https://www.zbmath.org/authors/?q=ai:vacca.giuseppeSummary: We discuss the \(p\)- and \(h p\)-versions of the virtual element method for the approximation of eigenpairs of elliptic operators with a potential term on polygonal meshes. An application of this model is provided by the Schrödinger equation with a pseudo-potential term. As an interesting byproduct, we present for the first time in literature an explicit construction of the stabilization of the mass matrix. We present in detail the analysis of the \(p\)-version of the method, proving exponential convergence in the case of analytic eigenfunctions. The theoretical results are supplied with a wide set of experiments. We also show numerically that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing \(h p\)-refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the \(h p\)-spaces.The conforming virtual element method for polyharmonic problems.https://www.zbmath.org/1452.653202021-02-12T15:23:00+00:00"Antonietti, Paola F."https://www.zbmath.org/authors/?q=ai:antonietti.paola-francesca"Manzini, G."https://www.zbmath.org/authors/?q=ai:manzini.gianmarco|manzini.giovanni|manzini.giorgio"Verani, Marco"https://www.zbmath.org/authors/?q=ai:verani.marcoSummary: In this work, we exploit the capability of virtual element methods in accommodating approximation spaces featuring high-order continuity to numerically approximate differential problems of the form \((- \Delta)^pu=f\), \(p\geq 1\). More specifically, we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems, and prove an abstract result that states the convergence of the method in suitable norms.Higher integrability for the singular porous medium system.https://www.zbmath.org/1452.350542021-02-12T15:23:00+00:00"Bögelein, Verena"https://www.zbmath.org/authors/?q=ai:bogelein.verena"Duzaar, Frank"https://www.zbmath.org/authors/?q=ai:duzaar.frank"Scheven, Christoph"https://www.zbmath.org/authors/?q=ai:scheven.christophThe authors establish in the fast diffusion range the higher integrability of the spatial gradient of weak solutions to porous medium systems. The result comes along with an explicit reverse Hölder inequality for the gradient. The novel feature in the proof is that a suitable intrinsic scaling for space-time cylinders (introduced originally by Gianazza and Schwarzacher) is combined with reverse Hölder inequalities and a Vitali covering argument within this geometry. The main result holds for the natural range of parameters suggested by other regularity results. This result applies to general fast diffusion systems and includes both, non-negative and signed solutions in the case of equations. Very interesting is the fact that the methods of proof are purely vectorial in their structure.
Reviewer: Vincenzo Vespri (Firenze)Ground states of spin-\(F\) Bose-Einstein condensates.https://www.zbmath.org/1452.351952021-02-12T15:23:00+00:00"Tian, Tonghua"https://www.zbmath.org/authors/?q=ai:tian.tonghua"Cai, Yongyong"https://www.zbmath.org/authors/?q=ai:cai.yongyong"Wu, Xinming"https://www.zbmath.org/authors/?q=ai:wu.xinming"Wen, Zaiwen"https://www.zbmath.org/authors/?q=ai:wen.zaiwenOn the Euler-Poincaré equation with non-zero dispersion.https://www.zbmath.org/1452.351532021-02-12T15:23:00+00:00"Li, Dong"https://www.zbmath.org/authors/?q=ai:li.dong"Yu, Xinwei"https://www.zbmath.org/authors/?q=ai:yu.xinwei"Zhai, Zhichun"https://www.zbmath.org/authors/?q=ai:zhai.zhichunSummary: We consider the Euler-Poincaré equation on \(\mathbb{R}^d\), \(d\geqq 2\). For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. This settles an open problem raised by \textit{D. Chae} and \textit{J.-G. Liu} [Commun. Math. Phys. 314, No. 3, 671--687 (2012; Zbl 1251.35075)]. Our analysis exhibits some new concentration mechanisms and hidden monotonicity formulas associated with the Euler-Poincaré flow. In particular we show an abundance of blowups emanating from smooth initial data with certain sign properties. No size restrictions are imposed on the data. We also showcase a class of initial data for which the corresponding solution exists globally in time.Solving the 4NLS with white noise initial data.https://www.zbmath.org/1452.351932021-02-12T15:23:00+00:00"Oh, Tadahiro"https://www.zbmath.org/authors/?q=ai:oh.tadahiro"Tzvetkov, Nikolay"https://www.zbmath.org/authors/?q=ai:tzvetkov.nikolay"Wang, Yuzhao"https://www.zbmath.org/authors/?q=ai:wang.yuzhaoSummary: We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the `random-resonant / nonlinear decomposition', which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model.https://www.zbmath.org/1452.651752021-02-12T15:23:00+00:00"Qiu, Wenlin"https://www.zbmath.org/authors/?q=ai:qiu.wenlin"Xu, Da"https://www.zbmath.org/authors/?q=ai:xu.da"Guo, Jing"https://www.zbmath.org/authors/?q=ai:guo.jing"Zhou, Jun"https://www.zbmath.org/authors/?q=ai:zhou.jun.2|zhou.jun.1|zhou.jun.3Summary: In this paper, we present a time two-grid algorithm based on the finite difference (FD) method for the two-dimensional nonlinear time-fractional mobile/immobile transport model. We establish the problem as a nonlinear fully discrete FD system, where the time derivative is discretized by the second-order backward difference formula (BDF) scheme, the Caputo fractional derivative is treated by means of \(L1\) discretization formula, and the spatial derivative is approximated by the central difference formula. For solving the nonlinear FD system more efficiently, a time two-grid algorithm is proposed, which consists of two steps: first, the nonlinear FD system on a coarse grid is solved by nonlinear iterations; second, the Newton iteration is utilized to solve the linearized FD system on the fine grid. The stability and convergence in \(L^2\)-norm are obtained for the two-grid FD scheme. Numerical results are consistent with the theoretical analysis. Meanwhile, numerical experiments show that the two-grid FD method is much more efficient than the general FD scheme for solving the nonlinear FD system.Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion.https://www.zbmath.org/1452.651662021-02-12T15:23:00+00:00"Liu, Xing"https://www.zbmath.org/authors/?q=ai:liu.xing"Deng, Weihua"https://www.zbmath.org/authors/?q=ai:deng.weihuaSummary: In this paper, we study the numerical schemes for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion. The main challenges of the numerical schemes come from the singularity in the time direction. When \(0 < H < 0.5\), a change of variables \(\partial (t^{2H} )=2Ht^{2H-1}\partial t\) avoids the singularity of numerical computation at \(t = 0\), which naturally results in nonuniform time discretization and greatly improves the computational efficiency. For \(H > 0.5\), the time span dependent numerical scheme and nonuniform time discretization are introduced to ensure the effectiveness of the calculation and the computational efficiency. The stability and convergence of the numerical schemes are demonstrated by using Fourier method. By numerically solving the corresponding Fokker-Planck equation, we obtain the mean squared displacement of stochastic processes, which conforms to the characteristics of the tempered fractional Brownian motion.A stochastic gradient method with mesh refinement for PDE-constrained optimization under uncertainty.https://www.zbmath.org/1452.625992021-02-12T15:23:00+00:00"Geiersbach, Caroline"https://www.zbmath.org/authors/?q=ai:geiersbach.caroline"Wollner, Winnifried"https://www.zbmath.org/authors/?q=ai:wollner.winnifriedBlow-up results for a semilinear parabolic differential inequality in an exterior domain.https://www.zbmath.org/1452.350472021-02-12T15:23:00+00:00"Jleli, Mohamed"https://www.zbmath.org/authors/?q=ai:jleli.mohamed"Samet, Bessem"https://www.zbmath.org/authors/?q=ai:samet.bessemSummary: We are concerned with a semilinear parabolic differential inequality posed in an exterior domain of \(\mathbb{R}^N\), \(N \geqslant 3\). Some blow-up and existence results are established for the considered problem. The novetly of this paper lies in considering a nontrivial Dirichlet boundary condition, which depends both on time and space. Indeed, to the best of our knowledge, in all articles which deal with the blow-up of solutions in exterior domains, the considered boundary condition is trivial or depends only on space.Numerical analysis of a three-species chemotaxis model.https://www.zbmath.org/1452.920072021-02-12T15:23:00+00:00"Bürger, Raimund"https://www.zbmath.org/authors/?q=ai:burger.raimund"Ordoñez, Rafael"https://www.zbmath.org/authors/?q=ai:ordonez.rafael"Sepúlveda, Mauricio"https://www.zbmath.org/authors/?q=ai:sepulveda.mauricio-a"Villada, Luis Miguel"https://www.zbmath.org/authors/?q=ai:villada.luis-miguelThe authors consider a chemotaxis system with reaction terms describing evolution of three species: a prey, a predator and a super-predator, coupled with three elliptic equations for diffusion of chemicals. A finite volume numerical scheme is proposed, and convergence of approximating solutions is shown. Discrete Sobolev type embedding inequalities and a space-time \(L^1\) type compactness argument are used.
Reviewer: Piotr Biler (Wrocław)Partial differential equations. An unhurried introduction.https://www.zbmath.org/1452.350022021-02-12T15:23:00+00:00"Tolstykh, Vladimir A."https://www.zbmath.org/authors/?q=ai:tolstykh.vladimir-aThe topics covered in this book are the standard topics of a one-semester undergraduate course in partial differential equations (PDEs).
The contents comprise eight chapters devoted to the following topics: Chapter 1 introduces basic concepts and notation. Chapter 2 is devoted to the technique of change of variables in PDEs.
Chapter 3 deals with first-order linear PDEs with constant coefficients. Chapter 4 concerns the method of characteristics for first-order semilinear PDEs. First-order quasilinear PDEs are introduced in Chapter 5 while the description of solution sets is done in Chapter 6. The method of characteristics for this class of PDEs is considered in Chapter 7. Chapter 8 deals with second-order semilinear PDEs. The text is supported by an appendix on inverse function theorems and functional dependence.
The main text is enriched with some examples on the usage of Maple and the online resource Wolfram Alpha for solving problems related to the theory of PDEs. The presentation is clear and the book is written so as to be accessible to a wide audience.
Reviewer: Ahmed Lesfari (El Jadida)An integral equation-based numerical method for the forced heat equation on complex domains.https://www.zbmath.org/1452.652722021-02-12T15:23:00+00:00"Fryklund, Fredrik"https://www.zbmath.org/authors/?q=ai:fryklund.fredrik"Kropinski, Mary Catherine A."https://www.zbmath.org/authors/?q=ai:kropinski.mary-catherine-a"Tornberg, Anna-Karin"https://www.zbmath.org/authors/?q=ai:tornberg.anna-karinSummary: Integral equation-based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.Difference potentials method for models with dynamic boundary conditions and bulk-surface problems.https://www.zbmath.org/1452.651552021-02-12T15:23:00+00:00"Epshteyn, Yekaterina"https://www.zbmath.org/authors/?q=ai:epshteyn.yekaterina"Xia, Qing"https://www.zbmath.org/authors/?q=ai:xia.qingSummary: In this work, we consider parabolic models with dynamic boundary conditions and parabolic bulk-surface problems in 3D. Such partial differential equations-based models describe phenomena that happen both on the surface and in the bulk/domain. These problems may appear in many applications, ranging from cell dynamics in biology, to grain growth models in polycrystalline materials. Using difference potentials framework, we develop novel numerical algorithms for the approximation of the problems. The constructed algorithms efficiently and accurately handle the coupling of the models in the bulk and on the surface, approximate 3D irregular geometry in the bulk by the use of only Cartesian meshes, employ fast Poisson solvers, and utilize spectral approximation on the surface. Several numerical tests are given to illustrate the robustness of the developed numerical algorithms.Gradient estimate of positive eigenfunctions of sub-Laplacian on complete pseudo-Hermitian manifolds.https://www.zbmath.org/1452.350702021-02-12T15:23:00+00:00"Ren, Yibin"https://www.zbmath.org/authors/?q=ai:ren.yibinThe paper's main theme is a classical Cheng-Yau gradient estimate for positive harmonic functions in the setting of complete noncompact pseudo-Hermitian manifolds with bounded geometric conditions. The operator considered is a sub-Laplacian, therefore standard Riemannian techniques are not applicable. The results include an estimate of the greatest lower bound for the \(L^{2}\)-spectrum of the sub-Laplacian. As a typical application of the Cheng-Yau estimate, the author proves the Liouville theorem of positive pseudo-harmonic functions on complete noncompact Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.
Reviewer: Maria Gordina (Storrs)Error analysis of the reduced RBF model based on POD method for time-fractional partial differential equations.https://www.zbmath.org/1452.652732021-02-12T15:23:00+00:00"Ghaffari, Rezvan"https://www.zbmath.org/authors/?q=ai:ghaffari.rezvan"Ghoreishi, Farideh"https://www.zbmath.org/authors/?q=ai:ghoreishi.faridehSummary: In this paper, we present a new reduced order model based on radial basis functions (RBFs) and proper orthogonal decomposition (POD) methods for fractional advection-diffusion equations with a Caputo fractional derivative in time. In the proposed scheme, the number of basis functions in the usual RBFs method reduces by the POD technique. Therefore, the computational cost of the RBF-POD method decreases in comparison with usual RBFs method, while the accuracy completely maintains. In the sequel, we provide a complete error analysis in the \(L_2\) norm between the exact solution and the RBFs solution, as well as between the exact solution and the proposed RBF-POD model by using the properties of the native space and projection operators. Also, the obtained error estimation is used to choose the number of POD bases for constructing the RBF-POD model with the required accuracy. Numerical examples are given to confirm the accuracy and efficiency of the proposed scheme.A proof of the instability of AdS for the Einstein-null dust system with an inner mirror.https://www.zbmath.org/1452.830092021-02-12T15:23:00+00:00"Moschidis, Georgios"https://www.zbmath.org/authors/?q=ai:moschidis.georgiosSummary: In 2006, Dafermos and Holzegel formulated the so-called AdS instability conjecture, stating that there exist arbitrarily small perturbations to AdS initial data which, under evolution by the Einstein vacuum equations for \(\Lambda<0\) with reflecting boundary conditions on conformal infinity \(\mathcal{I} \), lead to the formation of black holes. The numerical study of this conjecture in the simpler setting of the spherically symmetric Einstein-scalar field system was initiated by \textit{P. Bizon} and \textit{A. Rostworowski} [``Weakly turbulent instability of anti-de Sitter spacetime'', Phys. Rev. Lett. 107, No. 3, Article ID 031102, 4 p. (2011; \url{doi:10.1103/PhysRevLett.107.031102})], followed by a vast number of numerical and heuristic works by several authors.
In this paper, we provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the spherically symmetric Einstein-massless Vlasov system, in the case when the Vlasov field is moreover supported only on radial geodesics. This system is equivalent to the Einstein-null dust system, allowing for both ingoing and outgoing dust. In order to overcome the breakdown of this system occurring once the null dust reaches the center \(r=0\), we place an inner mirror at \(r=r_0>0\) and study the evolution of this system on the exterior domain \(\{r\ge r_0\} \). The structure of the maximal development and the Cauchy stability properties of general initial data in this setting are studied in our companion paper [``The Einstein-null dust system in spherical symmetry with an inner mirror: structure of the maximal development and Cauchy stability'', Preprint, \url{arXiv:1704.08685}].
The statement of the main theorem is as follows: We construct a family of mirror radii \(r_{0\varepsilon}>0\) and initial data \(\mathcal{S}_{\varepsilon} \), \( \varepsilon\in(0,1]\), converging, as \(\varepsilon\rightarrow0\), to the AdS initial data \(\mathcal{S}_0\) in a suitable norm, such that, for any \(\varepsilon\in(0,1]\), the maximal development \((\mathcal{M}_{\varepsilon},g_{\varepsilon})\) of \(\mathcal{S}_{\varepsilon}\) contains a black hole region. Our proof is based on purely physical space arguments and involves the arrangement of the null dust into a large number of beams which are successively reflected off \(\{r=r_{0\varepsilon}\}\) and \(\mathcal{I} \), in a configuration that forces the energy of a certain beam to increase after each successive pair of reflections. As \(\varepsilon\rightarrow0\), the number of reflections before a black hole is formed necessarily goes to \(+\infty \). We expect that this instability mechanism can be applied to the case of more general matter fields.Eigenvalue bounds for non-self-adjoint Schrödinger operators with nontrapping metrics.https://www.zbmath.org/1452.351222021-02-12T15:23:00+00:00"Guillarmou, Colin"https://www.zbmath.org/authors/?q=ai:guillarmou.colin"Hassell, Andrew"https://www.zbmath.org/authors/?q=ai:hassell.andrew"Krupchyk, Katya"https://www.zbmath.org/authors/?q=ai:krupchyk.katyaSummary: We study eigenvalues of non-self-adjoint Schrödinger operators on nontrapping asymptotically conic manifolds of dimension \(n\ge 3\). Specifically, we are concerned with the following two types of estimates. The first one deals with Keller-type bounds on individual eigenvalues of the Schrödinger operator with a complex potential in terms of the \(L^p\)-norm of the potential, while the second one is a Lieb-Thirring-type bound controlling sums of powers of eigenvalues in terms of the \(L^p\)-norm of the potential. We extend the results of \textit{R. L. Frank} [Bull. Lond. Math. Soc. 43, No. 4, 745--750 (2011; Zbl 1228.35158)], \textit{R. L. Frank} and \textit{J. Sabin} [Am. J. Math. 139, No. 6, 1649--1691 (2017; Zbl 1388.42018)], and \textit{R. L. Frank} and \textit{B. Simon} [J. Spectr. Theory 7, No. 3, 633--658 (2017; Zbl 1386.35061)] on the Keller- and Lieb-Thirring-type bounds from the case of Euclidean spaces to that of nontrapping asymptotically conic manifolds. In particular, our results are valid for the operator \(\Delta_g+V\) on \(\mathbb{R}^n\) with \(g\) being a nontrapping compactly supported (or suitably short-range) perturbation of the Euclidean metric and \(V\in L^p\) complex-valued.Cell density and cell size dynamics during \textit{in vitro} tissue growth experiments: implications for mathematical models of collective cell behaviour.https://www.zbmath.org/1452.920172021-02-12T15:23:00+00:00"Binder, Benjamin J."https://www.zbmath.org/authors/?q=ai:binder.benjamin-james"Simpson, Matthew J."https://www.zbmath.org/authors/?q=ai:simpson.matthew-jSummary: We present a detailed experimental data set describing a tissue growth experiment where a population of cells is initially distributed uniformly, at low density, on a two-dimensional substrate, and grows to eventually form a confluent monolayer. Using image processing tools, we provide precise information about temporal changes in the number of cells, the location of cells and the total area occupied by cells. This information shows that the increase in area occupied by the cell population is affected by both the increase in cell number as well as an increase in the average size of the cells. We show that standard approaches to interpret such experiments, where the cell size is typically treated as a constant, can lead to errors. Furthermore we show that a standard, discrete, random walk model of biological cell motility and cell proliferation should not be used to represent our experimental data set since this standard model treats all cells as having a constant size that does not change with time. Instead, we introduce a generalization of the standard model which allows agents in the random walk model to move, proliferate and grow in size, and we show that the data produced by this more general model is consistent with our experimental data set.Mathematical and numerical analysis of the generalized complex-frequency eigenvalue problem for two-dimensional optical microcavities.https://www.zbmath.org/1452.653182021-02-12T15:23:00+00:00"Spiridonov, Alexander O."https://www.zbmath.org/authors/?q=ai:spiridonov.alexander-o"Oktyabrskaya, Alina"https://www.zbmath.org/authors/?q=ai:oktyabrskaya.alina"Karchevskii, Evgenii M."https://www.zbmath.org/authors/?q=ai:karchevskii.evgenii"Nosich, Alexander I."https://www.zbmath.org/authors/?q=ai:nosich.alexander-iSpectral theory of pseudodifferential operators of degree 0 and an application to forced linear waves.https://www.zbmath.org/1452.350262021-02-12T15:23:00+00:00"Colin de Verdière, Yves"https://www.zbmath.org/authors/?q=ai:colin-de-verdiere.yvesSummary: We extend the results of our paper ``Attractors for two-dimensional waves with homogeneous Hamiltonians of degree 0,'' written with Laure Saint-Raymond [\textit{Y. C. de Verdière} and \textit{L. Saint-Raymond}, Commun. Pure Appl. Math. 73, No. 2, 421--462 (2020; Zbl 1442.35331)], to the case of forced linear wave equations in any dimension. We prove that, in dimension 2, if the foliation on the boundary at infinity of the energy shell is Morse-Smale, we can apply Mourre's theory and hence get the asymptotics of the forced solution. We also characterize the wavefront sets of the limit Schwartz distribution using radial propagation estimates.When does a perturbed Moser-Trudinger inequality admit an extremal?https://www.zbmath.org/1452.350092021-02-12T15:23:00+00:00"Thizy, Pierre-Damien"https://www.zbmath.org/authors/?q=ai:thizy.pierre-damienSummary: We are interested in several questions raised mainly by \textit{G. Mancini} and \textit{L. Martinazzi} [Calc. Var. Partial Differ. Equ. 56, No. 4, Paper No. 94, 26 p. (2017; Zbl 1382.35010)] (see also work of \textit{J. B. McLeod} and \textit{L. A. Peletier} [Arch. Ration. Mech. Anal. 106, No. 3, 261--285 (1989; Zbl 0687.46017)] and \textit{A. R. Pruss} [Can. Math. Bull. 39, No. 2, 227--237 (1996; Zbl 0855.49008)]). We consider the perturbed Moser-Trudinger inequality \(I_\alpha^g(\Omega)\) at the critical level \(\alpha=4\pi \), where \(g\), satisfying \(g(t)\to 0\) as \(t\to +\infty \), can be seen as a perturbation with respect to the original case \(g\equiv 0\). Under some additional assumptions, ensuring basically that \(g\) does not oscillate too fast as \(t\to +\infty \), we identify a new condition on \(g\) for this inequality to have an extremal. This condition covers the case \(g\equiv 0\) studied by \textit{L. Carleson} and \textit{S.-Y. A. Chang} [Bull. Sci. Math., II. Sér. 110, 113--127 (1986; Zbl 0619.58013)], \textit{M. Struwe} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5, No. 5, 425--464 (1988; Zbl 0664.35022)], and \textit{M. Flucher} [Comment. Math. Helv. 67, No. 3, 471--497 (1992; Zbl 0763.58008)]. We prove also that this condition is sharp in the sense that, if it is not satisfied, \(I_{4\pi}^g(\Omega)\) may have no extremal.A note on preconditioning for the \(3\times 3\) block saddle point problem.https://www.zbmath.org/1452.650542021-02-12T15:23:00+00:00"Xie, Xian"https://www.zbmath.org/authors/?q=ai:xie.xian"Li, Hou-Biao"https://www.zbmath.org/authors/?q=ai:li.houbiaoSummary: In this paper, we mainly propose some new preconditioners for a special class of \(3\times 3\) block saddle point problems, which can arise from the time-dependent Maxwell equations and some other practical problems. Firstly, the solvability of this kind of problem is investigated under suitable assumptions. Then, we analyze the corresponding eigenvalues of the new preconditioned matrices presented in this work. Furthermore, we show that proposed preconditioned matrices only have at most three distinct eigenvalues. Finally, two numerical examples are also carried to verify the effectiveness of proposed preconditioners.A Petrov-Galerkin finite element method for simulating chemotaxis models on stationary surfaces.https://www.zbmath.org/1452.920112021-02-12T15:23:00+00:00"Zhao, Shubo"https://www.zbmath.org/authors/?q=ai:zhao.shubo"Xiao, Xufeng"https://www.zbmath.org/authors/?q=ai:xiao.xufeng"Zhao, Jianping"https://www.zbmath.org/authors/?q=ai:zhao.jianping"Feng, Xinlong"https://www.zbmath.org/authors/?q=ai:feng.xinlongThe authors propose a modified Galerkin method for solving chemotaxis systems on surfaces. This method satisfies a discrete maximum principle and a discrete mass conservation property. A stability analysis is done, and numerical simulations are provided for blowing up solutions.
Reviewer: Piotr Biler (Wrocław)A probabilistic algorithm approximating solutions of a singular PDE of porous media type.https://www.zbmath.org/1452.650232021-02-12T15:23:00+00:00"Belaribi, Nadia"https://www.zbmath.org/authors/?q=ai:belaribi.nadia"Cuvelier, François"https://www.zbmath.org/authors/?q=ai:cuvelier.francois"Russo, Francesco"https://www.zbmath.org/authors/?q=ai:russo.francesco.2Summary: The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in \(L^1(\mathbb{R})\). In some recent papers of Blanchard et alia and Barbu et alia, the solution was represented by the solution of a non-linear stochastic differential equation in law if the initial condition is a bounded integrable function. We first extend this result, at least when \(\beta\) is continuous and the initial condition is only integrable with some supplementary technical assumption. The main purpose of the article consists in introducing and implementing a stochastic particle algorithm to approach the solution to (PDE) which also fits in the case when \(\beta\) is possibly irregular, to predict some long-time behavior of the solution and in comparing with some recent numerical deterministic techniques.Concentration compactness. Functional-analytic theory of concentration phenomena.https://www.zbmath.org/1452.350032021-02-12T15:23:00+00:00"Tintarev, Cyril"https://www.zbmath.org/authors/?q=ai:tintarev.kyrilThe convergence or structural representation of sequences in Banach spaces without given compact embedding is an important subject in the study of partial differential equations. This book mainly focuses on the concentration compactness in Banach spaces, including the embeddings of Besov and Triebel-Lizorkin spaces, embeddings associated with the Moser-Trudinger inequality, the Strichartz embedding for the nonlinear Schrödinger equation, the affine Sobolev inequality, and the profile decomposition to functional spaces that do not have a nontrivial group of invariance.
The book can be constructionally divided into two parts: the first one is preliminary and contains the basic notions of the theory and examples of compactness, technical preliminaries concerning Delta-convergence and its realization in Sobolev and other scale-invariant function spaces, some known results on cocompactness relative to the rescaling group, in Besov and Triebel-Lizorkin spaces, and cocompactness of an embedding of the Moser-Trudinger-type relative to a different group of logarithmic dilations. The second part includes five independent chapters, which discuss some further cocompact embeddings and profile decompositions, the defect of compactness for sequences restricted to different subsequences, the profile decompositions for nonreflexive spaces and for Sobolev spaces without invariance, and a small selection of applications of concentration methods to semilinear elliptic equations.
The book provides a helpful tool to the study of functional analysis, nonlinear analysis and applications and partial differential equations and so on.
Reviewer: Shuangjie Peng (Wuhan)On almost periodic viscosity solutions to Hamilton-Jacobi equations.https://www.zbmath.org/1452.350672021-02-12T15:23:00+00:00"Panov, Evgeny Yu."https://www.zbmath.org/authors/?q=ai:panov.evgeniy-yuSummary: We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with Bohr almost periodic initial data remains to be spatially almost periodic and the additive subgroup generated by its spectrum does not increase in time. In the case of one space variable and a non-degenerate hamiltonian we prove the decay property of almost periodic viscosity solutions when time \(t \to +\infty\). For convex hamiltonian we also provide another proof of this property using the Hopf-Lax-Oleinik formula. For periodic solutions the more general result is proved on unconditional asymptotic convergence of a viscosity solution to a traveling wave.Numerical simulations for the energy-supercritical nonlinear wave equation.https://www.zbmath.org/1452.351052021-02-12T15:23:00+00:00"Murphy, Jason"https://www.zbmath.org/authors/?q=ai:murphy.jason"Zhang, Yanzhi"https://www.zbmath.org/authors/?q=ai:zhang.yanzhiEffective fronts of polytope shapes.https://www.zbmath.org/1452.350212021-02-12T15:23:00+00:00"Jing, Wenjia"https://www.zbmath.org/authors/?q=ai:jing.wenjia"Tran, Hung V."https://www.zbmath.org/authors/?q=ai:tran.hung-vinh"Yu, Yifeng"https://www.zbmath.org/authors/?q=ai:yu.yifengSummary: We study the periodic homogenization of first order front propagations. Based on PDE methods, we provide a simple proof that, for \(n\geq 3\), the class of centrally symmetric polytopes with rational coordinates and nonempty interior is admissible as effective fronts, which was also established by \textit{I. Babenko} and \textit{F. Balacheff} [Manuscr. Math. 119, No. 3, 347--358 (2006; Zbl 1082.05509)] and \textit{M. Jotz} [Differ. Geom. Appl. 27, No. 4, 543--550 (2009; Zbl 1171.58004)] in the form of stable norms as an extension of \textit{G. A. Hedlund}'s classical result [Ann. Math. (2) 33, 719--739 (1932; Zbl 0006.32601)]. Besides, we obtain the optimal convergence rate of the homogenization problem for this class.Difference methods of the solution of local and non-local boundary value problems for loaded equation of thermal conductivity of fractional order.https://www.zbmath.org/1452.800192021-02-12T15:23:00+00:00"Beshtokov, M. H."https://www.zbmath.org/authors/?q=ai:beshtokov.murat-khamidbievich"Khudalov, M. Z."https://www.zbmath.org/authors/?q=ai:khudalov.m-zSummary: We study local and non-local boundary value problems for a one-dimensional space-loaded differential equation of thermal conductivity with variable coefficients with a fractional Caputo derivative, as well as difference schemes approximating these problems on uniform grids. For the solution of local and non-local boundary value problems by the method of energy inequalities, a priori estimates in differential and difference interpretations are obtained, which implies the uniqueness and stability of the solution from the initial data and the right side, as well as the convergence of the solution of the difference problem to the solution of the corresponding differential problem at the rate of \(O(h^2+\tau^2)\).
For the entire collection see [Zbl 1444.93003].Hypoelliptic mean field games -- a case study.https://www.zbmath.org/1452.350692021-02-12T15:23:00+00:00"Feleqi, Ermal"https://www.zbmath.org/authors/?q=ai:feleqi.ermal"Gomes, Diogo"https://www.zbmath.org/authors/?q=ai:gomes.diogo-luis-aguiar"Tada, Teruo"https://www.zbmath.org/authors/?q=ai:tada.teruoSummary: We study hypoelliptic mean-field games (MFG) that arise in stochastic control problems of degenerate diffusions. Here, we consider MFGs with quadratic Hamiltonians and prove the existence and uniqueness of solutions. Our main tool is the Hopf-Cole transform that converts the MFG into an eigenvalue problem. We prove the existence of a principal eigenvalue and a positive eigenfunction, which are then used to construct the unique solution to the original MFG.Solving the inverse heat conduction boundary problem for composite materials.https://www.zbmath.org/1452.800162021-02-12T15:23:00+00:00"Tanana, V. P."https://www.zbmath.org/authors/?q=ai:tanana.vitaly-pavlovich"Sidikova, A. I."https://www.zbmath.org/authors/?q=ai:sidikova.anna-ivanovnaSummary: The paper deals with the problem of determining the boundary condition in the heat equation consisting of homogeneous parts with different thermal properties. As boundary conditions, the Dirichlet condition at the left end of the rod (at \(x=0)\) corresponding to the heating of this end and the linear condition of the third kind at the right end (at \(x=1)\) corresponding to the cooling when interacting with the environment are considered. In a point of discontinuity of heat transfer properties (at \(x=x_0)\) conditions of continuity for temperature and heat flow are set. In the inverse problem, the boundary condition at the left is considered unknown over the entire infinite time interval. To find it, the value of the direct problem solution at the point of \(x_0\), that is, the point of the rod division into two homogeneous sections, is specified. In this paper, an analytical study of the direct problem was carried out, which allowed us to apply the time Fourier transform to the inverse boundary value problem. The inverse heat conduction boundary problem was solved using the projection-regularization method and order-accurate error estimates of this solution were obtained.
For the entire collection see [Zbl 1444.93003].Approximation of Hamilton-Jacobi equations with the Caputo time-fractional derivative.https://www.zbmath.org/1452.352332021-02-12T15:23:00+00:00"Camilli, Fabio"https://www.zbmath.org/authors/?q=ai:camilli.fabio"Duisembay, Serikbolsyn"https://www.zbmath.org/authors/?q=ai:duisembay.serikbolsynSummary: We investigate the numerical approximation of Hamilton-Jacobi equations with the Caputo time-fractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finite difference scheme for the approximation of the Hamiltonian. We show that the approximation scheme so obtained is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the Hamilton-Jacobi equation.Optimal order finite difference/local discontinuous Galerkin method for variable-order time-fractional diffusion equation.https://www.zbmath.org/1452.652532021-02-12T15:23:00+00:00"Wei, Leilei"https://www.zbmath.org/authors/?q=ai:wei.leilei"Yang, Yanfang"https://www.zbmath.org/authors/?q=ai:yang.yanfangSummary: In this paper, an accurate numerical method is presented to solve a class of variable-order fractional diffusion problem. The problem first is discretized by a finite difference method in temporal direction, and then a local discontinuous Galerkin method in space. The stability and \(L^2\) convergence of the proposed scheme are derived for all variable-order \(\alpha (t) \in (0, 1)\). We prove that the scheme is of accuracy-order \(O(\tau + h^{k + 1})\), where \(\tau, h\) and \(k\) are temporal step sizes, spatial step sizes and the degree of piecewise \(P^k\) polynomials, respectively. Some numerical experiments are provided to verify the theoretical analysis and high-accuracy of the proposed method.Decoupled modified characteristic FEMs for fully evolutionary Navier-Stokes-Darcy model with the Beavers-Joseph interface condition.https://www.zbmath.org/1452.652272021-02-12T15:23:00+00:00"Cao, Luling"https://www.zbmath.org/authors/?q=ai:cao.luling"He, Yinnian"https://www.zbmath.org/authors/?q=ai:he.yinnian"Li, Jian"https://www.zbmath.org/authors/?q=ai:li.jian.1"Yang, Di"https://www.zbmath.org/authors/?q=ai:yang.di.2|yang.di.1Summary: In this paper, we develop the numerical theory of decoupled modified characteristic FEMs for the fully evolutionary Navier-Stokes-Darcy model with the Beavers-Joseph interface condition. Based on lagging interface coupling terms, the system is decoupled, which means that the Navier-Stokes equations and the Darcy equation are solved in each time step, respectively. In particular, the Navier-Stokes equations are solved by the modified characteristic FEMs, which overcome the computational inefficiency and analytical difficulties caused by the nonlinear term. Then we prove the optimal \(L^2\)-norm error convergence order of the solutions by mathematical induction, whose proof implies the uniform \(L^\infty\)-boundedness of the fully discrete velocity solution. Finally some numerical tests are presented to show high efficiency of this method.Weak maximum principle for biharmonic equations in quasiconvex Lipschitz domains.https://www.zbmath.org/1452.350532021-02-12T15:23:00+00:00"Zhuge, Jinping"https://www.zbmath.org/authors/?q=ai:zhuge.jinpingThe validity of the weak maximum principle \[\|\nabla u\|_{L^\infty(\Omega)}\leq C\|\nabla u\|_{L^\infty(\partial \Omega)}\] for the
biharmonic equation \(\Delta^2u=0\) in a bounded Lipschitz domain \(\Omega\subset \mathbb{R}^d\) is investigated. It is known that in dimension \(2\) or \(3\),
the weak maximum principle for the biharmonic equation holds with no additional conditions on \(\Omega\), while it holds if \(\Omega\) is convex or \(C^1\) in
dimension greater than or equal to 4. In this latter case, it is also known that the weak maximum principle fails if \(\Omega\) contains the exterior of a
cone with small aperture.
Here, the author proves that the weak maximum principle for the biharmonic equation is still valid under a more general
quasi-convexity condition which is satisfied both by convex and \(C^1\) domains and allows domains containing the exterior of cones with sufficiently large
aperture (thus, in this sense, it is a sharp condition). In particular, the author gives the following notion of quasi-convexity: given \(\delta,R>0\) and
\(\sigma\in (0,1)\), a Lipschitz domain \(\Omega\) is said \((\delta,\sigma,R)\)-quasiconvex if, for any \(r\in (0,R)\) and \(Q\in
\partial \Omega\), one has \[|\Omega\cap B_r(Q)|\geq \sigma |B_r(Q)\] and there exists a convex domain \(V=V(Q,r)\) such that \[\Omega\cap B_r(Q)\subset V \ \
\ \text{and} \ \ \ d(\partial (\Omega\cap B_r(Q)),\partial V):=\sup_{x\in \partial (\Omega\cap B_r(Q))}\inf_{y\in \partial V}|x-y|\leq \delta r.\] Then, he
proves that there exists \(\delta_0>0\) depending only on the dimension \(d\) and the Lipschitz constant of \(\Omega\) such that the weak maximum principle for
the biharmonic equation is valid if \(\Omega\) is \((\delta,\sigma,R)\)-quasiconvex with \(\delta \in (0,\delta_0)\). The main ingredients of the proof are: Meyer's
estimate for biharmonic functions, the reverse Hölder inequality, and Shen's real variable method.
Reviewer: Giovanni Anello (Messina)Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation.https://www.zbmath.org/1452.760302021-02-12T15:23:00+00:00"Xu, Xiaojing"https://www.zbmath.org/authors/?q=ai:xu.xiaojing"Xue, Liutang"https://www.zbmath.org/authors/?q=ai:xue.liutangSummary: In this paper we consider the Yudovich type solution of the 2D inviscid Boussinesq system with critical and supercritical dissipation. For the critical case, we show that the system admits a global and unique Yudovich type solution; for the supercritical case, we prove the local and unique existence of Yudovich type solution, and the global result under a smallness condition of \({{\theta}_0}\). We also give a refined blowup criterion in the supercritical case.Semi-classical propagation of singularities for the Stokes system.https://www.zbmath.org/1452.351262021-02-12T15:23:00+00:00"Sun, Chenmin"https://www.zbmath.org/authors/?q=ai:sun.chenminSummary: We study the quasi-mode of Stokes system posed on a smooth bounded domain \(\Omega\) with Dirichlet boundary condition. We prove that the semi-classical defect measure associated with a sequence of solutions concentrates on the bicharacteristics of Laplacian as a matrix-valued Radon measure. Moreover, we show that the support of the measure is invariant under the Melrose-Sjöstrand flow.Three-commutators revisited.https://www.zbmath.org/1452.352382021-02-12T15:23:00+00:00"Da Lio, Francesca"https://www.zbmath.org/authors/?q=ai:da-lio.francesca"Rivière, Tristan"https://www.zbmath.org/authors/?q=ai:riviere.tristanSummary: We present a class of pseudo-differential elliptic systems with anti-self-dual potentials on \(\mathbb{R}\) satisfying compensation phenomena similar to the ones discovered by the second author for elliptic systems with anti-symmetric potentials. These compensation phenomena are based on new ``multi-commutator'' structures generalizing the 3-commutators recently introduced by the authors.Almost minimizers of the one-phase free boundary problem.https://www.zbmath.org/1452.350062021-02-12T15:23:00+00:00"De Silva, D."https://www.zbmath.org/authors/?q=ai:de-silva.daniela"Savin, O."https://www.zbmath.org/authors/?q=ai:savin.ovidiu-vSummary: We consider almost minimizers to the one-phase energy functional and we prove their optimal Lipschitz regularity and partial regularity of their free boundary. These results were recently obtained by David and Toro, and David, Engelstein, and Toro. Our proofs provide a different method based on a non-infinitesimal notion of viscosity solutions that we introduced.Nonlinear stability of homothetically shrinking Yang-Mills solitons in the equivariant case.https://www.zbmath.org/1452.350352021-02-12T15:23:00+00:00"Glogić, Irfan"https://www.zbmath.org/authors/?q=ai:glogic.irfan"Schörkhuber, Birgit"https://www.zbmath.org/authors/?q=ai:schorkhuber.birgitSummary: We study the heat flow for Yang-Mills connections on \(\mathbb{R}^d \times SO(d)\) It is well-known that in dimensions \(5\le d\le 9\) this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove. We prove the nonlinear asymptotic stability of the Weinkove solution under small equivariant perturbations and thus extend a result by the second author and Donninger for \(d = 5\) to higher dimensions. At the same time, we provide a general framework for proving stability of self-similar blowup solutions to a large class of semilinear heat equations in arbitrary space dimension \(d\ge 3\) including a robust and simple method for solving the underlying spectral problems.Fully nonlinear integro-differential equations with deforming kernels.https://www.zbmath.org/1452.350552021-02-12T15:23:00+00:00"Caffarelli, Luis"https://www.zbmath.org/authors/?q=ai:caffarelli.luis-a"Teymurazyan, Rafayel"https://www.zbmath.org/authors/?q=ai:teymurazyan.rafayel"Urbano, José Miguel"https://www.zbmath.org/authors/?q=ai:urbano.jose-miguelIn this interesting paper, the authors develop a regularity theory for integro-differential equations with kernels deforming in space like sections of a convex solution of a Monge-Ampère equation. Such a kind of equations appear in stochastic control problems, for example if, in a competitive stochastic game, two players are allowed to choose from different strategies at every step in order to maximize the expected value at the first exit point of a domain. The authors prove a non-local version of the Aleksandrov-Bakelman-Pucci estimate and a Harnack inequality, and derive Hölder and \(C^{1,\alpha}\) regularity results for solutions.
Reviewer: Vincenzo Vespri (Firenze)Hamilton-Jacobi in metric spaces with a homological term.https://www.zbmath.org/1452.350662021-02-12T15:23:00+00:00"Bessi, Ugo"https://www.zbmath.org/authors/?q=ai:bessi.ugoSummary: The Hamilton-Jacobi equation on metric spaces has been studied by several authors; following the approach of Gangbo and Swiech, we show that the final value problem for the Hamilton-Jacobi equation has a unique solution even if we add a homological term to the Hamiltonian. In metric measure spaces which satisfy the \(RCD (K, \infty)\) condition one can define a Laplacian which shares many properties with the ordinary Laplacian on \(\mathbb{R}^n\) in particular, it is possible to formulate a viscous Hamilton-Jacobi equation. We show that, if the homological term is sufficiently regular, the viscous Hamilton-Jacobi equation has a unique solution also in this case.Free boundary problems involving singular weights.https://www.zbmath.org/1452.350572021-02-12T15:23:00+00:00"Lamboley, Jimmy"https://www.zbmath.org/authors/?q=ai:lamboley.jimmy"Sire, Yannick"https://www.zbmath.org/authors/?q=ai:sire.yannick"Teixeira, Eduardo V."https://www.zbmath.org/authors/?q=ai:teixeira.eduardo-v-oSummary: In this paper we initiate the investigation of free boundary minimization problems ruled by general singular operators with \(A_2\) weights. We show existence, boundedness, and continuity of minimizers. The key novelty is a sharp \(C^{1 + \gamma}\) regularity result for solutions at their singular free boundary points. We also show a corresponding nondegeneracy estimate.Quaternionic structure and analysis of some Kramers-Fokker-Planck operators.https://www.zbmath.org/1452.352132021-02-12T15:23:00+00:00"Said, Mona Ben"https://www.zbmath.org/authors/?q=ai:said.mona-ben"Nier, Francis"https://www.zbmath.org/authors/?q=ai:nier.francis"Viola, Joe"https://www.zbmath.org/authors/?q=ai:viola.joeSummary: The present article is concerned with global subelliptic estimates for Kramers-Fokker-Planck operators with polynomials of degree less than or equal to two. The constants appearing in those estimates are accurately formulated in terms of the coefficients, especially when those are large.\( \sigma \)-evolution models with low regular time-dependent non-effective structural damping.https://www.zbmath.org/1452.352102021-02-12T15:23:00+00:00"Vargas, Edson Cilos jun."https://www.zbmath.org/authors/?q=ai:vargas.edson-cilos-jun"da Luz, Cleverson Roberto"https://www.zbmath.org/authors/?q=ai:da-luz.cleverson-robertoSummary: In this work we study decay rates for a \(\sigma \)-evolution equation in \(\mathbb R^n\) under effects of a damping term represented by the action of a fractional Laplacian operator and a time-dependent coefficient, \(b ( t ) ( - \Delta )^\theta u_t ( t , x )\). We consider that \(b\) is `confined' in the curve \(g ( t ) = ( 1 + t )^\alpha \ln^\gamma ( 1 + t )\) for large \(t \geqslant t_0\) and without any control on \(\frac{d}{d t} b ( t )\).Analysis and applications: the mathematical work of Elias Stein.https://www.zbmath.org/1452.320012021-02-12T15:23:00+00:00"Fefferman, Charles"https://www.zbmath.org/authors/?q=ai:fefferman.charles-louis"Ionescu, Alex"https://www.zbmath.org/authors/?q=ai:ionescu.alex"Tao, Terence"https://www.zbmath.org/authors/?q=ai:tao.terence-c"Wainger, Stephen"https://www.zbmath.org/authors/?q=ai:wainger.stephenSummary: This article discusses some of Elias M. Stein's seminal contributions to analysis.On the degenerate Cahn-Hilliard equation: global existence and entropy estimates of weak solutions.https://www.zbmath.org/1452.351522021-02-12T15:23:00+00:00"Jihui, Wu"https://www.zbmath.org/authors/?q=ai:jihui.wu"Shu, Wang"https://www.zbmath.org/authors/?q=ai:shu.wangSummary: In this paper, we consider a class of the degenerate Cahn-Hilliard equation with a smooth double-well potential. By applying a semi-discretization technique and asymptotic analysis method on non-degenerate Cahn-Hilliard equation, we obtain the existence and regularity of weak solutions to the Cahn-Hilliard equation with degenerate mobility. Moreover, we define a new entropy and obtain the global entropy estimates.Semiclassical resolvent estimates for short-range \(L^\infty\) potentials. II.https://www.zbmath.org/1452.351962021-02-12T15:23:00+00:00"Vodev, Georgi"https://www.zbmath.org/authors/?q=ai:vodev.georgiSummary: We prove semiclassical resolvent estimates for real-valued potentials \(V \in L^\infty ( \mathbb R^n )\), \(n \geqslant 3\), of the form \(V = V_L + V_S\), where \(V_L\) is a long-range potential which is \(C^1\) with respect to the radial variable, while \(V_S\) is a short-range potential satisfying \(V_S ( x) = \mathcal O ( \langle x \rangle^{- \delta})\) with \(\delta > 1\).Asymptotic behavior of ground states of generalized pseudo-relativistic Hartree equation.https://www.zbmath.org/1452.351602021-02-12T15:23:00+00:00"Belchior, P."https://www.zbmath.org/authors/?q=ai:belchior.p"Bueno, H."https://www.zbmath.org/authors/?q=ai:bueno.hamilton-p"Miyagaki, O. H."https://www.zbmath.org/authors/?q=ai:miyagaki.olimpio-hiroshi"Pereira, G. A."https://www.zbmath.org/authors/?q=ai:pereira.gilberto-aSummary: With appropriate hypotheses on the nonlinearity \(f\), we prove the existence of a ground state solution \(u\) for the problem
\[
\sqrt{- \Delta + m^2} u + V u = \left( W \ast F ( u ) \right) f ( u )\;\; \text{in } \mathbb R^N ,
\]
where \(V\) is a bounded potential, not necessarily continuous, and \(F\) the primitive of \(f\). We also show that any of this problem is a classical solution. Furthermore, we prove that the ground state solution has exponential decay.A posteriori analysis of the spectral element discretization of a non linear heat equation.https://www.zbmath.org/1452.652642021-02-12T15:23:00+00:00"Abdelwahed, Mohamed"https://www.zbmath.org/authors/?q=ai:abdelwahed.mohamed"Chorfi, Nejmeddine"https://www.zbmath.org/authors/?q=ai:chorfi.nejmeddineSummary: The paper deals with a posteriori analysis of the spectral element discretization of a non linear heat equation. The discretization is based on Euler's backward scheme in time and spectral discretization in space. Residual error indicators related to the discretization in time and in space are defined. We prove that those indicators are upper and lower bounded by the error estimation.Notice of retraction: ``Generalization of the Phragmèn-Lindelöf theorems for subfunctions''.https://www.zbmath.org/1452.310072021-02-12T15:23:00+00:00"Qiao, Lei"https://www.zbmath.org/authors/?q=ai:qiao.lei.1"Pan, Guoshuang"https://www.zbmath.org/authors/?q=ai:pan.guoshuangThe Kähler-Ricci mean curvature flow of a strictly area decreasing map between Riemann surfaces.https://www.zbmath.org/1452.530762021-02-12T15:23:00+00:00"Pan, Shujing"https://www.zbmath.org/authors/?q=ai:pan.shujingExtended Gevrey regularity via the short-time Fourier transform.https://www.zbmath.org/1452.460282021-02-12T15:23:00+00:00"Teofanov, Nenad"https://www.zbmath.org/authors/?q=ai:teofanov.nenad"Tomić, Filip"https://www.zbmath.org/authors/?q=ai:tomic.filipLet \(\tau > 0\) and \(\sigma > 1\) and set \(M_p^{\tau,\sigma} = p^{\tau p^\sigma}.\) The extended Gevrey class (of Roumieu type) \({\mathcal E}_{\{\tau,\sigma\}}(U)\) on an open subset \(U\subset {\mathbb R}^d\) consists of those smooth functions \(\phi\in C^\infty(U)\) with the property that for every compact subset \(K\subset U\) there is \(h >0\) such that
\[
\sup_{\alpha\in {\mathbb N}^d}\sup_{x\in K}\frac{\left|\partial^\alpha\phi(x)\right|}{h^{|\alpha|^\sigma}M_{|\alpha|}^{\tau,\sigma}} < \infty.\] For \(\tau > 1\) and \(\sigma = 1\) we recover the usual Gevrey classes. The authors present a version of the Paley-Wiener theorem for extended Gevrey regularity in terms of the short time Fourier transform (STFT). Also an appropriate notion of wave front set related to these classes is characterized via decay estimates of the STFT.
For the entire collection see [Zbl 1443.35001].
Reviewer: Antonio Galbis (Valencia)Variational integral and some inequalities of a class of quasilinear elliptic system.https://www.zbmath.org/1452.300242021-02-12T15:23:00+00:00"Lu, Yueming"https://www.zbmath.org/authors/?q=ai:lu.yu"Lian, Pan"https://www.zbmath.org/authors/?q=ai:lian.panSummary: This paper is concerned with properties for a class of degenerate elliptic equations in Clifford analysis. Here we obtain a direct proof of the existence and uniqueness for the Dirac equations by the method of variational integral. Also, we get the Poincaré inequalities for the case \(q<1\).On the measurability of stochastic Fourier integral operators.https://www.zbmath.org/1452.352712021-02-12T15:23:00+00:00"Oberguggenberger, Michael"https://www.zbmath.org/authors/?q=ai:oberguggenberger.michael-b"Schwarz, Martin"https://www.zbmath.org/authors/?q=ai:schwarz.martin-jun|schwarz.martin-dThe paper is related to the study of the hyperbolic equations from the stochastic point of view. This problem is relevant, because of the applications to modeling waves in random media, and as well difficult, since randomness in the coefficients of the equations affects in a nonlinear way the stochastic features of the solutions. Here the authors begin to consider Fourier integral operators (FIOs) from the stochastic perspective, assuming random phase and amplitude functions. Results of continuity and measurability are proved. FIOs are basic tools to express solutions of the hyperbolic equations, see, for example, [\textit{M. Mascarello} and \textit{L. Rodino}, Partial differential equations with multiple characteristics. Berlin: Akademie Verlag (1997; Zbl 0888.35001)]. Applications of the general results of the authors to the hyperbolic problems are then natural. As first examples, at the end of the paper transport and wave equations with spatially random speed are investigated.
For the entire collection see [Zbl 1443.35001].
Reviewer: Luigi Rodino (Torino)A new numerical scheme for discrete constrained total variation flows and its convergence.https://www.zbmath.org/1452.652312021-02-12T15:23:00+00:00"Giga, Yoshikazu"https://www.zbmath.org/authors/?q=ai:giga.yoshikazu"Sakakibara, Koya"https://www.zbmath.org/authors/?q=ai:sakakibara.koya"Taguchi, Kazutoshi"https://www.zbmath.org/authors/?q=ai:taguchi.kazutoshi"Uesaka, Masaaki"https://www.zbmath.org/authors/?q=ai:uesaka.masaakiSummary: In this paper, we propose a new numerical scheme for a spatially discrete model of total variation flows whose values are constrained to a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex; hence it is hard to calculate a minimizer of the functional with the manifold constraint even if it exists. We overcome this difficulty by ``localization technique'' using the exponential map and prove a finite-time error estimate. Finally, we show a few numerical results for the target manifolds \(S^2\) and \(SO(3)\).Lower a posteriori error estimates on anisotropic meshes.https://www.zbmath.org/1452.653462021-02-12T15:23:00+00:00"Kopteva, Natalia"https://www.zbmath.org/authors/?q=ai:kopteva.natalia|kopteva.natalia.1Summary: Lower a posteriori error bounds obtained using the standard bubble function approach are reviewed in the context of anisotropic meshes. A numerical example is given that clearly demonstrates that the short-edge jump residual terms in such bounds are not sharp. Hence, for linear finite element approximations of the Laplace equation in polygonal domains, a new approach is employed to obtain essentially sharper lower a posteriori error bounds and thus to show that the upper error estimator in the recent paper [\textit{N. Kopteva}, Numer. Math. 137, No. 3, 607--642 (2017; Zbl 1379.65084)] is efficient on partially structured anisotropic meshes.A structure-preserving discontinuous Galerkin scheme for the Fisher-KPP equation.https://www.zbmath.org/1452.652252021-02-12T15:23:00+00:00"Bonizzoni, Francesca"https://www.zbmath.org/authors/?q=ai:bonizzoni.francesca"Braukhoff, Marcel"https://www.zbmath.org/authors/?q=ai:braukhoff.marcel"Jüngel, Ansgar"https://www.zbmath.org/authors/?q=ai:jungel.ansgar"Perugia, Ilaria"https://www.zbmath.org/authors/?q=ai:perugia.ilariaSummary: An implicit Euler discontinuous Galerkin scheme for the Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation for population densities with no-flux boundary conditions is suggested and analyzed. Using an exponential variable transformation, the numerical scheme automatically preserves the positivity of the discrete solution. A discrete entropy inequality is derived, and the exponential time decay of the discrete density to the stable steady state in the \(L^1\) norm is proved if the initial entropy is smaller than the measure of the domain. The discrete solution is proved to converge in the \(L^2\) norm to the unique strong solution to the time-discrete Fisher-KPP equation as the mesh size tends to zero. Numerical experiments in one space dimension illustrate the theoretical results.A domain mapping approach for elliptic equations posed on random bulk and surface domains.https://www.zbmath.org/1452.653302021-02-12T15:23:00+00:00"Church, Lewis"https://www.zbmath.org/authors/?q=ai:church.lewis"Djurdjevac, Ana"https://www.zbmath.org/authors/?q=ai:djurdjevac.ana"Elliott, Charles M."https://www.zbmath.org/authors/?q=ai:elliott.charles-mSummary: In this article, we analyse the domain mapping method approach to approximate statistical moments of solutions to linear elliptic partial differential equations posed over random geometries including smooth surfaces and bulk-surface systems. In particular, we present the necessary geometric analysis required by the domain mapping method to reformulate elliptic equations on random surfaces onto a fixed deterministic surface using a prescribed stochastic parametrisation of the random domain. An abstract analysis of a finite element discretisation coupled with a Monte-Carlo sampling is presented for the resulting elliptic equations with random coefficients posed over the fixed curved reference domain and optimal error estimates are derived. The results from the abstract framework are applied to a model elliptic problem on a random surface and a coupled elliptic bulk-surface system and the theoretical convergence rates are confirmed by numerical experiments.Non-local initial problem for second order time-fractional and space-singular equation.https://www.zbmath.org/1452.352412021-02-12T15:23:00+00:00"Karimov, Erkinjon"https://www.zbmath.org/authors/?q=ai:karimov.erkinjon-tulkinovich"Mamchuev, Murat"https://www.zbmath.org/authors/?q=ai:mamchuev.murat-osmanovich"Ruzhansky, Michael"https://www.zbmath.org/authors/?q=ai:ruzhansky.michael-vSummary: In this work, we consider an initial problem for second order partial differential equations with Caputo fractional derivatives in the time-variable and Bessel operator in the space-variable. For non-local boundary conditions, we present a solution of this problem in an explicit form representing it by the Fourier-Bessel series. The obtained solution is written in terms of multinomial Mittag-Leffler functions and first kind Bessel functions.Eigenvalues and resonances of Dirac operators with dilation analytic potentials diverging at infinity.https://www.zbmath.org/1452.810982021-02-12T15:23:00+00:00"Ito, Hiroshi T."https://www.zbmath.org/authors/?q=ai:ito.hiroshi-tThe work is very useful for relativist physicists, which specify Dirac equation for the non-relativistic limit, where 1) One has resonances 2) the Pauli approximation is working perfectly. The author is making this transition carefully, and rigorously mathematically. The most important part of the work is the first. Such resonances could arises in many cases, e.g. in the double wells case, or in the simplest case of mass fermions in a Schwarzschild field. Moreover, the results of the work can be applied for any coordinates sets, including Eddington-Finkelstein coordinates for the Schwarzschild singularity, which can cover the space-time region inside the Black Hole Horison.
Reviewer: Alex B. Gaina (Chisinau)Comparative study of macroscopic traffic flow models at road junctions.https://www.zbmath.org/1452.350982021-02-12T15:23:00+00:00"Goatin, Paola"https://www.zbmath.org/authors/?q=ai:goatin.paola"Rossi, Elena"https://www.zbmath.org/authors/?q=ai:rossi.elenaThis paper deals with multilane traffic models. The authors qualitatively compare the solutions of a multilane model with those produced by the classical Lighthill-Whitham-Richards equation with suitable coupling conditions at simple road junctions. They develop several numerical experiments using Godunov and upwind schemes.
Reviewer: Giuseppe Maria Coclite (Bari)An algorithm for recovering the characteristics of the initial state of supernova.https://www.zbmath.org/1452.652092021-02-12T15:23:00+00:00"Kabanikhin, S. I."https://www.zbmath.org/authors/?q=ai:kabanikhin.sergei-i"Kulikov, I. M."https://www.zbmath.org/authors/?q=ai:kulikov.igor-m"Shishlenin, M. A."https://www.zbmath.org/authors/?q=ai:shishlenin.maxim-aSummary: An optimization method for the numerical solution of the inverse problem of recovering the initial state of a supernova is proposed. The gradient of the inverse problem objective functional is constructed. The solution of the direct and adjoint problems is based on a combination of the large particle method, Godunov's method, and piecewise parabolic method on a local stencil. Results of the verification of the proposed numerical method and the results of numerical solution of the Evrard and Sedov collapse problems are presented.Long-term dynamical behavior of the wave model with locally distributed frictional and viscoelastic damping.https://www.zbmath.org/1452.351042021-02-12T15:23:00+00:00"Li, Chan"https://www.zbmath.org/authors/?q=ai:li.chan"Liang, Jin"https://www.zbmath.org/authors/?q=ai:liang.jin"Xiao, Ti-Jun"https://www.zbmath.org/authors/?q=ai:xiao.ti-junSummary: We investigate the long-term dynamical behavior of the partially viscoelastic wave equation subject to a localized frictional damping, given by
\[
u_{tt}-\Delta u+\int_0^tg(t-s)\operatorname{div}(a(x)\nabla u(s))ds+b(x)g_1(u_t)=0,\text{ in }\Omega\times\mathbb{R}^+,
\]
where \(g\) denotes the memory kernel, \(a(x)\in C^1(\overline{\Omega})\), \(b(x)\in L^\infty(\Omega)\) are nonnegative functions satisfying the assumption
\[
a(x)+b(x)\geq 2\delta>0,\quad \forall x\in\omega_0,
\]
\(\omega_0\) is a subset of \(\Omega\), and \(b(x)g_1(u_t)\) denotes the frictional damping. Under as less as possible restrictions imposed on memory kernel \(g(\cdot)\) and some geometric condition on the subset \(\omega_0\), we show that there does not exist bifurcation and chaos for this physical model and actually the energy of the solution for the equation above decays definitely to zero with uniform decay rate as the time goes to infinity. Moreover, such a uniform decay rate is determined by the solution of an ordinary differential equation.A finite element approach for vector- and tensor-valued surface PDEs.https://www.zbmath.org/1452.653522021-02-12T15:23:00+00:00"Nestler, Michael"https://www.zbmath.org/authors/?q=ai:nestler.michael"Nitschke, Ingo"https://www.zbmath.org/authors/?q=ai:nitschke.ingo"Voigt, Axel"https://www.zbmath.org/authors/?q=ai:voigt.axelSummary: We derive a Cartesian componentwise description of the covariant derivative of tangential tensor fields of any degree on Riemannian manifolds. This allows to reformulate any vector- and tensor-valued surface PDE in a form suitable to be solved by established tools for scalar-valued surface PDEs. We consider piecewise linear Lagrange surface finite elements on triangulated surfaces and validate the approach by a vector- and a tensor-valued surface Helmholtz problem on an ellipsoid. We experimentally show optimal (linear) order of convergence for these problems. The full functionality is demonstrated by solving a surface Landau-de Gennes problem on the Stanford bunny. All tools required to apply this approach to other vector- and tensor-valued surface PDEs are provided.Extension and trace for nonlocal operators.https://www.zbmath.org/1452.460232021-02-12T15:23:00+00:00"Bogdan, Krzysztof"https://www.zbmath.org/authors/?q=ai:bogdan.krzysztof"Grzywny, Tomasz"https://www.zbmath.org/authors/?q=ai:grzywny.tomasz"Pietruska-Pałuba, Katarzyna"https://www.zbmath.org/authors/?q=ai:pietruska-paluba.katarzyna"Rutkowski, Artur"https://www.zbmath.org/authors/?q=ai:rutkowski.arturSummary: We prove an optimal extension and trace theorem for Sobolev spaces of nonlocal operators. The extension is given by a suitable Poisson integral and solves the corresponding nonlocal Dirichlet problem. We give a Douglas-type formula for the quadratic form of the Poisson extension.Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography.https://www.zbmath.org/1452.351492021-02-12T15:23:00+00:00"Fjordholm, Ulrik S."https://www.zbmath.org/authors/?q=ai:fjordholm.ulrik-skre"Mishra, Siddhartha"https://www.zbmath.org/authors/?q=ai:mishra.siddhartha"Tadmor, Eitan"https://www.zbmath.org/authors/?q=ai:tadmor.eitanSummary: We consider the shallow water equations with non-flat bottom topography. The smooth solutions of these equations are energy conservative, whereas weak solutions are energy stable. The equations possess interesting steady states of lake at rest as well as moving equilibrium states. We design energy conservative finite volume schemes which preserve (i) the lake at rest steady state in both one and two space dimensions, and (ii) one-dimensional moving equilibrium states. Suitable energy stable numerical diffusion operators, based on energy and equilibrium variables, are designed to preserve these two types of steady states. Several numerical experiments illustrating the robustness of the energy preserving and energy stable well-balanced schemes are presented.Single-point blow-up in the Cauchy problem for the higher-dimensional Keller-Segel system.https://www.zbmath.org/1452.352272021-02-12T15:23:00+00:00"Winkler, Michael"https://www.zbmath.org/authors/?q=ai:winkler.michaelThe main result of the paper for the doubly parabolic Keller-Segel model is that for each reasonably regular nonnegative (but nonzero) radially symmetric initial condition \(u_0\) there exists a sequence of data approximating \(u_0\) such that solutions with those initial conditions blow up in \(L^\infty\) in a finite time (and bounded by, say, \(1\)).
This result can be viewed as a generic property of radially symmetric solutions that blow up at the single point (the origin of the space) emanating from a dense set of initial data.
The analysis involves a modification of the energy functional and the corresponding dissipation of energy functional.
Reviewer: Piotr Biler (Wrocław)Nonlinear perturbations of evolution systems in scales of Banach spaces.https://www.zbmath.org/1452.352202021-02-12T15:23:00+00:00"Friesen, Martin"https://www.zbmath.org/authors/?q=ai:friesen.martin"Kutoviy, Oleksandr"https://www.zbmath.org/authors/?q=ai:kutoviy.oleksandr-vFractional generalization of the Fermi-Pasta-Ulam-Tsingou media and theoretical analysis of an explicit variational scheme.https://www.zbmath.org/1452.652022021-02-12T15:23:00+00:00"Macías-Díaz, J. E."https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardoSummary: In this work, we consider a partial differential equation that extends the well-known Fermi-Pasta-Ulam-Tsingou chains from nonlinear dynamics. The continuous model under consideration includes the presence of both a damping term and a polynomial function in terms of Riesz space-fractional derivatives. Initial and boundary conditions on a closed and bounded interval are considered in this work. The mathematical model has a fractional Hamiltonian which is conserved when the damping coefficient is equal to zero, and dissipated otherwise. Motivated by these facts, we propose a finite-difference method to approximate the solutions of the continuous model. The method is an explicit scheme which is based on the use of fractional centered differences to approximate the fractional derivatives of the model. A discretized form of the Hamiltonian is also proposed in this work, and we prove analytically that the method is capable of conserving or dissipating the discrete energy under the same conditions that guarantee the conservation or dissipation of energy of the continuous model. We show that solutions of the discrete model exist and are unique under suitable regularity conditions on the reaction function. We establish rigorously the properties of consistency, stability and convergence of the method. To that end, novel technical results are mathematically proved. Computer simulations that assess the capability of the method to preserve the energy are provided for illustration purposes.A full coupled drift-diffusion-Poisson simulation of a GFET.https://www.zbmath.org/1452.820362021-02-12T15:23:00+00:00"Nastasi, Giovanni"https://www.zbmath.org/authors/?q=ai:nastasi.giovanni"Romano, Vittorio"https://www.zbmath.org/authors/?q=ai:romano.vittorioUsually, graphene field-effect transistors (GFETs) are investigated by adopting reduced 1D models of the Poisson equation with some averaging procedure. In this paper, a full 2D simulation is presented for a top-gated GFET including a full 2D Poisson equation and a drift-diffusion model with mobilities by a suitable discontinuous Galerkin (DG) method. The mobilities are deduced from the direct solution of the Boltzmann equations for charge transport in graphene by a DG method. An extensive numerical simulation is performed based on the semiclassical Boltzmann equations, including electron-phonon scatterings, electron-impurities scatterings, and scattering with the remote phonon of the substrate, taking into account both intra and inter-band scatterings. The numerical results confirm the main features of such a device for the limited range of gate-source voltages and for the current-offstate, which are noise-free and allow to determine in an accurate way the low field mobility.
Reviewer: Bülent Karasözen (Ankara)Convex envelopes on trees.https://www.zbmath.org/1452.352302021-02-12T15:23:00+00:00"Del Pezzo, Leandro M."https://www.zbmath.org/authors/?q=ai:del-pezzo.leandro-m"Frevenza, Nicolas"https://www.zbmath.org/authors/?q=ai:frevenza.nicolas"Rossi, Julio D."https://www.zbmath.org/authors/?q=ai:rossi.julio-danielSummary: We introduce two notions of convexity for an infinite regular tree. For these two notions we show that given a continuous boundary datum there exists a unique convex envelope on the tree and characterize the equation that this envelope satisfies. We also relate the equation with two versions of the Laplacian on the tree. Moreover, for a function defined on the tree, the convex envelope turns out to be the solution to the obstacle problem for this equation.Rapidly rotating white dwarfs.https://www.zbmath.org/1452.850032021-02-12T15:23:00+00:00"Strauss, Walter A."https://www.zbmath.org/authors/?q=ai:strauss.walter-alexander"Wu, Yilun"https://www.zbmath.org/authors/?q=ai:wu.yilunGround state and sign-changing solutions for fractional Schrödinger-Poisson system with critical growth.https://www.zbmath.org/1452.350282021-02-12T15:23:00+00:00"Ye, Chaoxia"https://www.zbmath.org/authors/?q=ai:ye.chaoxia"Teng, Kaimin"https://www.zbmath.org/authors/?q=ai:teng.kaiminSummary: In this paper, we study the following fractional Schrödinger-Poisson system with critical growth
\[
\begin{aligned}
(-\Delta)^s u+ u + K(x) \phi (x) u &= a(x) |u|^{p-2} u + |u|^{2^*_s-2}u \;\; \text{in} \; \mathbb R^3,\\
(-\Delta)^t \phi &= K(x) u^2 \;\; \text{in } \mathbb R^3,\\
\end{aligned}
\]
where \(p\in (4,2^*_s)\), \(s\in (\frac{3}{4},1)\), \(t\in(0,1)\), \(2s+2t>3 \) and \(2^*_s = \frac{6}{3-2s}\). Under some suitable assumptions on \(K(x)\) and \(a(x)\), the ground state solutions and sign-changing solutions are obtained.Remarks on some dissipative sine-Gordon equations.https://www.zbmath.org/1452.350412021-02-12T15:23:00+00:00"Goubet, O."https://www.zbmath.org/authors/?q=ai:goubet.olivierSummary: In this short note, we discuss some results concerning the long-time behaviour of solutions to dissipative sine-Gordon equation in a bounded domain of \(\mathbb R^n\). We first prove that without forcing term and under some assumptions on the domain then the global attractor reduces to \(\{0\}\). We then rewrite the sine-Gordon equation as a Schrödinger equation with smoothing; this allows us to provide another proof for the so-called asymptotical smoothing effect for this dissipative wave equation.Reduced models and uncertainty quantification.https://www.zbmath.org/1452.652592021-02-12T15:23:00+00:00"Yue, Yao"https://www.zbmath.org/authors/?q=ai:yue.yao"Feng, Lihong"https://www.zbmath.org/authors/?q=ai:feng.lihong"Benner, Peter"https://www.zbmath.org/authors/?q=ai:benner.peter"Pulch, Roland"https://www.zbmath.org/authors/?q=ai:pulch.roland"Schöps, Sebastian"https://www.zbmath.org/authors/?q=ai:schops.sebastianSummary: Uncertainty quantification analyses the variability of system outputs with respect to process variations and thus represents a useful tool for robust design. Often statistics of a dynamical system's outputs are quantities of interest. A sampling of the outputs requires many transient simulations. Due to the complexity of systems in nanoelectronics, methods of model order reduction (MOR) are applied for accelerating the uncertainty quantification. We consider coupled problems or multiphysics systems. We employ parametric MOR techniques to build parameter-dependent reduced-order models, which can be used for fast computations at all parameter samples. Sampling-based techniques like the Latin hypercube method, for example, or quadrature rules yield the parameter values. We apply this approach to an electrothermal coupled system. Furthermore, we illustrate a co-simulation technique with different quadrature grids for the subsystems. Now just some parts of a coupled problem are substituted by parametric MOR, if the others cannot be reduced efficiently. This method is applied to a circuit-electromagnetic coupled system.
For the entire collection see [Zbl 1433.78001].Smoothing estimates for Boltzmann equation with full-range interactions: spatially inhomogeneous case.https://www.zbmath.org/1452.351302021-02-12T15:23:00+00:00"Chen, Yemin"https://www.zbmath.org/authors/?q=ai:chen.yemin"He, Lingbing"https://www.zbmath.org/authors/?q=ai:he.lingbingSummary: In this paper, we study the regularity of the solution to the Boltzmann equation with full-range interactions but for the spatially inhomogeneous case. Under the initial regularity assumption on the solution itself, we show that the solution will become immediately smooth for all the variables as long as the time is far way from zero. Our strategy relies upon the new upper and lower bounds for the collision operator established in [\textit{Y. Chen} and \textit{L. He}, Arch. Ration. Mech. Anal. 201, No. 2, 501--548 (2011; Zbl 1318.76018)], a hypo-elliptic estimate for the transport equation and the element energy method.The effect of fractional calculus on the formation of quantum-mechanical operators.https://www.zbmath.org/1452.810962021-02-12T15:23:00+00:00"Chung, Won Sang"https://www.zbmath.org/authors/?q=ai:chung.won-sang"Zare, Soroush"https://www.zbmath.org/authors/?q=ai:zare.soroush"Hassanabadi, Hassan"https://www.zbmath.org/authors/?q=ai:hassanabadi.hassan"Maghsoodi, Elham"https://www.zbmath.org/authors/?q=ai:maghsoodi.elhamSummary: In this paper, the deformation of the ordinary quantum mechanics is formulated based on the idea of conformable fractional calculus. Some properties of fractional calculus and fractional elementary functions are investigated. The fractional wave equation in \(1 + 1\) dimension and fractional version of the Lorentz transformation are discussed. Finally, the fractional quantum mechanics is formulated; infinite potential well problem, density of states for the ideal gas, and quantum harmonic oscillator problem are discussed.Asymptotic analysis of an advection-diffusion equation involving interacting boundary and internal layers.https://www.zbmath.org/1452.350132021-02-12T15:23:00+00:00"Amirat, Youcef"https://www.zbmath.org/authors/?q=ai:amirat.youcef-ait"Münch, Arnaud"https://www.zbmath.org/authors/?q=ai:munch.arnaudSummary: As \(\varepsilon\) goes to zero, the unique solution of the scalar advection-diffusion equation \(y_t^\varepsilon - \varepsilon y_{x x}^\varepsilon + M y_x^\varepsilon = 0\), \((x,t) \in(0,1)\times (0,T)\) with Dirichlet boundary conditions exhibits a boundary layer of size \(\mathcal{O}(\varepsilon)\) and an internal layer of size \(\mathcal{O}(\sqrt{ \varepsilon})\). If the time \(T\) is large enough, these thin layers, where the solution \(y^{} \epsilon\) displays rapid variations, intersect and interact with each other. Using the method of matched asymptotic expansions, we show how we can construct an explicit approximation \(\tilde{P}^\varepsilon\) of the solution \(y^{} \epsilon\) satisfying \(\| y^\varepsilon - \tilde{P}^\varepsilon \|_{L^\infty ( 0 , T ; L^2 ( 0 , 1 ) )} = \mathcal{O}( \varepsilon^{3 / 2})\) and \(\| y^\varepsilon - \tilde{P}^\varepsilon \|_{L^2 ( 0 , T ; H^1 ( 0 , 1 ) )} = \mathcal{O}(\varepsilon)\) for all \(\epsilon\) small enough.Blow-up for Joseph-Egri equation: theoretical approach and numerical analysis.https://www.zbmath.org/1452.351712021-02-12T15:23:00+00:00"Korpusov, Maxim O."https://www.zbmath.org/authors/?q=ai:korpusov.maksim-olegovich"Lukyanenko, Dmitry V."https://www.zbmath.org/authors/?q=ai:lukyanenko.dmitrii-vitalevich"Panin, Alexander A."https://www.zbmath.org/authors/?q=ai:panin.aleksandr-anatolevichSummary: This work develops the theory of the blow-up phenomena for Joseph-Egri equation. The existence of the nonextendable solution of two initial-boundary value problems (on a segment and a half-line) is demonstrated. Sufficient conditions of the finite-time blow-up of these solutions, as well as the analytical estimates of the blow-up time, are obtained. A numerical method that allows to precise the blow-up moment for specified initial data is proposed.A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms.https://www.zbmath.org/1452.351062021-02-12T15:23:00+00:00"Palmieri, Alessandro"https://www.zbmath.org/authors/?q=ai:palmieri.alessandroSummary: In this note, two blow-up results are proved for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower order terms both in the subcritical case and in the critical case when the damping and the mass terms make both equations in some sense ``wave-like.'' In the proof of the subcritical case, an iteration argument is used. This approach is based on a coupled system of nonlinear ordinary integral inequalities and lower bound estimates for the spatial integral of the nonlinearities. In the critical case, we employ a test function-type method that has been developed recently by Ikeda-Sobajima-Wakasa and relies strongly on a family of certain self-similar solutions of the adjoint linear equation. Therefore, as critical curve in the \(p-q\) plane of the exponents of the power nonlinearities for this weakly coupled system, we conjecture a shift of the critical curve for the corresponding weakly coupled system of semilinear wave equations.Nonnegative solutions to reaction-diffusion system with cross-diffusion and nonstandard growth conditions.https://www.zbmath.org/1452.350742021-02-12T15:23:00+00:00"Arumugam, Gurusamy"https://www.zbmath.org/authors/?q=ai:arumugam.gurusamy"Tyagi, Jagmohan"https://www.zbmath.org/authors/?q=ai:tyagi.jagmohanSummary: We establish the existence of nonnegative weak solutions to nonlinear reaction-diffusion system with cross-diffusion and nonstandard growth conditions subject to the homogeneous Neumann boundary conditions. We assume that the diffusion operators satisfy certain monotonicity condition and nonstandard growth conditions and prove that the existence of weak solutions using Galerkin's approximation technique.Solution blow-up for a fractional in time acoustic wave equation.https://www.zbmath.org/1452.350462021-02-12T15:23:00+00:00"Jleli, Mohamed"https://www.zbmath.org/authors/?q=ai:jleli.mohamed"Kirane, Mokhtar"https://www.zbmath.org/authors/?q=ai:kirane.mokhtar"Samet, Bessem"https://www.zbmath.org/authors/?q=ai:samet.bessemSummary: We consider the fractional in time acoustic wave equation
\[
\frac{ 1}{ c_0^2} \partial_{0 | t}^\alpha u - \Delta u = \frac{ \varepsilon}{ c_0^4 \rho_0} \partial_{0 | t}^\alpha u^2 ,\]
where \(1< \alpha <2\), \(\partial_{0 | t}^\alpha\) is the Caputo fractional derivative of order \(\alpha\), \(u=u(t,x)\), \(t>0\), \(x \in \mathbb{R}^3\), is the pressure in the medium, \( \varepsilon\) is the nonlinear acoustic parameter, \( \rho_0\) is the equilibrium density in the medium, and \(c_0\) is the equilibrium sound velocity. We study a Cauchy problem for this equation and a mixed boundary value problem in a bounded domain. For each problem, sufficient conditions for the blow-up of solutions are derived. Moreover, we provide a class of initial data for which there are no classical solutions even locally in time. Our approach is based on the nonlinear capacity method.Eigenvalues of magnetohydrodynamic mean-field dynamo models: bounds and reliable computation.https://www.zbmath.org/1452.352142021-02-12T15:23:00+00:00"Boegli, Sabine"https://www.zbmath.org/authors/?q=ai:bogli.sabine"Tretter, Christiane"https://www.zbmath.org/authors/?q=ai:tretter.christianeThe authors analyze the linear stability of three magnetohydrodynamic (MHD)
mean-field dynamo models used in astrophysics. After some computations,
they end with the problem written in spherical coordinates as: \[\partial
_{t}\left(
\begin{array}{c}
S \\
T
\end{array}
\right) =\left(
\begin{array}{cc}
\Delta & \alpha \\
-\alpha \Delta -\frac{1}{r}\alpha ^{\prime }\partial _{r}r+\omega ^{\prime
}\sin \theta \partial _{\theta } & \Delta
\end{array}
\right) \left(
\begin{array}{c}
S \\
T
\end{array}
\right) -\omega \partial _{\varphi }\left(
\begin{array}{c}
S \\
T
\end{array}
\right) ,\] for scalar-valued functions \(S\) and \(T\) which satisfy the
normalizations \(\int_{0}^{\pi }\int_{0}^{2\pi }S(r,\theta ,\varphi ;t)\sin
\theta d\varphi d\theta =0=\int_{0}^{\pi }\int_{0}^{2\pi }T(r,\theta
,\varphi ;t)\sin \theta d\varphi d\theta \) for every \(r\in \lbrack 0,1]\).
Then the authors expand the functions \(S\) and \(T\) in spherical harmonics \(
Y_{l}^{m}\): \(S(r,\theta ,\varphi ;t)=\sum_{m=-\infty }^{\infty
}\sum_{l=\left\vert m\right\vert }^{\infty }\frac{x_{l,m}(r)}{r}
Y_{l}^{m}(\theta ,\varphi )e^{\lambda _{l,m}t}\) and a similar expression for
\(T\) with coefficients \(y_{l,m}(r)\) and they prove that \(\lambda _{l,m}\) are
independent of \(l\) and that \(y_{l,m}(r)\) and \(x_{l,m}(r)\) satisfy an
infinite and coupled system of ordinary equations with coefficients \(\alpha\),
\(\alpha ^{\prime }\) and \(\omega ^{\prime }\). They consider three different
dynamo models: the \(\alpha ^{2}\)-model, in which the \(\alpha \)-effect is
assumed to dominate and the term with \(\omega ^{\prime }\) is neglected, the \(
\alpha \omega \)-model, in which the \(\omega \)-effect is assumed to dominate
and the terms with \(\alpha \) and \(\alpha ^{\prime }\) are neglected, and the \(
\alpha ^{2}\omega \)-model, for which both effects are kept and no term is
neglected. In each case, the authors prove antidynamo theorems, as they
derive thresholds for the helical turbulence function \(\alpha \) and the
rotational shear function \(\omega \) below which no MHD dynamo action can
occur for the linear models. They establish upper bounds for the real part
of the spectrum and they prove resolvent estimates. They use interval
truncation and finite section methods to regularize the singular
differential expressions and the infinite number of coupled equations, and
they prove that these methods are spectrally exact. The paper ends with the
presentation of numerical examples.
Reviewer: Alain Brillard (Riedisheim)Numerical treatment of non-linear fourth-order distributed fractional sub-diffusion equation with time-delay.https://www.zbmath.org/1452.651722021-02-12T15:23:00+00:00"Nandal, Sarita"https://www.zbmath.org/authors/?q=ai:nandal.sarita"Narain Pandey, Dwijendra"https://www.zbmath.org/authors/?q=ai:pandey.dwijendra-narainSummary: In this paper, we proposed a linearized second-order numerical technique for non-linear fourth-order distributed fractional sub-diffusion equation with time delay. Time fractional derivative is represented using Caputo derivative and further approximated using \(L2-1_\sigma\) formula which gives second-order temporal convergence and compact difference operator is employed for spatial dimensions. The proposed method is unconditionally stable and convergent to the analytical solution with the order of convergence \(O(\tau^2+h^4+(\Delta\alpha)^4)\) improvement in earlier work. Non-linear terms are linearized with the help of Taylor's series. At the last, we provided a few examples to show the efficiency of the compact difference scheme to support the theoretical results and also presented comparison with \(L1\)-approximation of Caputo fractional derivative.The Sobolev stability threshold of 2D hyperviscosity equations for shear flows near Couette flow.https://www.zbmath.org/1452.760682021-02-12T15:23:00+00:00"Luo, Xiang"https://www.zbmath.org/authors/?q=ai:luo.xiangSummary: We consider the 2D hyperviscosity equations on \(\mathbb{T} \times \mathbb{R} \). We show that if the initial data of 2D hyperviscosity equations are \(\epsilon \)-close to the shear flows \((U(y),0)\), which are sufficiently small perturbations of Couette flow \((y,0)\), then the solution will stay \(\epsilon \)-close to \(( e^{- \nu t \partial_y^4} U(y), 0)\) for all \(t>0\), where \(\epsilon \ll \nu^{\frac{ 1}{ 2}}\) and \(\nu\) denotes the kinematic viscosity coefficient. What is more, by the mixing-enhanced effect, the solutions converge to decaying shear flows for \(t \gg \nu^{- \frac{ 1}{ 5}} \), which is faster than the heat-equation timescale. Hence, we conclude that the stability threshold of 2D hyperviscosity equations with initial data \((U(y),0)\) is not worse than \(\nu^{\frac{ 1}{ 2}} \).The finite element method for fractional diffusion with spectral fractional Laplacian.https://www.zbmath.org/1452.352402021-02-12T15:23:00+00:00"Hu, Ye"https://www.zbmath.org/authors/?q=ai:hu.ye"Cheng, Fang"https://www.zbmath.org/authors/?q=ai:cheng.fangSummary: This paper focuses on the finite element method for Caputo-type parabolic equation with spectral fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0,1) and the spatial derivative is the spectral fractional Laplacian. The time discretization is based on the Hadamard finite-part integral (or the finite-part integral in the sense of Hadamard), where the piecewise linear interpolation polynomials are used. The spatial fractional Laplacian is lifted to the local spacial derivative by using the Caffarelli-Silvestre extension, where the finite element method is used. Full-discretization scheme is constructed. The convergence and error estimates are obtained. Finally, numerical experiments are presented which support the theoretical results.Subsonic flows in a multi-dimensional nozzle.https://www.zbmath.org/1452.762002021-02-12T15:23:00+00:00"Du, Lili"https://www.zbmath.org/authors/?q=ai:du.lili"Xin, Zhouping"https://www.zbmath.org/authors/?q=ai:xin.zhouping"Yan, Wei"https://www.zbmath.org/authors/?q=ai:yan.weiSummary: In this paper, we study the global subsonic irrotational flows in a multi-dimensional \((n \geqq 2)\) infinitely long nozzle with variable cross sections. The flow is described by the inviscid potential equation, which is a second order quasilinear elliptic equation when the flow is subsonic. First, we prove the existence of the global uniformly subsonic flow in a general infinitely long nozzle for arbitrary dimension with sufficiently small incoming mass flux and obtain the uniqueness of the global uniformly subsonic flow. Then we show that there exists a critical value of the incoming mass flux such that a global uniformly subsonic flow exists uniquely, provided that the incoming mass flux is less than the critical value. This gives a positive answer to the problem of Bers on global subsonic irrotational flows in infinitely long nozzles for arbitrary dimension [\textit{L. Bers}, Mathematical aspects of subsonic and transonic gas dynamics. New York: John Wiley \& Sons, Inc (1958; Zbl 0083.20501)]. Finally, under suitable asymptotic assumptions of the nozzle, we obtain the asymptotic behavior of the subsonic flow in far fields by means of a blow-up argument. The main ingredients of our analysis are methods of calculus of variations, the Moser iteration techniques for the potential equation and a blow-up argument for infinitely long nozzles.Global existence and decay of solutions of a singular nonlocal viscoelastic system with a nonlinear source term, nonlocal boundary condition, and localized damping term.https://www.zbmath.org/1452.350942021-02-12T15:23:00+00:00"Boulaaras, Salah"https://www.zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud"Mezouar, Nadia"https://www.zbmath.org/authors/?q=ai:mezouar.nadiaSummary: The paper deals with the existence of a global solution of a singular one-dimensional viscoelastic system with a nonlinear source term, nonlocal boundary condition, and localized frictional damping \(a(x)u_t\) using the potential well theory. Furthermore, the general decay result is proved. We construct a suitable Lyapunov functional and make use of the perturbed energy method.An overdetermined problem for a class of anisotropic equations in a cylindrical domain.https://www.zbmath.org/1452.351182021-02-12T15:23:00+00:00"Barbu, Luminiţa"https://www.zbmath.org/authors/?q=ai:barbu.luminita"Nicolescu, Adrian Eracle"https://www.zbmath.org/authors/?q=ai:nicolescu.adrian-eracleSummary: In this paper, we are going to investigate an overdetermined problem for a general class of anisotropic equations on a cylindrical domain \(\Omega \subset \mathbb{R}^N, N \geq 2\). Our aim is to show that if the overdetermined problem admits a solution in a suitable weak sense, then the underlying domain \(\Omega\) and the corresponding solution \(u\) satisfy some symmetry properties. This result represents the anisotropic extension to one of the main results obtained by LE Payne and GA Philippin in their previous paper.Shock formation in the compressible Euler equations and related systems.https://www.zbmath.org/1452.351372021-02-12T15:23:00+00:00"Chen, Geng"https://www.zbmath.org/authors/?q=ai:chen.geng"Young, Robin"https://www.zbmath.org/authors/?q=ai:young.robin-l"Zhang, Qingtian"https://www.zbmath.org/authors/?q=ai:zhang.qingtianA Beale-Kato-Majda criterion for three-dimensional compressible viscous heat-conductive flows.https://www.zbmath.org/1452.762042021-02-12T15:23:00+00:00"Sun, Yongzhong"https://www.zbmath.org/authors/?q=ai:sun.yongzhong"Wang, Chao"https://www.zbmath.org/authors/?q=ai:wang.chao.3|wang.chao.1|wang.chao.2"Zhang, Zhifei"https://www.zbmath.org/authors/?q=ai:zhang.zhifeiSummary: We prove a blow-up criterion in terms of the upper bound of \((\rho , \rho ^{-1}, \theta )\) for a strong solution to three dimensional compressible viscous heat-conductive flows. The main ingredient of the proof is an a priori estimate for a quantity independently introduced in [\textit{B. Haspot}, ``Regularity of weak solutions of the compressible isentropic Navier-Stokes equation, Preprint, \url{arXiv:1001.1581}] and [\textit{Y. Sun} et al., J. Math. Pures Appl. (9) 95, No. 1, 36--47 (2011; Zbl 1205.35212), whose divergence can be viewed as the effective viscous flux.An analysis for heat equations arises in diffusion process using new Yang-Abdel-Aty-Cattani fractional operator.https://www.zbmath.org/1452.352422021-02-12T15:23:00+00:00"Kumar, Sunil"https://www.zbmath.org/authors/?q=ai:kumar.sunil"Ghosh, Surath"https://www.zbmath.org/authors/?q=ai:ghosh.surath"Samet, Bessem"https://www.zbmath.org/authors/?q=ai:samet.bessem"Goufo, Emile Franc Doungmo"https://www.zbmath.org/authors/?q=ai:doungmo-goufo.emile-francSummary: The heat equation is parabolic partial differential equation and occurs in the characterization of diffusion progress. In the present work, a new fractional operator based on the Rabotnov fractional-exponential kernel is considered. Next, we conferred some fascinating and original properties of nominated new fractional derivative with some integral transform operators where all results are significant. The fundamental target of the proposed work is to solve the multidimensional heat equations of arbitrary order by using analytical approach homotopy perturbation transform method and residual power series method, where new fractional operator has been taken in new Yang-Abdel-Aty-Cattani (YAC) sense. The obtained results indicate that solution converges to the original solution in language of generalized Mittag-Leffler function. Three numerical examples are discussed to draw an effective attention to reveal the proficiency and adaptability of the recommended methods on new YAC operator.On the determination of ischemic regions in the monodomain model of cardiac electrophysiology from boundary measurements.https://www.zbmath.org/1452.352532021-02-12T15:23:00+00:00"Beretta, Elena"https://www.zbmath.org/authors/?q=ai:beretta.elena"Cavaterra, Cecilia"https://www.zbmath.org/authors/?q=ai:cavaterra.cecilia"Ratti, Luca"https://www.zbmath.org/authors/?q=ai:ratti.lucaBoundary stabilization of a microbeam model.https://www.zbmath.org/1452.350362021-02-12T15:23:00+00:00"Guzmán, Patricio"https://www.zbmath.org/authors/?q=ai:guzman.patricioSummary: In this paper, we study the boundary stabilization of the deflection of a clamped-free microbeam, which is modeled by a sixth-order hyperbolic equation. We design a boundary feedback control, simpler than the one designed in Vatankhah et al, that forces the energy associated to the deflection to decay exponentially to zero as the time goes to infinity. The rate in which the energy exponentially decays is explicitly given.On the method of preliminary group classification applied to the nonlinear heat equation \(u_t = f(x, u_x) u_{x x} + g(x, u_x)\).https://www.zbmath.org/1452.350102021-02-12T15:23:00+00:00"Edelstein, Rochelle M."https://www.zbmath.org/authors/?q=ai:edelstein.rochelle-m"Govinder, Keshlan S."https://www.zbmath.org/authors/?q=ai:govinder.keshlan-sathasivaSummary: We apply the method of preliminary group classification to a specific class of the nonlinear heat equation, ie, \( u_t = f(x, u_x) u_{x x} + g(x, u_x)\). This results in an optimal system of one-dimensional subalgebras, which will be used to find specific forms of the equation, which admit more symmetries and so allow us to find additional group invariant solutions. We extend this work by determining the potential symmetries of a closely related system, which gives rise to many new, additional solutions of the original equation.Weak vorticity formulation of 2D Euler equations with white noise initial condition.https://www.zbmath.org/1452.352692021-02-12T15:23:00+00:00"Flandoli, Franco"https://www.zbmath.org/authors/?q=ai:flandoli.francoThe enstrophy is an important invariant for the 2D Euler equations in vorticity form, giving rise to the enstrophy measure \(\mu\) (also called white noise measure) which is an invariant measure for 2D Euler dynamics. This measure is quite singular and is supported by Sobolev spaces of order lower than \(-1\). It is of great interest to study solutions to 2D Euler equations with stationary measure \(\mu\) (called white noise solutions) or, more generally, the solutions whose time-marginal laws are absolutely continuous w.r.t. \(\mu\). Using the method of Galerkin approximations, the existence of white noise solutions have been proven by \textit{S. Albeverio} and \textit{A. B. Cruzeiro} [Commun. Math. Phys. 129, No. 3, 431--444 (1990; Zbl 0702.76041)].
In the current paper, the author provides an alternative construction of white noise solutions by considering the point vortex system of 2D Euler equations on the torus with random initial conditions and random vortex intensities. It is shown that, as the number of point vortices goes to infinity, the associated random vorticities have a subsequence converging weakly to a limit, solving the weak vorticity formulation of 2D Euler equations in the sense of J.-M. Delort and S. Schochet. Moreover, if the law of the initial random vorticity \(\omega_0\) has a bounded continuous density \(\rho_0\) w.r.t. \(\mu\), then the author proves that there exists a stochastic process \(\{\omega_t\}_{t\in [0,T]}\) such that: (1) \(law(\omega_t)\ll \mu\) for any \(t\in [0,T]\), with a density \(\rho_t\) bounded by \(\|\rho_0\|_\infty\) and satisfing an infinite dimensional continuity equation; (2) \(\omega_\cdot\) solves the weak vorticity formulation of 2D Euler equation. The paper also contains many discussions of related open problems and finishes with a justification of the point vortex model on 2D torus.
Reviewer: Dejun Luo (Beijing)Well-posedness of the nonlinear Schrödinger equation on the half-plane.https://www.zbmath.org/1452.351882021-02-12T15:23:00+00:00"Himonas, A. Alexandrou"https://www.zbmath.org/authors/?q=ai:himonas.a-alexandrou"Mantzavinos, Dionyssios"https://www.zbmath.org/authors/?q=ai:mantzavinos.dionyssiosGlobal relaxation and nonrelativistic limit of nonisentropic Euler-Maxwell systems.https://www.zbmath.org/1452.350322021-02-12T15:23:00+00:00"Chao, Na"https://www.zbmath.org/authors/?q=ai:chao.na"Yang, Yongfu"https://www.zbmath.org/authors/?q=ai:yang.yongfuSummary: The aim of this paper is to investigate smooth solutions to Cauchy (or periodic) problem for a nonisentropic Euler-Maxwell system with small parameters. For initial data close to constant equilibrium states, we prove the global-in-time convergence of the Euler-Maxwell system as parameters go to zero. The limit systems are the drift-diffusion system and the nonisentropic Euler-Poisson system, respectively.A course on tug-of-war games with random noise. Introduction and basic constructions.https://www.zbmath.org/1452.910022021-02-12T15:23:00+00:00"Lewicka, Marta"https://www.zbmath.org/authors/?q=ai:lewicka.marta-wang-dehuaThis is a very technical book, addressed to specialists, and to be used, in my opinion, in some graduate or post graduate course at the universitary level. The aim of the book is to present a systematic overview of the basic constructions and results pertaining to the recently emerged field of tug-of-war games, as seen from an analyst's perspective. To a large extent, this book represents the author's own study itinerary, aiming at precision and completeness of a classroom text.
Just in the first page of the Introduction one encounters references to physics, partial differential equations, stochastic processes, etc.
After a linear and a nonlinear motivation, an idea or definition of what a tug-of-war game with random noise is appears on p. 5--6 of the Introduction. It is indeed very technical, but it describes the subject matter to be dealt with throughout the manuscript. As a matter of fact, the content of the book is also described on pp. 7--8 of the Introduction.
On p. 9, the prerrequisites are also commented. The presentation of the book aims to be self-contained. The reader should be familiar to differential and integral calculus in several variables, measure theory, probability and partial differential equations.
The main Chapters of the book analyze first the linear case (Chapter 2) and then different issues of tug-of-war with noise (Chapters 3 to 6).
It is noticeable that after these chapters several appendices (A-B-C) have been added to cover necessary background. In my opinion, perhaps these appendices (probability, Brownian motion and partial differential equations) should have appeared at the beginning of the book as ``preliminaries'', since otherwise a non-specialist reader (but interested, even tangentially, on the topics analyzed in the book) can be lost from the very beginning, and forced to go again and again to these appendices.
A lot of complementary exercises have been posed throughout the book, in each chapter. A final appendix D provides the reader with the solutions to some selected exercises.
Reviewer: Esteban Induraín (Pamplona)A complete Lie symmetry classification of a class of (1+2)-dimensional reaction-diffusion-convection equations.https://www.zbmath.org/1452.352112021-02-12T15:23:00+00:00"Cherniha, Roman"https://www.zbmath.org/authors/?q=ai:cherniha.roman-m"Serov, Mykola"https://www.zbmath.org/authors/?q=ai:serov.mykola-i"Prystavka, Yulia"https://www.zbmath.org/authors/?q=ai:prystavka.yuliaSummary: A class of nonlinear reaction-diffusion-convection equations describing various processes in physics, biology, chemistry etc. is under study in the case of time and two space variables. The group of equivalence transformations is constructed, which is applied for deriving a Lie symmetry classification for the class of such equations by the well-known algorithm. It is proved that the algorithm leads to 32 reaction-diffusion-convection equations admitting nontrivial Lie symmetries. Furthermore a set of form-preserving transformations for this class is constructed in order to reduce this number of the equations and obtain a complete Lie symmetry classification. As a result, the so called canonical list of all inequivalent equations admitting nontrivial Lie symmetry (up to any point transformations) and their Lie symmetries are derived. The list consists of 22 equations and it is shown that any other reaction-diffusion-convection equation admitting a nontrivial Lie symmetry is reducible to one of these 22 equations. As a nontrivial example, the symmetries derived are applied for the reduction and finding exact solutions in the case of the porous-Fisher type equation with the Burgers term.Pinning boundary conditions for phase-field models.https://www.zbmath.org/1452.651632021-02-12T15:23:00+00:00"Lee, Hyun Geun"https://www.zbmath.org/authors/?q=ai:lee.hyun-geun"Yang, Junxiang"https://www.zbmath.org/authors/?q=ai:yang.junxiang"Kim, Junseok"https://www.zbmath.org/authors/?q=ai:kim.junseokSummary: In this paper, we present pinning boundary conditions for two- (2D) and three-dimensional (3D) phase-field models. For the 2D and axisymmetric domains in the neighborhood of the pinning boundaries, we apply an odd-function-type treatment and use a local gradient of the phase-field for points away from the pinning boundaries. For the 3D domain, we propose a simple treatment that fixes the values on the ghost grid points beyond the discrete computational domain. As examples of the phase-field models, we consider the Allen-Cahn and conservative Allen-Cahn equations with the pinning boundary conditions. We present various numerical experiments to demonstrate the performance of the proposed pinning boundary treatment. The computational results confirm the efficiency of the proposed method.A BKM's criterion of smooth solution to the incompressible viscoelastic flow.https://www.zbmath.org/1452.760162021-02-12T15:23:00+00:00"Qiu, Hua"https://www.zbmath.org/authors/?q=ai:qiu.hua"Fang, Shaomei"https://www.zbmath.org/authors/?q=ai:fang.shaomeiSummary: In this paper, we study the regularity criterion of smooth solution to the Oldroyd model in \(\mathbb R^n\) \((n = 2,3)\). We obtain a Beale-Kato-Majda-type criterion in terms of vorticity in two and three space dimensions, namely, the solution \((u(t, x), F(t, x))\) does not develop singularity until \(t = T\) provided that \(\nabla \times u \in L^1(0, T; \dot{B}_{\infty,\infty}^0(\mathbb R^n))\) in the case \(n = 2, 3\).Discrete Darboux transformation for Ablowitz-Ladik systems derived from numerical discretization of Zakharov-Shabat scattering problem.https://www.zbmath.org/1452.651792021-02-12T15:23:00+00:00"Vaibhav, Vishal"https://www.zbmath.org/authors/?q=ai:vaibhav.vishalSummary: The numerical discretization of the Zakharov-Shabat Scattering problem using integrators based on the implicit Euler method, trapezoidal rule and the split-Magnus method yield discrete systems that qualify as Ablowitz-Ladik systems. These discrete systems are important on account of their layer-peeling property which facilitates the differential approach of inverse scattering. In this paper, we study the Darboux transformation at the discrete level by following a recipe that closely resembles the Darboux transformation in the continuous case. The viability of this transformation for the computation of multisoliton potentials is investigated and it is found that irrespective of the order of convergence of the underlying discrete framework, the numerical scheme thus obtained is of first order with respect to the step size.On the stochastic nonlinear Schrödinger equations at critical regularities.https://www.zbmath.org/1452.351922021-02-12T15:23:00+00:00"Oh, Tadahiro"https://www.zbmath.org/authors/?q=ai:oh.tadahiro"Okamoto, Mamoru"https://www.zbmath.org/authors/?q=ai:okamoto.mamoruSummary: We consider the Cauchy problem for the defocusing stochastic nonlinear Schrödinger equations (SNLS) with an additive noise in the mass-critical and energy-critical settings. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by \textit{Á. Bényi} et al. [Trans. Am. Math. Soc., Ser. B 2, 1--50 (2015; Zbl 1339.35281)] to the current stochastic PDE setting, we present a concise argument to establish global well-posedness of the mass-critical and energy-critical SNLS.Three-dimensional finite element model to study calcium distribution in astrocytes in presence of VGCC and excess buffer.https://www.zbmath.org/1452.652352021-02-12T15:23:00+00:00"Jha, Brajesh Kumar"https://www.zbmath.org/authors/?q=ai:jha.brajesh-kumar"Jha, Amrita"https://www.zbmath.org/authors/?q=ai:jha.amrita"Adlakha, Neeru"https://www.zbmath.org/authors/?q=ai:adlakha.neeruSummary: The role of astrocytes in physiological processes is always a matter of interest for biologists, mathematicians and computer scientists. Similar to neurons, astrocytes propagate \(\text{Ca}^{2+}\) over long distances in response to stimulation and release gliotransmitters \(\text{Ca}^{2+}\)-dependent manner to modulate various important brain functions. There are various processes and parameters that affect the cytoplasmic calcium concentration level of astrocytes like calcium buffering, influx via calcium channels, etc. Buffers bind with calcium ion \((\text{Ca}^{2+})\) and makes calcium bound buffers. Thus, it decreases the calcium concentration \([\text{Ca}^{2+}]\) level. \(\text{Ca}^{2+}\) enters into the cytosol through voltage gated calcium channel (VGCC) and thus it increases the concentration level. In view of above, a three-dimensional mathematical model is developed for combined study of the effect of buffer and VGCC on cytosolic calcium concentration in astrocytes. Finite element method is applied to find the solution using hexagonal elements. A computer programme is developed for entire problem to simulate the results. The obtained results show that high affinity buffer reveals the effect of VGCC and at low buffer concentration VGCC effects more significantly.The initial value problem for the quasi-linear partial integro-differential equation of higher order with a degenerate kernel.https://www.zbmath.org/1452.352312021-02-12T15:23:00+00:00"Yuldashev, Tursun Kamaldinovich"https://www.zbmath.org/authors/?q=ai:yuldashev.tursun-kamaldinovichSummary: High-order partial differential equations are of great interest when it comes to physical applications. Many problems of gas dynamics, elasticity theory and the theory of plates and shells are reduced to the consideration of high-order partial differential equations. This paper studies the one-valued solvability of the initial value problem for a nonlinear partial integro-differential equation of an arbitrary order with a degenerate kernel. The expression of higher-order partial differential equations as a superposition of first-order partial differential operators has allowed us to apply methods for solving first-order partial differential equations. First-order partial differential equations can be locally solved by the methods of the theory of ordinary differential equations, reducing them to a characteristic system. The existence and uniqueness of the solution to this problem is proved by the method of successive approximation. An estimate of convergence of the iterative Picard process is obtained. The stability of the solution from the second argument of the initial value problem is shown.Stability of linear multistep time iterations with the WENO5 discretization at discontinuities.https://www.zbmath.org/1452.652002021-02-12T15:23:00+00:00"Zhang, Jianying"https://www.zbmath.org/authors/?q=ai:zhang.jianyingSummary: The linear stability analysis on the WENO5 spatial discretization for solving the one-dimensional linear advection equation, combined with various fifth-order multistep methods, was presented in [\textit{M. Motamed} et al., J. Sci. Comput. 47, No. 2, 127--149 (2011; Zbl 1217.65180)]. The purpose of this work is to further investigate the mechanism of oscillations observed in these time integrators when simulating shock front propagation. In particular, extrapolated backward differentiation formula (eBDF5), explicit Adams methed (Adams5) and a predictor-corrector method (PC5) are selected for detailed performance comparison. We first analyze how the non-convex combinations involved in these multistep methods restrict the time step-size and lead to possible pointwise oscillations. Subsequently, the nonlinear weights in the WENO5 scheme are used as indicators to capture the evolution of discontinuities with time and determine the stability of the multistep methods. Numerical results are also provided to confirm the analysis and review the qualifications of these multistep methods for shock front tracking.Stability analysis of interface conditions for ocean-atmosphere coupling.https://www.zbmath.org/1452.651832021-02-12T15:23:00+00:00"Zhang, Hong"https://www.zbmath.org/authors/?q=ai:zhang.hong.2|zhang.hong.4|zhang.hong.1|zhang.hong.3|zhang.hong.5|zhang.hong"Liu, Zhengyu"https://www.zbmath.org/authors/?q=ai:liu.zhengyu"Constantinescu, Emil"https://www.zbmath.org/authors/?q=ai:constantinescu.emil-m"Jacob, Robert"https://www.zbmath.org/authors/?q=ai:jacob.robert-j-k|jacob.robert-lSummary: In this paper we analyze the stability of different coupling strategies for multidomain PDEs that arise in general circulation models used in climate simulations. We focus on fully coupled ocean-atmosphere models that are needed to represent and understand the complicated interactions of these two systems, becoming increasingly important in climate change assessment in recent years. Numerical stability issues typically arise because of different time-stepping strategies applied to the coupled PDE system. In particular, the contributing factors include using large time steps, lack of accurate interface flux, and single-iteration coupling. We investigate the stability of the coupled ocean-atmosphere models for various interface conditions such as the Dirichlet-Neumann condition and the bulk interface condition, which is unique to climate modeling. By analyzing a simplified model, we demonstrate how the parameterization of the bulk condition and other numerical and physical parameters affect the coupling stability and establish stability conditions for different coupling strategies.Method of Green's potentials for elliptic PDEs in domains with random apertures.https://www.zbmath.org/1452.653902021-02-12T15:23:00+00:00"Reshniak, Viktor"https://www.zbmath.org/authors/?q=ai:reshniak.viktor"Melnikov, Yuri"https://www.zbmath.org/authors/?q=ai:melnikov.yuri-aSummary: Problems with topological uncertainties appear in many fields ranging from nano-device engineering to the design of bridges. In many of such problems, a part of the domains boundaries is subjected to random perturbations making inefficient conventional schemes that rely on discretization of the whole domain. In this paper, we study elliptic PDEs in domains with boundaries comprised of a deterministic part and random apertures, and apply the method of modified potentials with Green's kernels defined on the deterministic part of the domain. This approach allows to reduce the dimension of the original differential problem by reformulating it as a boundary integral equation posed on the random apertures only. The multilevel Monte Carlo method is then applied to this modified integral equation and its optimal \(\epsilon^{-2}\) asymptotical complexity is shown. Finally, we provide the qualitative analysis of the proposed technique and support it with numerical results.Analysis of finite element approximations of Stokes equations with nonsmooth data.https://www.zbmath.org/1452.653342021-02-12T15:23:00+00:00"Durán, Ricardo"https://www.zbmath.org/authors/?q=ai:duran.ricardo-g"Gastaldi, Lucia"https://www.zbmath.org/authors/?q=ai:gastaldi.lucia"Lombardi, Ariel"https://www.zbmath.org/authors/?q=ai:lombardi.ariel-luisFinite element approximations of the Stokes equations with nonsmooth Dirichlet boundary data are analyzed. The discrete solution is defined by approximating the boundary datum by a smooth one and then a standard finite element method is applied to the regularized problem. Almost optimal order error estimates are proved for two regularizations in the
case of general data in fractional order Sobolev spaces and for the Lagrange interpolation for piecewise smooth data.
The finite element approximation is defined considering the boundary velocity data \(g \in L^2(\Gamma )\) and using some regularization of \(g\). In this way, the a priori error analysis is separated in two parts: the error due to the regularization and that due to the discretization. The first error, in general, is analyzed assuming a given approximation of \(g\) and considering some particular regularizations. For piecewise smooth boundary data, as in the case of the lid-driven cavity problem, an approximation to \(g\) its Lagrange interpolation is used at continuity points with some appropriate definition at the discontinuities. Furthermore, the authors analyze a posteriori error estimator of residual type that is equivalent to the appropriate norms of the error. Numerical examples show that an adaptive procedure based estimator produces optimal order error estimates for the lid-driven cavity problem. Some numerical examples are presented for the lid-driven cavity problem using two stable finite element methods: the Mini element and the Hood-Taylor element.
Reviewer: Bülent Karasözen (Ankara)A Beale-Kato-Majda criterion with optimal frequency and temporal localization.https://www.zbmath.org/1452.760502021-02-12T15:23:00+00:00"Luo, Xiaoyutao"https://www.zbmath.org/authors/?q=ai:luo.xiaoyutaoSummary: We obtain a Beale-Kato-Majda-type criterion with optimal frequency and temporal localization for the 3D Navier-Stokes equations. Compared to previous results our condition only requires the control of Fourier modes below a critical frequency, whose value is explicit in terms of time scales. As applications it yields a strongly frequency-localized condition for regularity in the space \(B^{-1}_{\infty ,\infty }\) and also a lower bound on the decaying rate of \(L^p\) norms \(2\le p <3\) for possible blowup solutions. The proof relies on new estimates for the cutoff dissipation and energy at small time scales which might be of independent interest.Spectral asymptotics of Laplacians related to one-dimensional graph-directed self-similar measures with overlaps.https://www.zbmath.org/1452.351252021-02-12T15:23:00+00:00"Ngai, Sze-Man"https://www.zbmath.org/authors/?q=ai:ngai.sze-man"Xie, Yuanyuan"https://www.zbmath.org/authors/?q=ai:xie.yuanyuanSummary: For the class of graph-directed self-similar measures on \(\mathbb{R} \), which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the spectral dimension of the Laplacians defined by these measures. For the class of finitely ramified graph-directed self-similar sets, the spectral dimension of the associated Laplace operators has been obtained by \textit{B. M. Hambly} and \textit{S. O. G. Nyberg} [Proc. Edinb. Math. Soc., II. Ser. 46, No. 1, 1--34 (2003; Zbl 1038.35046)]. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition.High-order time stepping schemes for semilinear subdiffusion equations.https://www.zbmath.org/1452.652522021-02-12T15:23:00+00:00"Wang, Kai"https://www.zbmath.org/authors/?q=ai:wang.kai.4|wang.kai.1|wang.kai|wang.kai.2|wang.kai.3"Zhou, Zhi"https://www.zbmath.org/authors/?q=ai:zhou.zhiHigh-order \(k\)-step backward differentiation formulae (BDFk) are developed for solving the initial-boundary value problem for the semilinear subdiffusion equation. The convolution quadrature is applied generated by BDFk to discretize the time-fractional derivative of order \(\alpha\in (0, 1)\) and the starting steps are modified in order to achieve an optimal convergence rate. The main result of the paper is to derive a convergence order \(\mathcal{O}(\Delta t^{\min (1,1 + 2\alpha -\epsilon)})\) for the corrected BDFk scheme without imposing further assumptions on the regularity of the solution. The optimal order convergence is achieved by splitting the nonlinear potential term into an irregular linear part and a smoother nonlinear part and using the generating function technique. Numerical examples are provided for the time-fractional Allen-Cahn equation to support the theoretical results.
Reviewer: Bülent Karasözen (Ankara)New approximations for solving the Caputo-type fractional partial differential equations.https://www.zbmath.org/1452.651762021-02-12T15:23:00+00:00"Ren, Jincheng"https://www.zbmath.org/authors/?q=ai:ren.jincheng"Sun, Zhi-zhong"https://www.zbmath.org/authors/?q=ai:sun.zhizhong"Dai, Weizhong"https://www.zbmath.org/authors/?q=ai:dai.weizhongSummary: Partial differential equations with the Caputo-type fractional derivative have been used in many engineering applications. Because the Caputo-type fractional derivative is an integral of the solution with respect to time, the numerical scheme for solving this type of fractional differential equations requires using the values of all previous time steps. This needs a large size of memory to store the necessary data when computing, which may lead to a memory problem in computer, particularly when solving systems of multi-dimensional fractional differential equations. For this purpose, this article presents a new approximation for solving the fractional differential equations using the Laplace transform method. The obtained differential equations will then be solved using some conventional two or three-level in time finite difference schemes, which reduce the computational cost significantly. For simplicity, the method is presented in 1D and is illustrated by several 1D and 2D examples. In practical, this method can be readily used for solving more complex cases within a reasonable accuracy.A novel least squares method for Helmholtz equations with large wave numbers.https://www.zbmath.org/1452.653422021-02-12T15:23:00+00:00"Hu, Qiya"https://www.zbmath.org/authors/?q=ai:hu.qiya"Song, Rongrong"https://www.zbmath.org/authors/?q=ai:song.rongrongThe authors are concerned with a Robin-type boundary value problem for Helmholtz equations on a 2D Lipschitz bounded domain. They are mainly concerned with the situation of large wave numbers and design a least squares method in order to reduce the so-called wave number pollution. To this end, an auxiliary unknown is introduced on the common interface of any two neighboring elements and a quadratic objective functional is defined by the jumps of the traces of the solutions of local Helmholtz equations across all the common interfaces. The minimization problem produces a Hermitian positive definite algebraic system for the auxiliary unknown. Then an approximate solution of the original Helmholtz equation is obtained by solving small local problems on the elements in a parallel manner.
Some numerical experiments are carried out. They suggest that the discretization method and the preconditioner are effective in solving large wave numbers Helmholtz problems on bounded domains.
Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)Numerical solutions of random mean square Fisher-KPP models with advection.https://www.zbmath.org/1452.352672021-02-12T15:23:00+00:00"Casabán, María Consuelo"https://www.zbmath.org/authors/?q=ai:casaban.maria-consuelo"Company, Rafael"https://www.zbmath.org/authors/?q=ai:company.rafael"Jódar, Lucas"https://www.zbmath.org/authors/?q=ai:jodar-sanchez.lucas-aSummary: This paper deals with the construction of numerical stable solutions of random mean square Fisher-Kolmogorov-Petrosky-Piskunov (Fisher-KPP) models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear inhomogeneous system of random differential equations. Then, by extending to the random framework, the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity, the results are illustrated by comparing the results with a test problem where the exact solution is known.Existence of a solution to the stochastic nonlocal Cahn-Hilliard Navier-Stokes model via a splitting-up method.https://www.zbmath.org/1452.352682021-02-12T15:23:00+00:00"Deugoué, G."https://www.zbmath.org/authors/?q=ai:deugoue.gabriel"Moghomye, B. Jidjou"https://www.zbmath.org/authors/?q=ai:moghomye.b-jidjou"Tachim Medjo, T."https://www.zbmath.org/authors/?q=ai:tachim-medjo.theodoreEuler equations for the estimation of dynamic discrete choice structural models.https://www.zbmath.org/1452.910782021-02-12T15:23:00+00:00"Aguirregabiria, Victor"https://www.zbmath.org/authors/?q=ai:aguirregabiria.victor"Magesan, Arvind"https://www.zbmath.org/authors/?q=ai:magesan.arvindSummary: We derive marginal conditions of optimality (i.e., Euler equations) for a general class of \textit{dynamic discrete choice} (DDC) structural models. These conditions can be used to estimate structural parameters in these models without having to solve for approximate value functions. This result extends to discrete choice models the GMM-Euler equation approach proposed by \textit{L. P. Hansen} and \textit{K. J. Singleton} [Econometrica 50, 1269--1286 (1982; Zbl 0497.62098)] for the estimation of dynamic continuous decision models. We first show that DDC models can be represented as models of continuous choice where the decision variable is a vector of choice probabilities. We then prove that the marginal conditions of optimality and the envelope conditions required to construct Euler equations are also satisfied in DDC models. The GMM estimation of these Euler equations avoids the curse of dimensionality associated to the computation of value functions and the explicit integration over the space of state variables. We present an empirical application and compare estimates using the GMM-Euler equations method with those from maximum likelihood and two-step methods.
For the entire collection see [Zbl 1298.91022].Incorporating boundary conditions in a stochastic volatility model for the numerical approximation of bond prices.https://www.zbmath.org/1452.651582021-02-12T15:23:00+00:00"Gómez-Valle, Lourdes"https://www.zbmath.org/authors/?q=ai:gomez-valle.lourdes"López-Marcos, Miguel Ángel"https://www.zbmath.org/authors/?q=ai:lopez-marcos.miguel-angel"Martínez-Rodríguez, Julia"https://www.zbmath.org/authors/?q=ai:martinez-rodriguez.juliaSummary: In this paper, we consider a two-factor interest rate model with stochastic volatility, and we assume that the instantaneous interest rate follows a jump-diffusion process. In this kind of problems, a two-dimensional partial integro-differential equation is derived for the values of zero-coupon bonds. To apply standard numerical methods to this equation, it is customary to consider a bounded domain and incorporate suitable boundary conditions. However, for these two-dimensional interest rate models, there are not well-known boundary conditions, in general. Here, in order to approximate bond prices, we propose new boundary conditions, which maintain the discount function property of the zero-coupon bond price. Then, we illustrate the numerical approximation of the corresponding boundary value problem by means of an alternative direction implicit method, which has been already applied for pricing options. We test these boundary conditions with several interest rate pricing models.Higher integrability in the obstacle problem for the fast diffusion equation.https://www.zbmath.org/1452.350842021-02-12T15:23:00+00:00"Cho, Yumi"https://www.zbmath.org/authors/?q=ai:cho.yumi"Scheven, Christoph"https://www.zbmath.org/authors/?q=ai:scheven.christophThe authors prove local higher integrability of the spatial gradient for solutions to obstacle problems of porous medium type in the fast diffusion case \(\frac{(n-2)}{(n+2)}\) \(< m < 1\). The result holds
for the natural range of exponents that is known from other regularity results for
porous medium type equations. The case of signed solutions is also covered. On a technical level, the problem is to construct a system of intrinsic cylinders that compensates the possible degeneracy in \(|u|\) and at
the same time, takes into account the size of the spatial gradient. This major problem was solved
by \textit{U. Gianazza} and \textit{S. Schwarzacher} [J. Funct. Anal. 277, No. 12, Article ID 108291, 57 p. (2019; Zbl 1423.35218)] who established the self-improving property of higher integrability for
the spatial gradient of non-negative solutions to the porous medium equation in the slow diffusion range
\(m > 1\). In the fast diffusion case \(m < 1\), the different behaviour of the singular
porous medium equation requires non-trivial adaptations of the previous techniques.
For the degenerate porous medium equation, the higher integrability was extended to obstacle problems by \textit{Y. Cho} and \textit{C. Scheven} [NoDEA, Nonlinear Differ. Equ. Appl. 26, No. 5, Paper No. 37, 44 p. (2019; Zbl 1423.35227)]. The present work is devoted to the question whether the result of the
previous work also applies in the singular case of the fast diffusion equation.
Reviewer: Vincenzo Vespri (Firenze)Principal eigenvalues of elliptic BVPs with glued Dirichlet-Robin mixed boundary conditions. Large potentials on the boundary conditions.https://www.zbmath.org/1452.351192021-02-12T15:23:00+00:00"Cano-Casanova, Santiago"https://www.zbmath.org/authors/?q=ai:cano-casanova.santiagoSummary: In this paper we perform a complete study about the existence, uniqueness and simplicity of the principal eigenvalue of a class of elliptic eigenvalue problems with glued Dirichlet-Robin boundary conditions on a component of the boundary of the domain and Dirichlet boundary conditions on the other component of the boundary. Moreover, this principal eigenvalue and its normalized principal eigenfunction are approached in \(\mathbb{R} \times H^1(\Omega)\) by principal eigen-pairs of boundary eigenvalue problems with classical mixed boundary conditions. In some sense, the glued mixed boundary conditions analyzed in this work may be regarded as classical mixed boundary conditions with large potentials in the Robin boundary condition. The principal eigenvalues analyzed in this paper, play a crucial role in analyzing the existence, uniqueness and asymptotic behavior of the positive solutions of certain kinds of semilinear elliptic boundary value problems with nonlinear boundary conditions and spatial heterogeneities. The main technical tools used to carry out the mathematical analysis of this work are variational and monotonicity techniques. The results in [\textit{H. Amann}, Isr. J. Math. 45, 225--254 (1983; Zbl 0535.35017)], [\textit{S. Cano-Casanova} and \textit{J. López-Gómez}, J. Differ. Equations 178, No. 1, 123--211 (2002; Zbl 1086.35073)] and [\textit{J. García-Melián} et al., NoDEA, Nonlinear Differ. Equ. Appl. 14, No. 5--6, 499--525 (2007; Zbl 1136.35045)] play a crucial role to obtain some of the results of this work.Development of a process model for the prediction of absorbent degradation during \(\mathrm{CO}_2\) capture. (Abstract of thesis).https://www.zbmath.org/1452.352172021-02-12T15:23:00+00:00"Dickinson, Jillian"https://www.zbmath.org/authors/?q=ai:dickinson.jillian(no abstract)Exponential Jacobi spectral method for hyperbolic partial differential equations.https://www.zbmath.org/1452.350882021-02-12T15:23:00+00:00"Youssri, Y. H."https://www.zbmath.org/authors/?q=ai:youssri.youssri-hassan"Hafez, R. M."https://www.zbmath.org/authors/?q=ai:hafez.ramy-mahmoudSummary: Herein, we have proposed a scheme for numerically solving hyperbolic partial differential equations (HPDEs) with given initial conditions. The operational matrix of differentiation for exponential Jacobi functions was derived, and then a collocation method was used to transform the given HPDE into a linear system of equations. The preferences of using the exponential Jacobi spectral collocation method over other techniques were discussed. The convergence and error analyses were discussed in detail. The validity and accuracy of the proposed method are investigated and checked through numerical experiments.Global Orlicz estimates for non-divergence elliptic operators with potentials satisfying a reverse Hölder condition.https://www.zbmath.org/1452.350522021-02-12T15:23:00+00:00"Truong, Le Xuan"https://www.zbmath.org/authors/?q=ai:le-xuan-truong."Dung, Tran Tri"https://www.zbmath.org/authors/?q=ai:dung.tran-tri"Trong, Nguyen Ngoc"https://www.zbmath.org/authors/?q=ai:trong.nguyen-ngoc"Tung, Nguyen Thanh"https://www.zbmath.org/authors/?q=ai:tung.nguyen-thanhThe aim of the authors is firstly to recall some basic facts about the critical function and Orlicz spaces, then, to prove a global Orlicz regularity for non-divergence form Schrödinger operators having coefficients in the new bounded mean oscillation class, in \(\mathbb R^n\).
The paper is a continuation to previous works of the authors and also is stressed that they improve estimate (1.5) previously obtained by Tang and Pan.
The results are, also, inspired by the pioneeristic papers by Chiarenza, Frasca and Longo.
To obtain the estimates of the second derivatives of the solutions, the authors adapt the ideas and techniques used by \textit{G. Pan} and \textit{L. Tang} [J. Funct. Anal. 270, No. 1, 88--133 (2016; Zbl 1342.35426)].
One of the main reasons for this adaption is that the method presented in the mentioned paper is particularly appropriate for BMO\(_\theta(\rho)\) coefficients, whereas the other methods used in several previous paper are applicable to VMO and BMO coefficients only.
More specifically, the authors prove a comparative inequality between the Hardy-Littlewood maximal function and the Fefferman-Stein maximal function (Lemma 3.2) and a Fefferman-Stein-type inequality in (Lemma 3.5).
These lemmata are the key ingredients for controlling the second derivatives of the solutions.
Reviewer: Maria Alessandra Ragusa (Catania)Haar wavelet collocation method for solving singular and nonlinear fractional time-dependent Emden-Fowler equations with initial and boundary conditions.https://www.zbmath.org/1452.653732021-02-12T15:23:00+00:00"Mohammadi, Amir"https://www.zbmath.org/authors/?q=ai:mohammadi.amir-amjad|mohammadi.amir|mohammadi.amir-hossein-mousavi"Aghazadeh, Nasser"https://www.zbmath.org/authors/?q=ai:aghazadeh.nasser"Rezapour, Shahram"https://www.zbmath.org/authors/?q=ai:rezapour.shahramSummary: In this paper, we have applied an iterative method to the singular and nonlinear fractional partial differential of Emden-Fowler equations types. Haar wavelets operational matrix of fractional integration will be used to solve the problem with the Picard technique. The singular equations turn to Sylvester equations that will be solved so that numerically solvable is very cost-effective. Moreover, the proposed technique is reliable enough to overcome the difficulty of the singular point at \(x = 0\). Numerical examples are providing to illustrate the efficiency and accuracy of the technique.Intrinsic formulations of the nonlinear Kirchhoff-Love-von Kármán plate theory.https://www.zbmath.org/1452.740712021-02-12T15:23:00+00:00"Geymonat, Giuseppe"https://www.zbmath.org/authors/?q=ai:geymonat.giuseppe"Krasucki, Françoise"https://www.zbmath.org/authors/?q=ai:krasucki.francoiseSummary: We use a special duality by perturbation approach in optimization to find two different bi-dual problems of the non-linear Kirchhoff-Love-von Kármán plate theory. The first one coincides with that found by means of the intrinsic approach of \textit{P. G. Ciarlet} and \textit{P. Ciarlet jun.} [Math. Models Methods Appl. Sci. 15, No. 2, 259--271 (2005; Zbl 1084.74006)], while the second is found by means of a complementary energy introduced by \textit{J. J. Telega} [C. R. Acad. Sci., Paris, Sér. II 308, No. 14, 1193--1198 (1989; Zbl 0659.73019)].Weak and strong solutions to the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system.https://www.zbmath.org/1452.351512021-02-12T15:23:00+00:00"Giorgini, Andrea"https://www.zbmath.org/authors/?q=ai:giorgini.andrea"Temam, Roger"https://www.zbmath.org/authors/?q=ai:temam.roger-mSummary: We study the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system in a bounded smooth domain in \(\mathbb{R}^d\), \(d=2, 3\). This model arises from the Diffuse Interface theory of binary mixtures accounting for density variation, capillarity effects at the interface and partial mixing. We consider the case of initial density away from zero and concentration-depending viscosity with free energy potential equal to either the Landau potential or the Flory-Huggins logarithmic potential. In this setting, we prove the existence of global weak solutions in two and three dimensions, and the existence of strong solutions with bounded and strictly positive density. The strong solutions are local in time in three dimensions and global in time in two dimensions.Asymptotic behavior for textiles in von-Kármán regime.https://www.zbmath.org/1452.350202021-02-12T15:23:00+00:00"Griso, Georges"https://www.zbmath.org/authors/?q=ai:griso.georges"Orlik, Julia"https://www.zbmath.org/authors/?q=ai:orlik.julia"Wackerle, Stephan"https://www.zbmath.org/authors/?q=ai:wackerle.stephanSummary: This paper is dedicated to the investigation of simultaneous homogenization and dimension reduction of textile structures as elasticity problem with an energy in the von-Kármán-regime. An extension for deformations is presented allowing to use the decomposition of plate-displacements. The limit problem in terms of displacements is derived with the help of the unfolding operator and yields in the limit the von-Kármán plate with linear elastic cell-problems. It is shown, that for homogeneous isotropic beams in the structure, the resulting plate is orthotropic. As application of the obtained limit plate we study the buckling behavior of orthotropic textiles.The periodic version of the Da Prato-Grisvard theorem and applications to the bidomain equations with FitzHugh-Nagumo transport.https://www.zbmath.org/1452.350862021-02-12T15:23:00+00:00"Hieber, Matthias"https://www.zbmath.org/authors/?q=ai:hieber.matthias"Kajiwara, Naoto"https://www.zbmath.org/authors/?q=ai:kajiwara.naoto"Kress, Klaus"https://www.zbmath.org/authors/?q=ai:kress.klaus"Tolksdorf, Patrick"https://www.zbmath.org/authors/?q=ai:tolksdorf.patrickSummary: In this article, the periodic version of the classical Da Prato-Grisvard theorem on maximal \({{L}}^p\)-regularity in real interpolation spaces is developed, as well as its extension to semilinear evolution equations. Applying this technique to the bidomain equations subject to ionic transport described by the models of FitzHugh-Nagumo, Aliev-Panfilov, or Rogers-McCulloch, it is proved that this set of equations admits a \textit{unique, strong} \( T\)-periodic solution in a neighborhood of stable equilibrium points provided it is innervated by \(T\)-periodic forces.Existence of densities for the 3D Navier-Stokes equations driven by Gaussian noise.https://www.zbmath.org/1452.761932021-02-12T15:23:00+00:00"Debussche, Arnaud"https://www.zbmath.org/authors/?q=ai:debussche.arnaud"Romito, Marco"https://www.zbmath.org/authors/?q=ai:romito.marcoSummary: We prove three results on the existence of densities for the laws of finite-dimensional functionals of the solutions of the stochastic Navier-Stokes equations in dimension~3. In particular, under very mild assumptions on the noise, we prove that finite-dimensional projections of the solutions have densities with respect to the Lebesgue measure which have some smoothness when measured in a Besov space. This is proved thanks to a new argument inspired by an idea introduced in [\textit{N. Fournier} and \textit{J. Printems}, Bernoulli 16, No.~2, 343--360 (2010; Zbl 1248.60062)].Properties of boundary-layer flow solutions for non-Newtonian fluids with non-linear terms of first and second-order derivatives.https://www.zbmath.org/1452.760092021-02-12T15:23:00+00:00"Al-Ashhab, Samer"https://www.zbmath.org/authors/?q=ai:al-ashhab.samer-sSummary: A third-order highly non-linear ODE that arises in applications of non-Newtonian boundary-layer fluid flow, governed by a power-law Ostwald-de Waele rheology, is considered. The model appears in many disciplines related to applied and engineering mathematics, in addition to engineering and industrial applications. The aim is to use a new set of variables, defined via the first and second-order derivatives of the dependent variable, to transform the problem to a bounded domain, where we study properties of solutions, discuss existence and uniqueness of solutions, and investigate some physical parameter values and limitations leading to non-existence of solutions.Flexibility of curves on a single-sheet hyperboloid.https://www.zbmath.org/1452.352092021-02-12T15:23:00+00:00"Maksimović, Miroslav D."https://www.zbmath.org/authors/?q=ai:maksimovic.miroslav-dSummary: Hyperbolic towers are towers in the shape of a single-sheet hyperboloid, and they are interesting in architecture. In this paper, we deal with the infinitesimal bending of a curve on a hyperboloid of one sheet; that is, we study the flexibility of the net-like structures used to make a hyperbolic tower. Visualization of infinitesimal bending has been carried out using \textit{Mathematica}, and some examples are presented and discussed.Monotonicity and symmetry of solutions to fractional \(p\)-Laplacian equations.https://www.zbmath.org/1452.352472021-02-12T15:23:00+00:00"Zhang, Yajie"https://www.zbmath.org/authors/?q=ai:zhang.yajie"Ma, Feiyao"https://www.zbmath.org/authors/?q=ai:ma.feiyao"Wo, Weifeng"https://www.zbmath.org/authors/?q=ai:wo.weifengSummary: We investigate the fractional \(p\)-Laplacian equation \((-\Delta)_p^s u = f (x, u, \nabla u)\). We obtain the monotonicity and symmetry of positive solutions of the fractional \(p\)-Laplacian equation on bounded and unbounded domains. Specially, for unbounded case, we present a new \textit{decay at infinity}.Large-time behavior of solutions for the \(1{D}\) viscous heat-conducting gas with radiation: the pure scattering case.https://www.zbmath.org/1452.762022021-02-12T15:23:00+00:00"Qin, Yuming"https://www.zbmath.org/authors/?q=ai:qin.yuming"Feng, Baowei"https://www.zbmath.org/authors/?q=ai:feng.baowei"Zhang, Ming"https://www.zbmath.org/authors/?q=ai:zhang.mingSummary: In this paper, assuming suitable hypotheses on the transport coefficients, we prove the large-time behavior, as time tends to infinity, of solutions in \(\mathcal H_2\) and \(\mathcal H_3\) (see below for their definitions) for the one-dimensional viscous heat-conducting gas with radiation.Modeling of high-frequency fields in irregular waveguides through boundary potentials.https://www.zbmath.org/1452.352032021-02-12T15:23:00+00:00"Yunakovsky, A. D."https://www.zbmath.org/authors/?q=ai:yunakovsky.a-dSummary: The work is devoted to the study of the possibilities of creating mathematical models broadband waveguide systems and mode converters in the millimeter and terahertz ranges. Their models should allow describing and studying the processes of microwave propagation using substantially (two orders of magnitude) large diffraction elements. To construct converters from gyro-BWT to emitters, as well as to study the influence of local disturbances, a model is used that takes into account only the envelopes of the high-frequency components of the signal propagating in a multimode waveguide (the so-called parabolic approximation). The envelope in this case, instead of the Helmholtz equation, satisfies the Schrödinger equation. The main idea is to define the solution of the Schrödinger equation as a boundary potential with respect to the external the boundary of the cylindrical region enclosing the irregular waveguide, and the boundary conditions should be checked for real -- i.e. actually the desired border. Inside the enclosing cylindrical oscillation region of the core, the representations of the solution are already available for calculations.Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension.https://www.zbmath.org/1452.920102021-02-12T15:23:00+00:00"Liu, Jia"https://www.zbmath.org/authors/?q=ai:liu.jia"Wang, Zhi-An"https://www.zbmath.org/authors/?q=ai:wang.zhianA fully parabolic chemotaxis system with two chemicals: a chemoattractant and a chemorepellent is considered in the one dimensional case, with the Neumann boundary conditions. Nontrivial steady states are studied, and the existence of global-in-time solutions is proved.
Reviewer: Piotr Biler (Wrocław)Asymptotic profiles of solutions and propagating terrace for a free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity.https://www.zbmath.org/1452.352612021-02-12T15:23:00+00:00"Kaneko, Yuki"https://www.zbmath.org/authors/?q=ai:kaneko.yuki"Matsuzawa, Hiroshi"https://www.zbmath.org/authors/?q=ai:matsuzawa.hiroshi"Yamada, Yoshio"https://www.zbmath.org/authors/?q=ai:yamada.yoshioThe authors the consider one-dimensional, one-phase free boundary problem
\[
\begin{aligned}
u_t &= u_{xx} + f(u),\,\ t > 0,\, 0 < x < h(t),\\
h(0) &= h_0, \, u(0,x) = u_0(x), \, 0 \leq x \leq h_0, \, u_x(t,0) = 0, \,t > 0,\\
u(t, h(t)) &= 0, \, h'(t) = -\mu u_x(t,h(t)), \, t > 0,\\
\end{aligned} \tag{1}
\]
here \(x = h(t)\) is an equation of free boundary, \(h_0, \ \mu_0\) are positive constants.
Under the conditions on \(f(u)\) and the assumption that the functions \(u(t,x), \ h(t)\) are the spreading solution to the problem (1),
the authors determine an asymptotic behavior of \(u(t,x), \ h(t)\) as \(t \to \infty\).
Reviewer: Galina Bizhanova (Almaty)Multiscale simulation for the system of radiation hydrodynamics.https://www.zbmath.org/1452.850042021-02-12T15:23:00+00:00"Sun, Wenjun"https://www.zbmath.org/authors/?q=ai:sun.wenjun"Jiang, Song"https://www.zbmath.org/authors/?q=ai:jiang.song"Xu, Kun"https://www.zbmath.org/authors/?q=ai:xu.kun"Cao, Guiyu"https://www.zbmath.org/authors/?q=ai:cao.guiyuSummary: This paper aims at the simulation of multiple scale physics for the system of radiation hydrodynamics. The system couples the fluid dynamic equations with the radiative heat transfer. The coupled system is solved by the gas-kinetic scheme (GKS) for the compressible inviscid Euler flow and the unified gas-kinetic scheme (UGKS) for the non-equilibrium radiative transfer, together with the momentum and energy exchange between these two phases. For the radiative transfer, due to the possible large variation of fluid opacity in different regions, the transport of photons through the flow system is simulated by the multiscale UGKS, which is capable of naturally capturing the transport process from the photon's free streaming to the diffusive wave propagation. Since both GKS and UGKS are finite volume methods, all unknowns are defined inside each control volume and are discretized consistently in the updates of hydrodynamic and radiative variables. For the coupled system, the scheme has the asymptotic preserving property, such as recovering the equilibrium diffusion limit for the radiation hydrodynamic system in the optically thick region, where the cell size is not limited by photon's mean free path. A few test cases, such as radiative shock wave problems, are used to validate the current approach.Particle trajectories in nonlinear Schrödinger models.https://www.zbmath.org/1452.760332021-02-12T15:23:00+00:00"Carter, John D."https://www.zbmath.org/authors/?q=ai:carter.john-d"Curtis, Christopher W."https://www.zbmath.org/authors/?q=ai:curtis.christopher-w"Kalisch, Henrik"https://www.zbmath.org/authors/?q=ai:kalisch.henrikSummary: The nonlinear Schrödinger equation is well known as a universal equation in the study of wave motion. In the context of wave motion at the free surface of an incompressible fluid, the equation accurately predicts the evolution of modulated wave trains with low to moderate wave steepness. While there is an abundance of studies investigating the reconstruction of the surface profile \(\eta\), and the fidelity of such profiles provided by the nonlinear Schrödinger equation as predictions of real surface water waves, very few works have focused on the associated flow field in the fluid. In the current work, it is shown that the velocity potential \(\phi\) can be reconstructed in a similar way as the free surface profile. This observation opens up a range of potential applications since the nonlinear Schrödinger equation features fairly simple closed-form solutions and can be solved numerically with comparatively little effort. In particular, it is shown that particle trajectories in the fluid can be described with relative ease not only in the context of the nonlinear Schrödinger equation, but also in higher-order models such as the Dysthe equation, and in models incorporating certain types of viscous effects.Inverse scattering by a random periodic structure.https://www.zbmath.org/1452.780152021-02-12T15:23:00+00:00"Bao, Gang"https://www.zbmath.org/authors/?q=ai:bao.gang"Lin, Yiwen"https://www.zbmath.org/authors/?q=ai:lin.yiwen"Xu, Xiang"https://www.zbmath.org/authors/?q=ai:xu.xiangThe authors deal with the scattering of a time-harmonic electromagnetic plane wave by a periodic structure. They present an efficient numerical method for solving the inverse scattering problem to determine a random periodic interface or obstacle. This is done by combining the Monte Carlo technique for sampling the probability space, a continuation method with respect to the wavenumber, and the Karhunen-Loève expansion of the random structure. Numerical results are included, and illustrate the reliability and efficiency of the method.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Threshold analysis of the three dimensional lattice Schrödinger operator with non-local potential.https://www.zbmath.org/1452.351232021-02-12T15:23:00+00:00"Muminov, Z. E."https://www.zbmath.org/authors/?q=ai:muminov.zahriddin-i"Alladustov, Sh. U."https://www.zbmath.org/authors/?q=ai:alladustov.sh-u"Lakaev, Sh. S."https://www.zbmath.org/authors/?q=ai:lakaev.shukhrat-saidakhmatovichLet \(L_2(\mathbb T^3)=(L_2(\mathbb T^3),<\cdot,\cdot>_{L_2(\mathbb T^3)})\) be the square-integrable functions on \(\mathbb T^3\), the three-dimensional torus, and equipped with the inner product \(<\cdot,\cdot>_{L_2(\mathbb T^3)}\). The one-particle Hamiltonian on \(L_2(\mathbb T^3)\) is defined by \(H_{\lambda\mu}=\mathcal{F}^*(-\Delta)\mathcal{F}-V\), where \(\mathcal{F}\) is the \(l_2(\mathbb Z^3)\)-valued Fourier transform on \(L_2(\mathbb T^3)\) and
\[
-\Delta=\sum_{s\in\mathbb Z^3:|s|=1}(T(0)-T(s)).
\]
Here \(T(y)\) is the shift operator defined on \(l_2(\mathbb Z^3)\) with \(y\in\mathbb T^3\) and
\[
V(f)(p)=\frac{1}{(2\pi)^3}\int_{\mathbb T^3}(\lambda+\mu(e^{i{<}x_0,p{>}_{L_2(\mathbb T^3)}} +e^{-i{<}x_0,s{>}_{L_2(\mathbb T^3)}})f(s)ds
\]
such that \(f\in L_2(\mathbb T^3)\), \((p,x_0)\in(\mathbb T^3)^2\), and \(\lambda\), \(\mu\) are real parameters.
The authors find two results. The first one states that \(H_{\lambda\mu}\) has no eigenvalues in \((-\infty,0]\) whenever \(P(\lambda,\mu)\) is strictly negative. Here \(P\) is a suitable function and \((\lambda,\mu)\) belongs to the graph of this function. The second one states that, if \(P(\lambda,\mu)>0\), then \(H_{\lambda\mu}\) has a simple eigenvalue in \((-\infty,0)\). Further, the authors examine threshold eigenvalues (resp., resonances) analysis of \(H_{\lambda\mu}\).
Reviewer: Mohammed El Aïdi (Bogotá)Erratum: ``Asymptotically compatible schemes for stochastic homogenization''.https://www.zbmath.org/1452.350232021-02-12T15:23:00+00:00"Sun, Qi"https://www.zbmath.org/authors/?q=ai:sun.qi"Du, Qiang"https://www.zbmath.org/authors/?q=ai:du.qiang"Ming, Ju"https://www.zbmath.org/authors/?q=ai:ming.juFluctuation relations for anomalous dynamics generated by time-fractional Fokker-Planck equations.https://www.zbmath.org/1452.352392021-02-12T15:23:00+00:00"Dieterich, Peter"https://www.zbmath.org/authors/?q=ai:dieterich.peter-simon"Klages, Rainer"https://www.zbmath.org/authors/?q=ai:klages.rainer"Chechkin, Aleksei V."https://www.zbmath.org/authors/?q=ai:chechkin.aleksei-vCritical counterexamples for linear wave equations with time-dependent propagation speed.https://www.zbmath.org/1452.351082021-02-12T15:23:00+00:00"Ghisi, Marina"https://www.zbmath.org/authors/?q=ai:ghisi.marina"Gobbino, Massimo"https://www.zbmath.org/authors/?q=ai:gobbino.massimoThe authors consider the wave equation from an abstract point of view, the Laplacian being replaced by $- A$, with $A$ self-adjoint nonnegative operator. A time-dependent propagation speed is considered and a dissipation term is allowed, represented by a power of the operator $A$. In the non-dissipative case, the classical results of \textit{F. Colombini} et al. [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 6, 511--559 (1979; Zbl 0417.35049)] state either well- posedness or ill-posedness of the Cauchy problem in Sobolev spaces and Gevrey Classes, according to the regularity of the speed coefficient. The effect of the presence of the dissipation term was studied in detail by \textit{M. Ghisi} and \textit{M. Gobbino} [J. Differ. Equations 260, No. 2, 1585--1621 (2016; Zbl 1336.35252)] and \textit{M. Ghisi} et al. [Trans. Am. Math. Soc. 368, No. 3, 2039--2079 (2016; Zbl 1342.35184)], showing that the dissipation can prevail and improve the results of well-posedness. The present paper is devoted to some critical cases left open in the above mentioned contributions. In particular, counterexamples are given providing a complete picture of results. The proofs are based on a precise estimate of the growth of solutions for a family of ordinary differential equations.
Reviewer: Luigi Rodino (Torino)Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate.https://www.zbmath.org/1452.350292021-02-12T15:23:00+00:00"Bernard, Étienne"https://www.zbmath.org/authors/?q=ai:bernard.etienne"Gabriel, Pierre"https://www.zbmath.org/authors/?q=ai:gabriel.pierre.1The rate of convergence to equilibrium is studied for solutions to the growth-fragmentation equation
\[
\partial_t f(t,x) + \partial_x (\tau(x) f(t,x)) = \int_0^1 B\left( \frac{x}{z} \right) f\left( t , \frac{x}{z} \right) \frac{\mathfrak{p}(dz)}{z} - B(x) f(t,x)
\]
for \((t,x)\in (0,\infty)^2\), supplemented with the boundary condition \(f(t,0)=0\) for \(t>0\) and an initial condition \(f(0,x)=f^{in}(x)\) for \(x>0\). The growth rate \(\tau\in C^1(0,\infty)\) is positive with \(1/\tau\in L^1(0,1)\) and there are \(\nu_0\le 1\) and \(\tau_1\ge \tau_0> 0\) such that
\[
\tau_0 \mathbf{1}_{(1,\infty)}(x) x^{\nu_0} \le \tau(x) \le \tau_1 \max\{1,x\}, \quad x>0,
\]
and the total fragmentation rate \(B\in C(0,\infty)\) is non-negative with connected support and there are \(0<\gamma_0\le \gamma_1\), \(0<B_0\le B_1\), and \(x_0>0\) such that
\[
B_0 \mathbf{1}_{(x_0,\infty)}(x) x^{\gamma_0} \le B(x) \le B_1 \max\{ 1 , x^{\gamma_1}\}, \quad x>0.
\]
Finally, the fragmentation kernel \(\mathfrak{p}\) is a finite positive measure on \((0,1)\) satisfying
\[
\int_0^1 z \mathfrak{p}(dz)=1\ \text{ and }\ \underline{\alpha} := \inf\left\{ \alpha\in \mathbb{R}: \int_0^1 z^\alpha \mathfrak{p}(dz) < \infty \right\} \in [-\infty,0].
\]
It is by now well-known that these assumptions guarantee that there are \(\lambda>0\) and a non-negative function \(G\in L^1(0,\infty)\) such that
\[
G\in \bigcap_{\alpha\ge 0} L^1((0,\infty),(1+x)^\alpha), \quad \|G\|_{L^1}=1,
\]
and \((t,x) \mapsto e^{\lambda t} G(x)\) is a particular solution to the growth-fragmentation equation. In addition, this solution is an attractor for the dynamics, in the sense that \(f(t) e^{-\lambda t}\) converges to \(C_* G\) as \(t\to\infty\) in a suitable topology with a well-identified constant \(C_*\) depending only on \(\tau\), \(B\), \(\mathfrak{p}\), and \(f^{in}\).
It is the question of temporal decay estimates for the difference \(f(t)e^{-\lambda t} - C_* G\) which is investigated in the paper under review. Specifically, it is shown that, given \(\alpha>\max\{1 , \underline{\alpha} + 2(\gamma_1-\gamma_0)\}\), there are \(M>0\) and \(\sigma>0\) such that
\[
\| f(t)e^{-\lambda t} - C_* G \|_{L^1((0,\infty),(1+x)^\alpha dx)} \le M e^{-\sigma t} \|f^{in}\|_{L^1((0,\infty),(1+x)^\alpha dx)}
\]
for all \(f^{in}\in L^1((0,\infty),(1+x)^\alpha dx)\) when, either \(\mathfrak{p}\) is absolutely continuous with respect to the Lebesgue measure on \((0,1)\), or \(\mathfrak{p}\) is supported in \((\epsilon,1-\epsilon)\) for some \(\epsilon\in (0,1)\) and \(\tau\) is constant. Besides, the optimality of the integrability of \(1/\tau\) on \((0,1)\) is discussed in the last section.
Owing to the linearity of the growth-fragmentation equation, the proof mainly relies on semigroup arguments. An intermediate result is that some moments, which are initially infinite, become instantaneously finite for positive times under the action of the evolution equation.
Reviewer: Philippe Laurençot (Toulouse)Symmetrization of MHD equations of incompressible viscoelastic polymer fluid.https://www.zbmath.org/1452.762662021-02-12T15:23:00+00:00"Blokhin, A. M."https://www.zbmath.org/authors/?q=ai:blokhin.aleksandr-mikhailovich"Goldin, A. Yu."https://www.zbmath.org/authors/?q=ai:goldin.a-yuSummary: Equations describing the flow of an incompressible viscoelastic polymer fluid in the presence of a magnetic field are considered. The symmetrization of this system of equations is discussed.An improved gradient bound for 2D MBE.https://www.zbmath.org/1452.350502021-02-12T15:23:00+00:00"Li, Dong"https://www.zbmath.org/authors/?q=ai:li.dong|li.dong.2"Wang, Fan"https://www.zbmath.org/authors/?q=ai:wang.fan"Yang, Kai"https://www.zbmath.org/authors/?q=ai:yang.kaiSummary: We consider a two dimensional molecular beam epitaxy equation with biharmonic dissipation. We give an improved gradient bound, which removes the logarithmic dependence on the dissipation coefficient previously shown in [\textit{D. Li} et al., J. Differ. Equations 262, No. 3, 1720--1746 (2017; Zbl 1364.35115)].Hybrid singularity for the oblique incidence response of a cold plasma.https://www.zbmath.org/1452.780082021-02-12T15:23:00+00:00"Lafitte, Olivier"https://www.zbmath.org/authors/?q=ai:lafitte.olivier-dThe authors study the electromagnetic response of an inhomogeneous cold plasma under the influence of a strong magnetic field in general slab geometry. The cold plasma model assumes that the ions and the electrons follow the classical law of motion. Collisions are taking into account through a dissipation parameter \(\nu\) (small with respect to the characteristic time of motion).
The behavior of the electromagnetic field in the neighborhood of the hybrid resonance is expressed for the oblique incident wave when \(\nu\to 0+\). In this case, a fully coupled system of ODEs of order 4 is obtained. The investigation is done by deducing from the complete system of ODEs to an integro-differential system whose differential part contains a regular singular point at \(\nu\to 0+\). The integro-differential system is then solved in the neighborhood of this regular singular point by using the variation of parameters method.
Reviewer: David Kapanadze (Tbilisi)Edged-based smoothed point interpolation method for acoustic radiation with perfectly matched layer.https://www.zbmath.org/1452.653872021-02-12T15:23:00+00:00"You, Xiangyu"https://www.zbmath.org/authors/?q=ai:you.xiangyu"Chai, Yingbin"https://www.zbmath.org/authors/?q=ai:chai.yingbin"Li, Wei"https://www.zbmath.org/authors/?q=ai:li.wei.11Summary: It is known that for exterior acoustics the domain-based finite element method always encounters the accuracy loss for high wavenumbers and the difficulty in dealing with the Sommerfeld radiation condition. In this work, by using the polynomial and radial bases with specific node-selection schemes, the point interpolation method with the generalized gradient smoothing technique is employed to improve the accuracy for acoustic computations. The appropriately softened acoustical stiffness of the discrete model is constructed based on the edge-based smoothing domains via the gradient smoothing technique. In addition, the perfectly matched layer technique enables the smoothed point interpolation method to investigate exterior acoustics. The complex coordinate transformation is utilized to determine the proper parameters for the perfectly matched layer through theoretical and numerical analyses. Several typical acoustic problems are analyzed, revealing that better accuracy is obtained by the present method compared with the finite element method, especially for high wavenumbers.Finite difference/spectral methods for the two-dimensional distributed-order time-fractional cable equation.https://www.zbmath.org/1452.652852021-02-12T15:23:00+00:00"Zheng, Rumeng"https://www.zbmath.org/authors/?q=ai:zheng.rumeng"Liu, Fawang"https://www.zbmath.org/authors/?q=ai:liu.fawang"Jiang, Xiaoyun"https://www.zbmath.org/authors/?q=ai:jiang.xiaoyun"Turner, Ian W."https://www.zbmath.org/authors/?q=ai:turner.ian-williamSummary: The cable equation plays a significant role in many areas of electrophysiology and in modeling neuronal dynamics. In recent years, considerable attention has been devoted to distributed-order differential equations because they appear to be more effective for modeling complex processes. In this work, a finite difference/Legendre spectral method is presented for the numerical simulation of the two-dimensional (2D) distributed-order time-fractional cable equation, where the finite difference method is employed in the temporal discretization and Legendre spectral method is adopted in the spatial discretization. The midpoint quadrature rule is used to approximate the distributed-order, such that the considered equation could be transformed into a multi-term fractional equation. The stability and convergence analysis of the proposed scheme is established, which illustrates that the numerical solution converges to the exact solution with order \(O(\tau^2+\sigma^2+N^{-s})\), where \(\tau,\sigma,N\) are the time step size, the step length in the approximation of the distributed-order and the polynomial degree, respectively. Furthermore, to demonstrate the versatility and applicability of our method, we provide numerical results that show good agreement with the theoretical analysis.A theory for spiral wave drift in reaction-diffusion-mechanics systems.https://www.zbmath.org/1452.350762021-02-12T15:23:00+00:00"Dierckx, Hans"https://www.zbmath.org/authors/?q=ai:dierckx.hans"Arens, Sander"https://www.zbmath.org/authors/?q=ai:arens.sander"Li, Bing-Wei"https://www.zbmath.org/authors/?q=ai:li.bing-wei"Weise, Louis D."https://www.zbmath.org/authors/?q=ai:weise.louis-d"Panfilov, Alexander V."https://www.zbmath.org/authors/?q=ai:panfilov.alexander-vSolutions with concentration for conservation laws with discontinuous flux and its applications to numerical schemes for hyperbolic systems.https://www.zbmath.org/1452.350952021-02-12T15:23:00+00:00"Aggarwal, Aekta"https://www.zbmath.org/authors/?q=ai:aggarwal.aekta"Sahoo, Manas Ranjan"https://www.zbmath.org/authors/?q=ai:sahoo.manas-ranjan"Sen, Abhrojyoti"https://www.zbmath.org/authors/?q=ai:sen.abhrojyoti"Vaidya, Ganesh"https://www.zbmath.org/authors/?q=ai:vaidya.ganeshSummary: Measure-valued weak solutions for conservation laws with discontinuous flux are proposed and explicit formulae have been derived. We propose convergent discontinuous flux-based numerical schemes for the class of hyperbolic systems that admit nonclassical \(\delta\)-shocks, by extending the theory of discontinuous flux for nonlinear conservation laws to scalar transport equation with a discontinuous coefficient. The article also discusses the concentration phenomenon of solutions along the line of discontinuity, for scalar transport equations with a discontinuous coefficient. The existence of the solutions for transport equation is shown using the vanishing viscosity approach and the asymptotic behavior of the solutions is also established. The performance of the numerical schemes for both scalar conservation laws and systems to capture the \(\delta\)-shocks effectively is displayed through various numerical experiments.Low-frequency dipolar electromagnetic scattering by a solid ellipsoid in lossless environment.https://www.zbmath.org/1452.780142021-02-12T15:23:00+00:00"Vafeas, Panayiotis"https://www.zbmath.org/authors/?q=ai:vafeas.panayiotisSummary: Electromagnetic wave scattering phenomena for target identification are important in many applications related to fundamental science and engineering. Here, we present an analytical formulation for the calculation of the magnetic and electric fields that scatter off a highly conductive ellipsoidal body, located within an otherwise homogeneous and isotropic lossless medium. The primary excitation source assumes a time-harmonic magnetic dipole, precisely fixed and arbitrarily orientated that operates at low frequencies and produces the incident fields. The scattering problem itself is modeled with respect to rigorous expansions of the electromagnetic fields at the low-frequency regime in terms of positive integral powers of the real wave number of the ambient. Obviously, the Rayleigh static term and a few dynamic terms are sufficient for the purpose of the present work, as the additional terms are neglected due to their minor contribution. Therein, the classical Maxwell's theory is suitably modified, leading to intertwined either Laplace's or Poisson's equations, accompanied by the impenetrable boundary conditions for the total fields and the limiting behavior at infinity. On the other hand, the complete spatial anisotropy of the three-dimensional space is secured via the introduction of the genuine ellipsoidal coordinate system, being appropriate for tackling incrementally such scattering boundary value problems. The nonaxisymmetric fields are obtained via infinite series expansions in terms of ellipsoidal harmonic eigenfunctions, providing handy closed-form solutions in a compact fashion, whose validity is verified by a straightforward reduction to simpler geometries of the metal object. The main idea is to demonstrate an efficient methodology, according to which the constructed analytical formulae can offer the appropriate environment for a fast numerical estimation of the scattered electromagnetic fields that could be useful for real data inversion.General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger equation.https://www.zbmath.org/1452.351982021-02-12T15:23:00+00:00"Ye, Rusuo"https://www.zbmath.org/authors/?q=ai:ye.rusuo"Zhang, Yi"https://www.zbmath.org/authors/?q=ai:zhang.yi.3Summary: General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger (NLS) equation are discussed via a matrix version of binary Darboux transformation. With this technique, searching for solutions of the Lax pair is transferred to find vector solutions of the associated linear differential equation system. From vanishing and nonvanishing seed solutions, general vector solutions of such linear differential equation system in terms of the canonical forms of the spectral matrix can be constructed by means of triangular Toeplitz matrices. Several explicit one-soliton solutions and two-soliton solutions are provided corresponding to different forms of the spectral matrix. Furthermore, dynamics and interactions of bright solitons, degenerate solitons, breathers, rogue waves, and dark solitons are also explored graphically.The complex Hamiltonian systems and quasi-periodic solutions in the derivative nonlinear Schrödinger equations.https://www.zbmath.org/1452.351802021-02-12T15:23:00+00:00"Chen, Jinbing"https://www.zbmath.org/authors/?q=ai:chen.jinbing"Zhang, Runsu"https://www.zbmath.org/authors/?q=ai:zhang.runsuSummary: The complex Hamiltonian systems with real-valued Hamiltonians are generalized to deduce quasi-periodic solutions for a hierarchy of derivative nonlinear Schrödinger (DNLS) equations. The DNLS hierarchy is decomposed into a family of complex finite-dimensional Hamiltonian systems by separating the temporal and spatial variables, and the complex Hamiltonian systems are then proved to be integrable in the Liouville sense. Due to the commutability of complex Hamiltonian flows, the relationship between the DNLS equations and the complex Hamiltonian systems is specified via the Bargmann map. The Abel-Jacobi variable is elaborated to straighten out the DNLS flows as linear superpositions on the Jacobi variety of an invariant Riemann surface. Finally, by using the technique of Riemann-Jacobi inversion, some quasi-periodic solutions are obtained for the DNLS equations in view of the Riemann theorem and the trace formulas.Periodic problem for the nonlinear damped wave equation with convective nonlinearity.https://www.zbmath.org/1452.351032021-02-12T15:23:00+00:00"Carreño-Bolaños, Rafael"https://www.zbmath.org/authors/?q=ai:carreno-bolanos.rafael"Juarez-Campos, Beatriz"https://www.zbmath.org/authors/?q=ai:juarez-campos.beatriz"Naumkin, Pavel I."https://www.zbmath.org/authors/?q=ai:naumkin.pavel-iSummary: We study the nonlinear damped wave equation with a linear pumping and a convective nonlinearity. We consider the solutions, which satisfy the periodic boundary conditions. Our aim is to prove global existence of solutions to the periodic problem for the nonlinear damped wave equation by applying the energy-type estimates and estimates for the Green operator. Moreover, we study the asymptotic profile of global solutions.Partial differential equations admitting a given nonclassical point symmetry.https://www.zbmath.org/1452.350112021-02-12T15:23:00+00:00"Pucci, Edvige"https://www.zbmath.org/authors/?q=ai:pucci.edvige"Saccomandi, Giuseppe"https://www.zbmath.org/authors/?q=ai:saccomandi.giuseppeSummary: Given a local one parameter Lie group of transformations \(G\), we determine the most general scalar partial differential equation in \((1 + 1)\)-independent variables of a given order admitting \(G\) as nonclassical symmetry in the Bluman and Cole sense.Solitary wave solutions to the Isobe-Kakinuma model for water waves.https://www.zbmath.org/1452.351482021-02-12T15:23:00+00:00"Colin, Mathieu"https://www.zbmath.org/authors/?q=ai:colin.mathieu"Iguchi, Tatsuo"https://www.zbmath.org/authors/?q=ai:iguchi.tatsuoSummary: We consider the Isobe-Kakinuma model for two-dimensional water waves in the case of a flat bottom. The Isobe-Kakinuma model is a system of Euler-Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe-Kakinuma model in the long wave regime. Numerical analysis for large amplitude solitary wave solutions is also provided and suggests the existence of a solitary wave of extreme form with a sharp crest.Numerical study of the transverse stability of the Peregrine solution.https://www.zbmath.org/1452.652762021-02-12T15:23:00+00:00"Klein, Christian"https://www.zbmath.org/authors/?q=ai:klein.christian"Stoilov, Nikola"https://www.zbmath.org/authors/?q=ai:stoilov.nikola-mSummary: We generalize a previously published numerical approach for the one-dimensional (1D) nonlinear Schrödinger (NLS) equation based on a multidomain spectral method on the whole real line in two ways: first, a fully explicit fourth-order method for the time integration, based on a splitting scheme and an implicit Runge-Kutta method for the linear part, is presented. Second, the 1D code is combined with a Fourier spectral method in the transverse variable both for elliptic and hyperbolic NLS equations. As an example we study the transverse stability of the Peregrine solution, an exact solution to the 1D NLS equation and thus a \(y\)-independent solution to the 2D NLS. It is shown that the Peregine solution is unstable agains all standard perturbations, and that some perturbations can even lead to a blow-up for the elliptic NLS equation.Sparse spectral and \(p\)-finite element methods for partial differential equations on disk slices and trapeziums.https://www.zbmath.org/1452.653762021-02-12T15:23:00+00:00"Snowball, Ben"https://www.zbmath.org/authors/?q=ai:snowball.ben"Olver, Sheehan"https://www.zbmath.org/authors/?q=ai:olver.sheehanSummary: Sparse spectral methods for solving partial differential equations have been derived in recent years using hierarchies of classical orthogonal polynomials on intervals, disks, and triangles. In this work, we extend this methodology to a hierarchy of nonclassical orthogonal polynomials on disk slices and trapeziums. This builds on the observation that sparsity is guaranteed due to the boundary being defined by an algebraic curve, and that the entries of partial differential operators can be determined using formulae in terms of (nonclassical) univariate orthogonal polynomials. We apply the framework to solving the Poisson, variable coefficient Helmholtz, and biharmonic equations. In this paper, we focus on constant Dirichlet boundary conditions, as well as zero Dirichlet and Neumann boundary conditions, with other types of boundary conditions requiring future work.The nonlinear Dirac equation in Bose-Einstein condensates. II: Relativistic soliton stability analysis.https://www.zbmath.org/1452.351862021-02-12T15:23:00+00:00"Haddad, L. H."https://www.zbmath.org/authors/?q=ai:haddad.l-h"D. Carr, Lincoln"https://www.zbmath.org/authors/?q=ai:carr.lincoln-dThe nonlinear Dirac equation in Bose-Einstein condensates. I: Relativistic solitons in armchair nanoribbon optical lattices.https://www.zbmath.org/1452.351872021-02-12T15:23:00+00:00"Haddad, L. H."https://www.zbmath.org/authors/?q=ai:haddad.l-h"Weaver, C. M."https://www.zbmath.org/authors/?q=ai:weaver.christina-m"Carr, Lincoln D."https://www.zbmath.org/authors/?q=ai:carr.lincoln-dSpectral analysis of the Schrödinger operator with a PT-symmetric periodic optical potential.https://www.zbmath.org/1452.811082021-02-12T15:23:00+00:00"Veliev, O. A."https://www.zbmath.org/authors/?q=ai:veliev.oktay-alishSummary: In this paper, we give a description of the spectral analysis of the Schrödinger operator \(L(q)\) with the potential \(q(x) = 4 \cos^2x + 4 iV\) sin \(2x\) for all \(V > 1/2\). First, we consider the Bloch eigenvalues and spectrum of \(L(q)\). Then, we investigate spectral singularities and essential spectral singularities (ESS). We prove that there exists a sequence \(V_2 < V_3 < \cdots\) such that the operator \(L(q)\) has no ESS and has ESS, respectively, if and only if \((V \neq V_k\) and \(V = V_k\) for \(k \geq 2\), where \(V_k \rightarrow \infty\) as \(k \rightarrow \infty, V_2\) is the second critical point. Using ESS, we classify the spectral expansion in term of the points \(V_k\) for \(k \geq 2\). Finally, we discuss the critical points \(V_k \), formulate some conjectures, and describe the changes in the spectrum of \(L(q)\) when \(V\) changes from 1/2 to \(\infty \).
{\copyright 2020 American Institute of Physics}Linear theory for beams with intermediate piers.https://www.zbmath.org/1452.340332021-02-12T15:23:00+00:00"Garrione, Maurizio"https://www.zbmath.org/authors/?q=ai:garrione.maurizio"Gazzola, Filippo"https://www.zbmath.org/authors/?q=ai:gazzola.filippoThe authors develop the full linear theory for hinged beams with intermediate piers
\[
u_{tt}+u_{xxxx}+\gamma u=0,\ x\in(-\pi,\pi),\, t>0,\]
\[u(-\pi, t)=u(\pi, t)=u(b\pi, t)=u(a\pi, t)=0, \,t\geq 0.
\]
The analysis starts with the variational setting and the study of the linear stationary problem. They provide well-posedness results and analyze the possible loss of regularity, due to the presence of the piers. Then they give a complete spectral theorem, explicitly determine the eigenvalues according to the position of the piers and exhibite the fundamental modes of oscillation. They also study a related second-order eigenvalue problem and show that it may display nonsmooth eigenfunctions and that the fourth-order problem cannot be seen as the square of a second-order problem.
Reviewer: Yanqiong Lu (Lanzhou)Transcritical flow past an obstacle.https://www.zbmath.org/1452.760072021-02-12T15:23:00+00:00"Grimshaw, R."https://www.zbmath.org/authors/?q=ai:grimshaw.roger-h-jSummary: It is well known that transcritical flow past an obstacle may generate undular bores propagating away from the obstacle. This flow has been successfully modelled in the framework of the forced Korteweg-de Vries equation, where numerical simulations and asymptotic analyses have shown that the unsteady undular bores are connected by a locally steady solution over the obstacle. In this paper we present an overview of the underlying theory, together with some recent work on the case where the obstacle has a large width.Mathematical foundations of computational electromagnetism.https://www.zbmath.org/1452.780012021-02-12T15:23:00+00:00"Assous, Franck"https://www.zbmath.org/authors/?q=ai:assous.franck"Ciarlet, Patrick"https://www.zbmath.org/authors/?q=ai:ciarlet.patrick-jun"Labrunie, Simon"https://www.zbmath.org/authors/?q=ai:labrunie.simonThis remarkable monograph is devoted to the study of some problems in electrodynamics investigated by the authors. The main contribution of the authors is their ability to present the problems in an appropriate form for their practical solution. The book consists of ten chapters.
Chapter 1 presents the physics framework of electromagnetism, in relation to the Maxwell equations and some related approximations.
Chapter 2 is entitled ``Basic applied functional analysis''. The authors recall a number of definitions and results on Lebesgue and Sobolev spaces. Then, they introduce more specialized Sobolev spaces, which are better suited to measuring solutions to the divergence and the rotation of fields.
In Chapter 3, the authors complement the classic results of the previous chapter. First, they review some recent results on the traces of vector fields -- especially the tangential trace of electromagnetic-like fields. Then, they focus on the extraction of potentials of rot-free and /or divergence-free fields.
In Chapter 4, the known definitions and results from the theory of Banach and Hilbert spaces are given.
Chapter 5 is devoted to establishing mathematical properties concerning the electromagnetic fields that are governed by the time-dependent Maxwell equations. They focus on uniqueness, existence, continuous dependence with respect to the data and regularity in terms of Sobolev spaces.
In Chapter 6, the authors study the approximate models derived from Maxwell equations. They investigate the models from Chapter 1 and rely on the mathematical tools introduced in Chapters 2, 3 and 4.
Chapter 7 is devoted to an alternative, second-order formulation of the Maxwell's equations. The authors think that this new formulation is especially relevant for computational applications, as it admits several variational formulations, which can be simulated by versatile finite element methods.
In Chapter 8, the time-harmonic Maxwell equations are studied. They derive from the time-dependent equations by assuming that the time dependence of the data and fields is proportional to $e^{-j\omega t}$, where $\omega\geq0$ is the circular frequency. When $\omega\geq0$ is unknown, then the time-harmonic problem becomes a free vibration problem of the electromagnetic field. One has to solve an eigenvalue problem, for which both the fields and circular frequency are unknown. One can refer to this problem as a Helmholtz-like problem, for which the only unknown are the fields.
In Chapter 9, the authors consider some special cases in which the three-dimensional Maxwell equations can be reformulated as two-dimensional models. In this case, the computational domain boils down to a subset of $\mathbb R^2$, with respect to a suitable coordinate system.
In Chapter 10, the authors investigate the coupled models from Section 1.3, namely, the Vlasov-Poisson system, the Vlasov-Maxwell system and the magneto-hydrodynamics system. The variety of existence and uniqueness results for several types of solution are presented.
The monograph ends with an index of function spaces and references.
In conclusion, it can be said that the monograph provides useful mathematical tools for investigations of some problems for electromagnetic fields and their computational realizations.
Reviewer: Vasil G. Angelov (Sofia)Stabilisation of a viscoelastic flexible marine riser under unknown spatiotemporally varying disturbance.https://www.zbmath.org/1452.350932021-02-12T15:23:00+00:00"Berkani, Amirouche"https://www.zbmath.org/authors/?q=ai:berkani.amirouche"Tatar, Nasser-eddine"https://www.zbmath.org/authors/?q=ai:tatar.nasser-eddine"Seghour, Lamia"https://www.zbmath.org/authors/?q=ai:seghour.lamiaSummary: In this work, we study a viscoelastic flexible marine riser problem with vessel dynamics subject to a distributed disturbance. The dynamic of the problem is modelled as a viscoelastic Euler-Bernoulli beam structure. Based on the multiplier method and some ideas introduced by \textit{N. Tatar} [J. Contemp. Math. Anal., Armen. Acad. Sci. 48, No. 6, 285--296 (2013; Zbl 1302.35261)], we shall suppress the riser's vibration in a certain manner that we will determine. In fact, we prove uniform stability of the system for a large class of relaxation functions. Moreover, the relationship between the behaviour of the relaxation function and the decay rate of the energy is established. This improves earlier work where a control on the top of the structure has been imposed and the ocean disturbance was ignored.Interaction solutions for Kadomtsev-Petviashvili equation with variable coefficients.https://www.zbmath.org/1452.351732021-02-12T15:23:00+00:00"Liu, Jian-Guo"https://www.zbmath.org/authors/?q=ai:liu.jian-guo.1"Zhu, Wen-Hui"https://www.zbmath.org/authors/?q=ai:zhu.wenhui"Zhou, Li"https://www.zbmath.org/authors/?q=ai:zhou.liOptimal estimates for hyperbolic Poisson integrals of functions in \(L^p\) with \(p > 1\) and radial eigenfunctions of the hyperbolic Laplacian.https://www.zbmath.org/1452.350072021-02-12T15:23:00+00:00"Chen, Jiaolong"https://www.zbmath.org/authors/?q=ai:chen.jiaolong"Kalaj, David"https://www.zbmath.org/authors/?q=ai:kalaj.davidSummary: Assume that \(p \in (1 , \infty ]\) and \(u = P_h [ \phi ]\), where \(\phi \in L^p (\mathbb{S}^{n - 1} , \mathbb{R}^n )\). Then for any \(x \in \mathbb{B}^n\), we obtain the sharp inequalities
\[
| u (x) | \leq \frac{ \mathbf{C}_q^{\frac{ 1}{ q}} (x )}{ (1 - | x |^2 )^{\frac{ n - 1}{ p}}} \| \phi \|_{L^p} \;\;\text{ and } \;\;| u (x) | \leq \frac{ \mathbf{C}_q^{\frac{ 1}{ q}}}{ (1 - | x |^2 )^{\frac{ n - 1}{ p}}} \| \phi \|_{L^p}
\]
for some function \(\mathbf{C}_q (x )\) and constant \(\mathbf{C}_q\) in terms of Gauss hypergeometric and Gamma functions, where \(q\) is the conjugate of \(p\). This result generalizes and extends some known results from harmonic mapping theory [\textit{D. Kalaj} and \textit{M. Marković}, Positivity 16, No. 4, 771--782 (2012; Zbl 1255.31003), Theorems 1.1 and 1.2] and [\textit{S. Axler} et al., Harmonic function theory. New York: Springer-Verlag (1992; Zbl 0765.31001), Proposition 6.16)]. The proofs are mainly based on certain characterizations of the radial eigenfunctions of the hyperbolic Laplacian \(\Delta_h\), which are of independent interest.Diophantine tori and pragmatic calculation of quasimodes for operators with integrable principal symbol.https://www.zbmath.org/1452.370712021-02-12T15:23:00+00:00"Anikin, A. Yu."https://www.zbmath.org/authors/?q=ai:anikin.a-yu"Dobrokhotov, S. Yu."https://www.zbmath.org/authors/?q=ai:dobrokhotov.sergei-yuThe authors study the asymptotic eigenfunctions (quasimodes) of a semi-classical Weyl pseudo-differential operator. According to a well-known method (see [\textit{V. Ivrii}, Microlocal analysis and precise spectral asymptotics. Berlin: Springer (1998; Zbl 0906.35003)]), under the assumption of complete integrability for the Hamiltonian, the phase space is foliated into a family of invariant tori, so that one can study the quasimodes. The authors observe that such an approach is not very efficient for practical calculations and they propose a new method for operators with integrable principal symbol. As an example, they consider the case of a two weakly coupled oscillators. A Diophantine torus is then chosen and their algorithm can be applied. By using a mathematical software like Wolfram Mathematica, the quasimodes can be explicitly constructed.
Reviewer: Luigi Rodino (Torino)Sharp Hardy-Leray inequality for curl-free fields with a remainder term.https://www.zbmath.org/1452.260162021-02-12T15:23:00+00:00"Hamamoto, Naoki"https://www.zbmath.org/authors/?q=ai:hamamoto.naoki"Takahashi, Futoshi"https://www.zbmath.org/authors/?q=ai:takahashi.futoshiThe Hardy-Leray inequality for a smooth vector field \(\mathbf{u}:\mathbb{R}^N \to \mathbb{R}^N\) is given by \[C_{N,\gamma} \int_{\mathbb{R}^N} \frac{|\mathbf{u}|^2}{|\mathbf{x}|^2}|\mathbf{x}|^{2\gamma}dx \le \int_{\mathbb{R}^N}|\nabla\mathbf{u}|^2|\mathbf{x}|^{2\gamma}dx ,\] where the constant \(C_{N,\gamma}=\left(\gamma + \frac{N}{2}-1 \right)^2\) is known to be sharp. For axisymmetric divergence-free fields, \textit{O. Costin} and \textit{V. Maz'ya} [Calc. Var. Partial Differ. Equ. 32, No. 4, 523--532 (2008; Zbl 1147.35122)] proved that the constant \(C_{N,\gamma}\) can be improved and replaced by a larger one (the axisymmetric assumption was removed by the first author).
In the previous paper [``Sharp Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields'', Math. Ann. 379, No. 1--2, 719--742 (2021; \url{doi:10.1007/s00208-019-01945-x})], the authors prove a similar result for the case of curl-free fields. In the paper under review, the authors propose another proof of the Hardy-Leray inequality for curl-free fields with the best constant. Moreover, the authors obtain an improved inequality with a remainder term. The unattainability of the best constant is a consequence of the new inequality. The sharp Rellich-Leray inequality
\[R_{N,\gamma} \int_{\mathbb{R}^N} \frac{|\mathbf{u}|^2}{|\mathbf{x}|^4}|\mathbf{x}|^{2\gamma}dx \le \int_{\mathbb{R}^N}|\Delta\mathbf{u}|^2|\mathbf{x}|^{2\gamma}dx ,\]
for curl-free vector fields with sharp constant and remainder term is also proved in this paper.
The proof is based on a decomposition of curl-free fields into radial and spherical parts.
Reviewer: Michael Perelmuter (Kyjiw)Corrigendum to: ``Quasiconvex elastodynamics: weak-strong uniqueness for measure-valued solutions''.https://www.zbmath.org/1452.350992021-02-12T15:23:00+00:00"Koumatos, Konstantino"https://www.zbmath.org/authors/?q=ai:koumatos.konstantino"Spirito, Stefano"https://www.zbmath.org/authors/?q=ai:spirito.stefanoFrom the text: We correct a gap in Theorem 5.1 of the authors' paper [ibid. 72, No. 6, 1288--1320 (2019; Zbl 1437.35489)].Characteristic boundary layers for mixed hyperbolic-parabolic systems in one space dimension and applications to the Navier-Stokes and MHD equations.https://www.zbmath.org/1452.351432021-02-12T15:23:00+00:00"Bianchini, Stefano"https://www.zbmath.org/authors/?q=ai:bianchini.stefano"Spinolo, Laura V."https://www.zbmath.org/authors/?q=ai:spinolo.laura-vSummary: We provide a detailed analysis of the boundary layers for mixed hyperbolic-parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so-called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so-called \textit{doubly characteristic} case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolic-parabolic system and for the hyperbolic system obtained by neglecting the second-order terms. Our analysis applies in particular to the compressible Navier-Stokes and MHD equations in Eulerian coordinates, with both positive and null electrical resistivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to nonconservative systems.Vortex layers of small thickness.https://www.zbmath.org/1452.351462021-02-12T15:23:00+00:00"Caflisch, R. E."https://www.zbmath.org/authors/?q=ai:caflisch.russel-e"Lombardo, M. C."https://www.zbmath.org/authors/?q=ai:lombardo.maria-carmela"Sammartino, M. M. L."https://www.zbmath.org/authors/?q=ai:sammartino.marco-maria-luigiSummary: We consider a 2D vorticity configuration where vorticity is highly concentrated around a curve and exponentially decaying away from it: the intensity of the vorticity is \(O(1/ \varepsilon )\) on the curve while it decays on an \(O( \varepsilon )\) distance from the curve itself. We prove that, if the initial datum is of vortex-layer type, Euler solutions preserve this structure for a time that does not depend on \(\varepsilon \). Moreover, the motion of the center of the layer is well approximated by the Birkhoff-Rott equation.Analysis on lump, lumpoff and rogue waves with predictability to a generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation.https://www.zbmath.org/1452.351742021-02-12T15:23:00+00:00"Liu, Wen-Hao"https://www.zbmath.org/authors/?q=ai:liu.wenhao"Zhang, Yu-Feng"https://www.zbmath.org/authors/?q=ai:zhang.yu-feng"Shi, Dan-Dan"https://www.zbmath.org/authors/?q=ai:shi.dandanGlobal bifurcation of rotating vortex patches.https://www.zbmath.org/1452.760382021-02-12T15:23:00+00:00"Hassainia, Zineb"https://www.zbmath.org/authors/?q=ai:hassainia.zineb"Masmoudi, Nader"https://www.zbmath.org/authors/?q=ai:masmoudi.nader"Wheeler, Miles H."https://www.zbmath.org/authors/?q=ai:wheeler.miles-hSummary: We rigorously construct continuous curves of rotating vortex patch solutions to the two-dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [30, 38] of singular limiting patches with \(90^\circ\) corners at which the relative fluid velocity vanishes. For solutions close to the disk, we prove that there are ``cat's-eyes''-type structures in the flow, and provide numerical evidence that these structures persist along the entire solution curves and are related to the formation of corners. We also show, for any rotating vortex patch, that the boundary is analytic as soon as it is sufficiently regular.Partial data inverse problem with \(L^{n/2}\) potentials.https://www.zbmath.org/1452.352542021-02-12T15:23:00+00:00"Chung, Francis J."https://www.zbmath.org/authors/?q=ai:chung.francis-j"Tzou, Leo"https://www.zbmath.org/authors/?q=ai:tzou.leoIn this technical work, the authors give a partial data result for Calderón's problem with unbounded potentials. It is an injectivity result that applies to Schrödinger's equation with potential \(q \in L^{n/2}\) and thereby the scalar conductivity equation with \(\gamma \in W^{2, n/2}\) under assumptions of boundary determination. The result is based on an explicit Green's function for the conjugated Laplacian with several desirable properties. The proofs proceed by pseudodifferential calculus.
Reviewer: Tommi Brander (Trondheim)Interactions of lump and solitons to generalized \((2+1)\)-dimensional Ito systems.https://www.zbmath.org/1452.351622021-02-12T15:23:00+00:00"Du, Xuan"https://www.zbmath.org/authors/?q=ai:du.xuan"Lou, Sen-Yue"https://www.zbmath.org/authors/?q=ai:lou.senyueAlmost automorphically and almost periodically forced circle flows of almost periodic parabolic equations on \(S^1\).https://www.zbmath.org/1452.350782021-02-12T15:23:00+00:00"Shen, Wenxian"https://www.zbmath.org/authors/?q=ai:shen.wenxian"Wang, Yi"https://www.zbmath.org/authors/?q=ai:wang.yi.1"Zhou, Dun"https://www.zbmath.org/authors/?q=ai:zhou.dunSummary: We consider the skew-product semiflow which is generated by a scalar reaction-diffusion equation
\[
u_t=u_{xx}+f/t,u,u_x),\, t>0,\, x\in S^1=\mathbb{R}/2\pi\mathbb{Z},
\]
where \(f\) is uniformly almost periodic in \(t\). The structure of the minimal set \(M\) is thoroughly investigated under the assumption that the center space \(V^c(M)\) associated with \(M\) is no more than 2-dimensional. Such situation naturally occurs while, for instance, \(M\) is hyperbolic or uniquely ergodic. It is shown in this paper that \(M\) is a 1-cover of the hull \(H(f)\) provided that \(M\) is hyperbolic (equivalently, \( \text{dim}V^c(M)=0)\). If \(\text{dim}V^c(M)=1\) (resp. \( \text{dim}V^c(M)=2\) with \(\text{dim}V^u(M)\) being odd), then either \(M\) is an almost 1-cover of \(H(f)\) and topologically conjugate to a minimal flow in \(\mathbb{R}\times H(f)\); or \(M\) can be (resp. residually) embedded into an almost periodically (resp. almost automorphically) forced circle-flow \(S^1\times H(f)\). When \(f(t,u,u_x)=f(t,u,-u_x)\) (which includes the case \(f=f(t,u))\), it is proved that any minimal set \(M\) is an almost 1-cover of \(H(f)\). In particular, any hyperbolic minimal set \(M\) is a 1-cover of \(H(f)\). Furthermore, if \(\text{dim}V^c(M)=1\), then \(M\) is either a 1-cover of \(H(f)\) or is topologically conjugate to a minimal flow in \(\mathbb{R}\times H(f)\). For the general spatially-dependent nonlinearity \(f=f(t,x,u,u_x)\), we show that any stable or linearly stable minimal invariant set \(M\) is residually embedded into \(\mathbb{R}^2\times H(f)\).Nonconforming quadrilateral finite element method for nonlinear Kirchhoff-type equation with damping.https://www.zbmath.org/1452.652482021-02-12T15:23:00+00:00"Shi, Dongyang"https://www.zbmath.org/authors/?q=ai:shi.dongyang"Wu, Yanmi"https://www.zbmath.org/authors/?q=ai:wu.yanmiSummary: Nonconforming quadrilateral \(EQ_1^{rot}\) finite element method (FEM) of nonlinear Kirchhoff-type equation with damping is studied on anisotropic meshes. Based on the property of the nonlocal term of this equation, unconditional optimal error estimates of \(O(h)\) and \(O(h + \tau^2)\) (\(h\), the spatial parameter, and \(\tau \), the time step) in the broken \(H^1\) norm are deduced for the semidiscrete and a linearized fully discrete schemes without any restrictions of \(\tau\) through a distinct approach compared with the methods used for other partial differential equations, respectively. Besides, the damping term appearing in the Kirchhoff-type equation is solved with a novel technique, which is the major difficulty in the theoretical analysis. Finally, some numerical results are provided to verify the theoretical analysis.Local proportional-integral boundary feedback stabilization for quasilinear hyperbolic systems of balance laws.https://www.zbmath.org/1452.930322021-02-12T15:23:00+00:00"Zhang, Liguo"https://www.zbmath.org/authors/?q=ai:zhang.liguo"Prieur, Christophe"https://www.zbmath.org/authors/?q=ai:prieur.christophe"Qiao, Junfei"https://www.zbmath.org/authors/?q=ai:qiao.junfeiSuperconvergence error estimate of a finite element method on nonuniform time meshes for reaction-subdiffusion equations.https://www.zbmath.org/1452.652472021-02-12T15:23:00+00:00"Ren, Jincheng"https://www.zbmath.org/authors/?q=ai:ren.jincheng"Liao, Hong-lin"https://www.zbmath.org/authors/?q=ai:liao.honglin"Zhang, Zhimin"https://www.zbmath.org/authors/?q=ai:zhang.zhiminSummary: In this paper, we consider superconvergence error estimates of finite element method approximation of Caputo's time fractional reaction-subdiffusion equations under nonuniform time meshes. For the standard Galerkin method we see that the optimal order error estimate of temporal direction cannot be derived from the weak formulation of the problem. We establish a time-space error splitting argument, which are called the temporal error and the spatial error, respectively. The temporal error is proved skillfully based on an improved discrete Grönwall inequality. We obtain the sharp temporal \(H^1\)-norm error estimates with respect to the convergence order of the approximate solution and \(H^1\)-norm superclose results are given in details. Furthermore, by virtue of the interpolated postprocessing techniques, the global \(H^1\)-norm superconvergence results are presented. Finally, we present some numerical results that give insight into the reliability of the theoretical analysis.Error estimates for an immersed finite element method for second order hyperbolic equations in inhomogeneous media.https://www.zbmath.org/1452.652202021-02-12T15:23:00+00:00"Adjerid, Slimane"https://www.zbmath.org/authors/?q=ai:adjerid.slimane"Lin, Tao"https://www.zbmath.org/authors/?q=ai:lin.tao"Zhuang, Qiao"https://www.zbmath.org/authors/?q=ai:zhuang.qiaoSummary: A group of partially penalized immersed finite element (PPIFE) methods for second-order hyperbolic interface problems were discussed in [\textit{Q. Yang}, Numer. Math., Theory Methods Appl. 11, No. 2, 272--298 (2018; Zbl 1424.65173)] where the author proved their optimal \(O(h)\) convergence in an energy norm under a sub-optimal piecewise \(H^3\) regularity assumption. In this article, we reanalyze the fully discrete PPIFE method presented in [Yang, loc. cit.]. Utilizing the error bounds given recently in [\textit{R. Guo} et al., Int. J. Numer. Anal. Model. 16, No. 4, 575--589 (2019; Zbl 1427.65361)] for elliptic interface problems, we are able to derive optimal a-priori error bounds for this PPIFE method not only in the energy norm but also in \(L^2\) norm under the standard piecewise \(H^2\) regularity assumption in the space variable of the exact solution, rather than the excessive piecewise \(H^3\) regularity. Numerical simulations for standing and travelling waves are presented, which corroboratively confirm the reported error analysis.A kinetic model for epidemic spread.https://www.zbmath.org/1452.920402021-02-12T15:23:00+00:00"Pulvirenti, Mario"https://www.zbmath.org/authors/?q=ai:pulvirenti.mario"Simonella, Sergio"https://www.zbmath.org/authors/?q=ai:simonella.sergioSummary: We present a Boltzmann equation for mixtures of three species of particles reducing to the Kermack-McKendrick (SIR) equations for the time evolution of the density of infected agents in an isolated population. The kinetic model is potentially more detailed and might provide information on space mixing of the agents.Pushing the boundaries: models for the spatial spread of ecosystem engineers.https://www.zbmath.org/1452.352222021-02-12T15:23:00+00:00"Lutscher, Frithjof"https://www.zbmath.org/authors/?q=ai:lutscher.frithjof"Fink, Justus"https://www.zbmath.org/authors/?q=ai:fink.justus"Zhu, Yingjie"https://www.zbmath.org/authors/?q=ai:zhu.yingjieSummary: Ecosystems engineers are species that can substantially alter their abiotic environment and thereby enhance their population growth. The net growth rate of obligate engineers is even negative unless they modify the environment. We derive and analyze a model for the spread and invasion of such species. Prior to engineering, the landscape consists of unsuitable habitat; after engineering, the habitat is suitable. The boundary between the two types of habitat is moved by the species through their engineering activity. Our model is a novel type of a reaction-diffusion free boundary problem. We prove the existence of traveling waves and give upper and lower bounds for their speeds. We illustrate how the speed depends on individual movement and engineering behavior near the boundary.Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model.https://www.zbmath.org/1452.651802021-02-12T15:23:00+00:00"Wang, Jiangshan"https://www.zbmath.org/authors/?q=ai:wang.jiangshan"Meng, Lingxiong"https://www.zbmath.org/authors/?q=ai:meng.lingxiong"Jia, Hongen"https://www.zbmath.org/authors/?q=ai:jia.hongenSummary: In this paper, we construct a modular grad-div stabilization method for the Navier-Stokes/Darcy model, which is based on the first order Backward Euler scheme. This method does not enlarge the accuracy of numerical solution, but also can improve mass conservation and relax the influence of parameters. Herein, we give stability analysis and error estimations. Finally, by some numerical experiment, the scheme our proposed is shown to be valid.Suspension bridges with non-constant stiffness: bifurcation of periodic solutions.https://www.zbmath.org/1452.350122021-02-12T15:23:00+00:00"Holubová, Gabriela"https://www.zbmath.org/authors/?q=ai:holubova.gabriela"Janoušek, Jakub"https://www.zbmath.org/authors/?q=ai:janousek.jakubSummary: We consider a modified version of a suspension bridge model with a spatially variable stiffness parameter to reflect the discrete nature of the placement of the bridge hangers. We study the qualitative and quantitative properties of this model and compare the cases of constant and non-constant coefficients. In particular, we show that for certain values of the stiffness parameter, the bifurcation occurs. Moreover, we can expect also the appearance of blowups, whose existence is closely connected with the so-called Fučík spectrum of the corresponding linear operator.On well-posedness and large time behavior for smectic-a liquid crystals equations in \(\mathbb{R}^3\).https://www.zbmath.org/1452.351592021-02-12T15:23:00+00:00"Zhao, Xiaopeng"https://www.zbmath.org/authors/?q=ai:zhao.xiaopeng"Zhou, Yong"https://www.zbmath.org/authors/?q=ai:zhou.yong.1Summary: The main purpose of this manuscript is to study the well-posedness and decay estimates for strong solutions to the Cauchy problem of 3D smectic-A liquid crystals equations. First, applying Banach fixed point theorem, we prove the local existence and uniqueness of strong solutions. Then, by establishing some nontrivial estimates with energy method and a standard continuity argument, we prove that there exists a unique global strong solution provided that the initial data are sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solutions.Class of Neumann-type problems for the polyharmonic equation in a ball.https://www.zbmath.org/1452.310142021-02-12T15:23:00+00:00"Karachik, V. V."https://www.zbmath.org/authors/?q=ai:karachik.valery-vSummary: A set of necessary solvability conditions for the class \(\mathcal{N}_k\) of Neumann-type problems for the polyharmonic equation with a polynomial right-hand side in the unit ball is obtained. These conditions have the form of the orthogonality of homogeneous harmonic polynomials to linear combinations of boundary functions with coefficients from the Neumann integer triangle perturbed by certain derivatives of the right-hand side of the equation.Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach.https://www.zbmath.org/1452.352192021-02-12T15:23:00+00:00"Du, Zengji"https://www.zbmath.org/authors/?q=ai:du.zengji"Liu, Jiang"https://www.zbmath.org/authors/?q=ai:liu.jiang"Ren, Yulin"https://www.zbmath.org/authors/?q=ai:ren.yulinThe authors prove the existence of traveling wave solutions of a Keller-Segel system in one space dimension. This fully parabolic chemotaxis model features slow cell diffusion and a general production of a chemoattractant term. Methods of singular perturbation theory and topological tools such as the Poincaré-Bendixson theorem are used.
Reviewer: Piotr Biler (Wrocław)An improved car-following model accounting for the time-delayed velocity difference and backward looking effect.https://www.zbmath.org/1452.651692021-02-12T15:23:00+00:00"Ma, Guangyi"https://www.zbmath.org/authors/?q=ai:ma.guangyi"Ma, Minghui"https://www.zbmath.org/authors/?q=ai:ma.minghui"Liang, Shidong"https://www.zbmath.org/authors/?q=ai:liang.shidong"Wang, Yansong"https://www.zbmath.org/authors/?q=ai:wang.yansong"Zhang, Yaozong"https://www.zbmath.org/authors/?q=ai:zhang.yaozongSummary: In order to explore the impacts of the time-delayed velocity difference and backward looking effect on traffic flow, this paper proposes an improved car-following model based on the full velocity difference model (FVDM) by accounting for the time-delayed velocity difference and backward looking effect. The linear stability condition of the proposed model is derived by taking advantage of the linear stability theory. The time-dependent Ginzburg-Landau (TDGL) equation and the modified Korteweg-de Vries (mKdV) equation are established based on the nonlinear theory to describe the evolution of the traffic density waves near the critical stability point. Moreover, the link between the TDGL and mKdV equations is also provided. Finally, the results from both the numerical simulation and the theoretical analysis show that the proposed model can not only strengthen the stability of traffic flow, but also suppress the traffic congestion.Reducibility of Schrödinger equation at high frequencies.https://www.zbmath.org/1452.811072021-02-12T15:23:00+00:00"Sun, Yingte"https://www.zbmath.org/authors/?q=ai:sun.yingteSummary: In this paper, we prove a reducibility result for a linear Schrödinger equation with a time quasi-periodic perturbation on \(\mathbb{T} \). In contrast with previous reducibility results of the Schrödinger equation, the assumption of the small amplitude of the time quasi-periodic perturbation is replaced by fast oscillating.
{\copyright 2020 American Institute of Physics}Boundary value problems for elliptic pseudodifferential equations in a multidimensional cone.https://www.zbmath.org/1452.352702021-02-12T15:23:00+00:00"Vasil'ev, V. B."https://www.zbmath.org/authors/?q=ai:vasilyev.vladimir-bSummary: We consider model boundary value problems for elliptic pseudodifferential equations in multidimensional cones. A result on the unique solvability and representation of solutions of some boundary value problems in suitable Sobolev-Slobodetskii spaces is obtained. A priori estimates of solutions are given.On the linear Boltzmann equation in evolutionary domains with an absorbing boundary.https://www.zbmath.org/1452.351312021-02-12T15:23:00+00:00"Salvarani, Francesco"https://www.zbmath.org/authors/?q=ai:salvarani.francescoExplicit descriptions of spectral properties of Laplacians on spheres \(\mathbb{S}^N\) \((N\ge 1)\): a review.https://www.zbmath.org/1452.580022021-02-12T15:23:00+00:00"Awonusika, Richard Olu"https://www.zbmath.org/authors/?q=ai:awonusika.richard-oluSummary: In their remarkable paper, Minakshisundaram and Pleijel established by using the parametrix for the heat equation the asymptotic expansion of the heat kernel on compact Riemannian manifolds. The result has since been extensively used in the spectral analysis of the Laplace-Beltrami operator, and in particular, in proving Weyl's law for the asymptotic distribution of eigenvalues and various direct and inverse problems in spectral geometry. However, the question of describing the explicit values of the corresponding heat trace coefficients associated with an arbitrary compact Riemannian manifold has remained an interesting task. In this paper, we review results on Minakshisundaram-Pleijel coefficients associated with the Laplacian on spheres \(\mathbb{S}^N\) \((N\ge 1)\) and other associated spectral invariants, namely, the Minakshisundaram-Pleijel zeta functions \& their residues, and the zeta-regularised determinants of the Laplacian on spheres. The results reviewed deal mainly with closed-form formulae for the afore-mentioned spectral invariants and the explicit values of the first few of these spectral invariants are given.Numerical approximations for fractional elliptic equations via the method of semigroups.https://www.zbmath.org/1452.352372021-02-12T15:23:00+00:00"Cusimano, Nicole"https://www.zbmath.org/authors/?q=ai:cusimano.nicole"Del Teso, Félix"https://www.zbmath.org/authors/?q=ai:del-teso.felix"Gerardo-Giorda, Luca"https://www.zbmath.org/authors/?q=ai:gerardo-giorda.lucaWe provide a novel approach to the numerical solution of the family of nonlocal elliptic equations \((-\Delta)^s u = f\) in \(\Omega\), subject to some homogeneous boundary conditions \(B\) on \(\partial \Omega\), where \(s \in (0,1)\), \(\Omega \subset \mathbb R^n\) is a bounded domain, and \((-\Delta)^s\) is the spectral fractional Laplacian associated to \(B\) on \(\partial \Omega\). We use the solution representation \((-\Delta)^{-s}f\) together with its singular integral expression given by the method of semigroups. By combining finite element discretizations for the heat semigroup with monotone quadratures for the singular integral we obtain accurate numerical solutions. Roughly speaking, given a datum \(f\) in a suitable fractional Sobolev space of order \(r \geq 0\) and the discretization parameter \(h > 0\), our numerical scheme converges as \(O(h^{r+2s})\), providing super quadratic convergence rates up to \(O(h^4)\) for sufficiently regular data, or simply \(O(h^{2s})\) for merely \(f \in L^2(\Omega)\). We also extend the proposed framework to the case of nonhomogeneous boundary conditions and support our results with some illustrative numerical tests.
Fractional Calculus has attracted a great interest from mathematicians and scientists in various research studies in the fields of science and engineering. Many research studies have been conducted on investigating fractional-order derivatives or integrals and providing analytical, approximate-analytical, or numerical solutions for solving fractional differential equations arising from scientific and engineering phenomena (we refer to [\textit{K. Ryszewska}, J. Math. Anal. Appl. 483, No. 2, Article ID 123654, 17 p. (2020; Zbl 1436.35323); \textit{L. C. F. Ferreira} et al., Bull. Sci. Math. 153, 86--117 (2019; Zbl 1433.35185); \textit{S. D. Taliaferro}, J. Math. Pures Appl. (9) 133, 287--328 (2020; Zbl 1437.35697); \textit{T. Ghosh} et al., Anal. PDE 13, No. 2, 455--475 (2020; Zbl 1439.35530); \textit{H. Dong} and \textit{D. Kim}, J. Funct. Anal. 278, No. 3, Article ID 108338, 66 p. (2020; Zbl 1427.35316); \textit{A. Ghanmi} and \textit{Z. Zhang}, Bull. Korean Math. Soc. 56, No. 5, 1297--1314 (2019; Zbl 1432.34012); \textit{M. Kaabar}, ``Novel methods for solving the conformable wave equations'', J. New Theory 2020, No. 31, 56--85 (2020); \textit{M. K. A. Kaabar} et al., ``New approximate-analytical solutions for the nonlinear fractional Schrödinger equation with second-order spatio-temporal dispersion via double Laplace transform method'', Preprint, \url{arXiv:2010.10977}; \textit{F. Martínez} et al., ``Note on the conformable boundary value problems: Sturm's theorems and Green's function'', Preprint (2020; \url{doi:10.20944/preprints202009.0440.v1}); \textit{F. Martínez} et al., ``Some new results on conformable fractional power series'', Asia Pac. J. Math. 7, No. 31, 1--14 (2020); \textit{F. Martínez} et al., ``New results on complex conformable integral'', AIMS Math. 5, No. 6, 7695--7710 (2020); \textit{Y. Gholami} and \textit{K. Ghanbari}, S\(\vec{\text{e}}\)MA J. 75, No. 2, 305--333 (2018; Zbl 1400.26012)], and [\textit{F. Martínez} et al., ``Note on the conformable fractional derivatives and integrals of complex-valued functions of a real variable'', IAENG Int. J. Appl. Math. 50, No. 3, 609--615 (2020)]). In this paper, the authors propose a new and novel numerical technique, using a well-known method called the method of semigroups, to provide numerical solutions to the nonlocal elliptic equations. In other words, the studied nonlocal fractional Poisson problem in the authors' paper can be expressed as follows: Given \(s\in(0,1)\), bounded domain \(\Omega\subset\Re^{n}\), and a spectral fractional Laplacian, denoted by \((-\Delta)^{s}\), then the fractional elliptic problem that is subject to some homogeneous boundary conditions, denoted by \(B(u)\) can be written as:
\((-\Delta)^{s}u=f\) in \(\Omega\) such that \(B(u)=0\) on \(\partial \Omega\). Authors have investigated their proposed problem using the following boundary conditions such as Neumann, Dirichlet, and Robin. The authors have used the solution representation, denoted by \((-\Delta)^{-s}f\) with its singular integral representation using the method of semigroups. For simplicity purposes, the authors have successfully applied the discretization for the variable, \(t\), in integration. In additions, the authors have provided an extension of the proposed problem in this paper by investigating a modified version of the proposed fractional elliptic problem such that the proposed nonlocal operator is coupled with non-homogeneous boundary conditions to validate all obtained results and to conduct more related numerical experiments. Finally, the authors have provided the reader with many helpful comments and suggestions for further possible research work on some suggested open research problems. Therefore, I would highly recommend all interested mathematicians and scientists to read this excellent research work to conduct related research studies or possibly improve the proposed numerical approach in the near future.
Reviewer: Mohammed Kaabar (Gelugor)Bound states and the potential parameter spectrum.https://www.zbmath.org/1452.810952021-02-12T15:23:00+00:00"Alhaidari, A. D."https://www.zbmath.org/authors/?q=ai:alhaidari.abdulaziz-d"Bahlouli, H."https://www.zbmath.org/authors/?q=ai:bahlouli.hocineSummary: In this article, we answer the following question: If the wave equation possesses bound states, but it is exactly solvable for only a single non-zero energy, can we find all bound state solutions (energy spectrum and associated wavefunctions)? To answer this question, we use the ``tridiagonal representation approach'' to solve the wave equation at the given energy by expanding the wavefunction in a series of energy-dependent square integrable basis functions in configuration space. The expansion coefficients satisfy a three-term recursion relation, which is solved in terms of orthogonal polynomials. Depending on the selected energy, we show that one of the potential parameters must assume a value from within a discrete set called the ``potential parameter spectrum'' (PPS). This discrete set is obtained from the spectrum of the above polynomials and can be either a finite or an infinite set. Inverting the relation between the energy and the PPS gives the bound state energy spectrum. Therefore, the answer to the above question is affirmative.
{\copyright 2020 American Institute of Physics}Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms.https://www.zbmath.org/1452.350342021-02-12T15:23:00+00:00"Giga, Yoshikazu"https://www.zbmath.org/authors/?q=ai:giga.yoshikazu"Mitake, Hiroyoshi"https://www.zbmath.org/authors/?q=ai:mitake.hiroyoshi"Tran, Hung V."https://www.zbmath.org/authors/?q=ai:tran.hung-vinhSummary: We study a level-set mean curvature flow equation with driving and source terms, and establish convergence results on the asymptotic behavior of solutions as time goes to infinity under some additional assumptions. We also study the associated stationary problem in details in a particular case, and establish Alexandrov's theorem in two dimensions in the viscosity sense, which is of independent interest.Fast imaging of scattering obstacles from phaseless far-field measurements at a fixed frequency.https://www.zbmath.org/1452.653112021-02-12T15:23:00+00:00"Zhang, Bo"https://www.zbmath.org/authors/?q=ai:zhang.bo"Zhang, Haiwen"https://www.zbmath.org/authors/?q=ai:zhang.haiwenTransport coefficients in the \(2\)-dimensional Boltzmann equation.https://www.zbmath.org/1452.762112021-02-12T15:23:00+00:00"Bobylev, Alexander"https://www.zbmath.org/authors/?q=ai:bobylev.alexandre-vasiljevitch"Esposito, Raffaele"https://www.zbmath.org/authors/?q=ai:esposito.raffaeleSummary: We show that a rarefied system of hard disks in a plane, described in the Boltzmann-Grad limit by the 2-dimensional Boltzmann equation, has bounded transport coefficients. This is proved by showing opportune compactness properties of the gain part of the linearized Boltzmann operator.Natural Daftardar-Jafari method for solving fractional partial differential equations.https://www.zbmath.org/1452.652952021-02-12T15:23:00+00:00"Jafari, H."https://www.zbmath.org/authors/?q=ai:jafari.hamed-houri|jafari.hamideh|jafari.habib|jafari.hossein"Ncube, M. N."https://www.zbmath.org/authors/?q=ai:ncube.m-n"Makhubela, L."https://www.zbmath.org/authors/?q=ai:makhubela.lSummary: In this paper we introduce a new method, the natural Daftardar-Jafari method for solving fractional differential equations. This method is a combination of the natural transform and an iterative technique. The fractional derivative is considered in the Caputo sense.The regularized Boussinesq equation: Instability of periodic traveling waves.https://www.zbmath.org/1452.351562021-02-12T15:23:00+00:00"Pava, Jaime Angulo"https://www.zbmath.org/authors/?q=ai:angulo-pava.jaime"Banquet, Carlos"https://www.zbmath.org/authors/?q=ai:banquet-brango.carlos"Silva, Jorge Drumond"https://www.zbmath.org/authors/?q=ai:silva.jorge-drumond"Oliveira, Filipe"https://www.zbmath.org/authors/?q=ai:oliveira.filipe-a-aSummary: In this work we study the linear instability of periodic traveling waves associated with a generalization of the Regularized Boussinesq equation. By using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles is obtained. With respect to applications of this approach, we prove the linear/nonlinear instability of cnoidal wave solutions for the modified Regularized Boussinesq equation and for a system of two coupled Boussinesq equations.Operating conditions for the hemodialysis treatment based on the volume averaging theory.https://www.zbmath.org/1452.352242021-02-12T15:23:00+00:00"Sano, Yoshihiko"https://www.zbmath.org/authors/?q=ai:sano.yoshihikoSummary: The effect of operating conditions on the clearance of a countercurrent hollow fiber dialyzer has been investigated by utilizing the membrane transport model based on the volume averaging theory. The three-dimensional numerical method for describing the mass transport phenomena within a hollow fiber membrane dialyzer has been proposed to estimate performances under the several volume flow rates for blood and dialysate phases. Clearances obtained from the present numerical simulation are compared against available set of experimental data to elucidate the validity of the present three-dimensional numerical method. A series of calculations reveal the effect of the volume flow rate for blood and dialysate phases on urea clearance under the several total ultrafiltration rates. Moreover, the removal efficiency, which is the ratio of the mass flow rate of urea removed from the blood phase within a dialyzer to that at the blood phase inlet, is introduced in order to estimate an appropriate volume flow rate for blood and dialysate phases in the hemodialysis treatment. The present study clearly indicates that the present numerical method is quite useful for determining the best clinical protocol of the hemodialysis treatment and developing new dialysis systems such as home hemodialysis, nocturnal dialysis and even wearable artificial kidney.Rotating waves in a model of delayed feedback optical system with diffraction.https://www.zbmath.org/1452.352322021-02-12T15:23:00+00:00"Budzinskiy, Stanislav"https://www.zbmath.org/authors/?q=ai:budzinskiy.stanislav-sSummary: We study a delayed parabolic functional differential equation on a circle that is coupled with an initial value problem for the Schrodinger equation. Such equations arise as models of nonlinear optical systems with a time-delayed feedback loop, when diffusion of molecular excitation and diffraction are taken into account. The goal of this paper is to prove the existence of spatially inhomogeneous rotating-wave solutions bifurcating from homogeneous equilibria. We pass to a rotating coordinate system and seek an inhomogeneous solution to an ordinary functional differential equation. We find the solution in the form of a small parameter expansion and explicitly compute the first-order coefficients. We also provide examples of parameters that satisfy the constraints imposed throughout the analysis.Good Wannier bases in Hilbert modules associated to topological insulators.https://www.zbmath.org/1452.820242021-02-12T15:23:00+00:00"Ludewig, Matthias"https://www.zbmath.org/authors/?q=ai:ludewig.matthias"Thiang, Guo Chuan"https://www.zbmath.org/authors/?q=ai:thiang.guo-chuanSummary: For a large class of physically relevant operators on a manifold with discrete group action, we prove general results on the (non-)existence of a basis of well-localized Wannier functions for their spectral subspaces. This turns out to be equivalent to the freeness of a certain Hilbert module over the group \(C^*\)-algebra canonically associated with the spectral subspace. This brings into play \(K\)-theoretic methods and justifies their importance as invariants of topological insulators in physics.
{\copyright 2020 American Institute of Physics}On the steady equations for compressible radiative gas.https://www.zbmath.org/1452.762012021-02-12T15:23:00+00:00"Kreml, Ondřej"https://www.zbmath.org/authors/?q=ai:kreml.ondrej"Nečasová, Šárka"https://www.zbmath.org/authors/?q=ai:necasova.sarka"Pokorný, Milan"https://www.zbmath.org/authors/?q=ai:pokorny.milanSummary: We study the equations describing the steady flow of a compressible radiative gas with newtonian rheology. Under suitable assumptions on the data that include the physically relevant situations (i.e., the pressure law for monoatomic gas, the heat conductivity growing with square root of the temperature), we show the existence of a variational entropy solution to the corresponding system of partial differential equations. Under additional restrictions, we also show the existence of a weak solution to this problem.A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler law.https://www.zbmath.org/1452.920392021-02-12T15:23:00+00:00"Khan, Hasib"https://www.zbmath.org/authors/?q=ai:khan.hasib"Gómez-Aguilar, J. F."https://www.zbmath.org/authors/?q=ai:gomez-aguilar.jose-francisco"Alkhazzan, Abdulwasea"https://www.zbmath.org/authors/?q=ai:alkhazzan.abdulwasea"Khan, Aziz"https://www.zbmath.org/authors/?q=ai:khan.azizSummary: The biological models for the study of human immunodeficiency virus (HIV) and its advanced stage acquired immune deficiency syndrome (AIDS) have been widely studied in last two decades. HIV virus can be transmitted by different means including blood, semen, preseminal fluid, rectal fluid, breast milk, and many more. Therefore, initiating HIV treatment with the TB treatment development has some advantages including less HIV-related losses and an inferior risk of HIV spread also having difficulties including incidence of immune reconstitution inflammatory syndrome (IRIS) because of a large pill encumbrance. It has been analyzed that patients with HIV have more weaker immune system and are susceptible to infections, for example, tuberculosis (TB). Keeping the importance of the HIV models, we are interested to consider an analysis of HIV-TB coinfected model in the Atangana-Baleanu fractional differential form. The model is studied for the existence, uniqueness of solution, Hyers-Ulam (HU) stability and numerical simulations with assumption of specific parameters.Global existence and convergence rates of smooth solutions for the full compressible MHD equations.https://www.zbmath.org/1452.762712021-02-12T15:23:00+00:00"Pu, Xueke"https://www.zbmath.org/authors/?q=ai:pu.xueke"Guo, Boling"https://www.zbmath.org/authors/?q=ai:guo.bolingSummary: In this paper, we consider the global smooth solutions and their decay for the full compressible magnetohydrodynamic equations in \(R^3\). We prove the global existence of smooth solutions near the constant state in Sobolev norms by energy method and show the convergence rates of the \(L^p\)-norm of these solutions to the constant state when the \(L^q\)-norm of the perturbation is bounded.The asymptotic limits of Riemann solutions for the isentropic drift-flux model of compressible two-phase flows.https://www.zbmath.org/1452.351022021-02-12T15:23:00+00:00"Shen, Chun"https://www.zbmath.org/authors/?q=ai:shen.chunThe author considers the 1D Euler system for isentropic two-phase flow. The model is the drift-flux. The initial data are of Riemann type. The velocities of two phaces are equal. The mixture pressure is proportional to some power of the density sum with the factor \(k\). The author considers the limit case, with the factor \(k\to 0\), and proves that the limit of the solution is consistent with the solution of the limit system, which is weakly hyperbolic. The convergence is established in the sense of distributions. However, different types of solutions, i.e., vacuum state and delta-shock are shown to be possible.
Reviewer: Ilya A. Chernov (Petrozavodsk)Capillarity and Archimedes' principle.https://www.zbmath.org/1452.760372021-02-12T15:23:00+00:00"McCuan, John"https://www.zbmath.org/authors/?q=ai:mccuan.john"Treinen, Ray"https://www.zbmath.org/authors/?q=ai:treinen.raySummary: We consider some of the complications that arise in attempting to generalize a version of Archimedes' principle concerning floating bodies to account for capillary effects. The main result provides a means to relate the floating position (depth in the liquid) of a symmetrically floating sphere in terms of other observable geometric quantities.
A similar result is obtained for an idealized case corresponding to a symmetrically floating infinite cylinder.
These results depend on a definition of equilibrium for capillary systems with floating objects which to our knowledge has not formally appeared in the literature. The definition, in turn, depends on a variational formula for floating bodies which was derived in a special case earlier [the first author, Pac. J. Math. 231, No. 1, 167--191 (2007; Zbl 1148.76012)] and is here generalized to account for gravitational forces.
A formal application of our results is made to the problem of a ball floating in an infinite bath asymptotic to a prescribed level. We obtain existence and nonuniqueness results.Reconstruction of the initial state from the data measured on a sphere for plasma-acoustic wave equations.https://www.zbmath.org/1452.652042021-02-12T15:23:00+00:00"Bae, Junsik"https://www.zbmath.org/authors/?q=ai:bae.junsik"Kwon, Bongsuk"https://www.zbmath.org/authors/?q=ai:kwon.bongsuk"Moon, Sunghwan"https://www.zbmath.org/authors/?q=ai:moon.sunghwanExistence and concentration of ground state solution to a nonlocal Schrödinger equation.https://www.zbmath.org/1452.811012021-02-12T15:23:00+00:00"Mao, Anmin"https://www.zbmath.org/authors/?q=ai:mao.anmin"Zhang, Qian"https://www.zbmath.org/authors/?q=ai:zhang.qianSummary: We study a class of Schrödinger-Kirchhoff system involving a critical exponent. We aim to find suitable conditions to assure the existence of a positive ground state solution of Nehari-Pohožaev type \(u_\epsilon\) with exponential decay at infinity for \(\epsilon \), and \(u_\epsilon\) concentrates around a global minimum point of \(V\) as \(\epsilon \rightarrow 0^+\). The nonlinear term includes the nonlinearity \(f(u) \sim |u|^{p -1}u\) for the well-studied case \(p \in \) [3, 5) and the less-studied case \(p \in (2, 3)\).
{\copyright 2020 American Institute of Physics}Solving a system of nonlinear fractional partial differential equations using the sinc-Muntz collocation method.https://www.zbmath.org/1452.652652021-02-12T15:23:00+00:00"Ajeel, Mahmood Shareef"https://www.zbmath.org/authors/?q=ai:ajeel.mahmood-shareef"Gachpazan, Morteza"https://www.zbmath.org/authors/?q=ai:gachpazan.morteza"Soheili, Ali Reza"https://www.zbmath.org/authors/?q=ai:soheili.alireza|soheili.ali-rezaSummary: We present a new numerical method for solving a system of nonlinear fractional partial differential equations (SNFPDEs). This technique is based on the Sinc functions and the fractional Muntz-Legendre polynomials together with the collocation method. The proposed approximation reduces the solution of the SNFPDEs to the solution of a system of nonlinear algebraic equations. In some numerical examples, we show that approximate solutions also agree with exact solutions.Existence of solutions for a class of \(p(x)\)-curl systems arising in electromagnetism without (A-R) type conditions.https://www.zbmath.org/1452.352002021-02-12T15:23:00+00:00"Afrouzi, Ghasem A."https://www.zbmath.org/authors/?q=ai:afrouzi.ghasem-alizadeh"Chung, Nguyen Thanh"https://www.zbmath.org/authors/?q=ai:nguyen-thanh-chung."Naghizadeh, Z."https://www.zbmath.org/authors/?q=ai:naghizadeh.zohrehSummary: In this paper, we study the existence and multiplicity of solutions for a class of of \(p(x)\)-curl systems arising in electromagnetism. Under suitable conditions on the nonlinearities which do not satisfy Ambrosetti-Rabinowitz (A-R) type conditions, we obtain some existence and multiplicity results for the problem by using the mountain pass theorem and fountain theorem. Our main results in this paper complement and extend some earlier ones concerning the \(p(x)\)-curl operator in [\textit{A. Bahrouni} and \textit{D. Repovš}, Complex Var. Elliptic Equ. 63, No. 2, 292--301 (2018; Zbl 1423.35124) and [\textit{M. Xiang} et al., J. Math. Anal. Appl. 448, No. 2, 1600--1617 (2017; Zbl 1358.35181)].Optimal design by adaptive mesh refinement on shape optimization of flow fields considering proper orthogonal decomposition.https://www.zbmath.org/1452.652452021-02-12T15:23:00+00:00"Nakazawa, Takashi"https://www.zbmath.org/authors/?q=ai:nakazawa.takashi"Nakajima, Chihiro"https://www.zbmath.org/authors/?q=ai:nakajima.chihiro-hSummary: This paper presents optimal design using Adaptive Mesh Refinement (AMR) with shape optimization method. The method suppresses time periodic flows driven only by the non-stationary boundary condition at a sufficiently low Reynolds number using Snapshot Proper Orthogonal Decomposition (Snapshot POD). For shape optimization, the eigenvalue in Snapshot POD is defined as a cost function. The main problems are non-stationary Navier-Stokes problems and eigenvalue problems of POD. An objective functional is described using Lagrange multipliers and finite element method. Two-dimensional cavity flow with a disk-shaped isolated body is adopted. The non-stationary boundary condition is defined on the top boundary and non-slip boundary condition respectively for the side and bottom boundaries and for the disk boundary. For numerical demonstration, the disk boundary is used as the design boundary. Using \(H^1\) gradient method for domain deformation, all triangles over a mesh are deformed as the cost function decreases. To avoid decreasing the numerical accuracy based on squeezing triangles, AMR is applied throughout the shape optimization process to maintain numerical accuracy equal to that of a mesh in the initial domain. The combination of eigenvalues that can best suppress the time periodic flow is investigated.Generating admissible space-time meshes for moving domains in \((d + 1)\) dimensions.https://www.zbmath.org/1452.653532021-02-12T15:23:00+00:00"Neumüller, Martin"https://www.zbmath.org/authors/?q=ai:neumuller.martin"Karabelas, Elias"https://www.zbmath.org/authors/?q=ai:karabelas.elias"Neumüller, Martin"https://www.zbmath.org/authors/?q=ai:neumuller.martin"Karabelas, Elias"https://www.zbmath.org/authors/?q=ai:karabelas.eliasSummary: In this paper, we present a discontinuous Galerkin finite element method for the solution of the transient Stokes equations on moving domains. For the discretization, we use an interior penalty Galerkin approach in space, and an upwind technique in time. The method is based on a decomposition of the space-time cylinder into finite elements. Our focus lies on three-dimensional moving geometries, thus we need to triangulate four dimensional objects. For this, we will present an algorithm to generate \((d + 1)\)-dimensional simplex space-time meshes, and we show under natural assumptions that the resulting space-time meshes are admissible. Further, we will show how one can generate a four-dimensional object resolving the domain movement. First numerical results for the transient Stokes equations on triangulations generated with the newly developed meshing algorithm are presented.
For the entire collection see [Zbl 1425.65008].Thermal rectification based on phonon hydrodynamics and thermomass theory.https://www.zbmath.org/1452.800112021-02-12T15:23:00+00:00"Dong, Yuan"https://www.zbmath.org/authors/?q=ai:dong.yuanSummary: The thermal diode is the fundamental device for phononics. There are various mechanisms for thermal rectification, e.g. different temperature dependent thermal conductivity of two ends, asymmetric interfacial resistance, and nonlocal behavior of phonon transport in asymmetric structures. The phonon hydrodynamics and thermomass theory treat the heat conduction in a fluidic viewpoint. The phonon gas flowing through the media is characterized by the balance equation of momentum, like the Navier-Stokes equation for fluid mechanics. Generalized heat conduction law thereby contains the spatial acceleration (convection) term and the viscous (Laplacian) term. The viscous term predicts the size dependent thermal conductivity. Rectification appears due to the MFP supersession of phonons. The convection term also predicts rectification because of the inertia effect, like a gas passing through a nozzle or diffuser.Blow-up in a parabolic-elliptic Keller-Segel system with density-dependent sublinear sensitivity and logistic source.https://www.zbmath.org/1452.350492021-02-12T15:23:00+00:00"Tanaka, Yuya"https://www.zbmath.org/authors/?q=ai:tanaka.yuya"Yokota, Tomomi"https://www.zbmath.org/authors/?q=ai:yokota.tomomiThis paper considers the parabolic-elliptic Keller-Segel system with density-dependent sublinear sensitivity and logistic source:
\[
\begin{cases}
u_{t}=\Delta u- \chi \nabla \cdot (u(u+1)^{\alpha-1} \nabla v)+ \lambda u- \mu u^{\kappa}, & x\in \Omega,\, t>0, \\
0= \Delta v-v +u, & x\in \Omega,\, t>0,
\end{cases}
\]
where \(\Omega:= B_{R}(0)\subset \mathbb{R}^{n}\) \((n\geq 3)\) is a ball with some \(R>0\) and \(\chi>0\), \(0<\alpha<1\), \(\lambda\in R\), \(\mu>0\) and \(\kappa>1\).
The main contribution of this paper is to find conditions for \(\alpha\) and \(\kappa\) such that there exist solutions that blow up in finite time in the case of weak-chemotactic sensitivity, that is, in the case \(0<\alpha<1\). The authors extend the results of \textit{M. Winkler} [Z. Angew. Math. Phys. 69, No. 2, Paper No. 40, 25 p. (2018; Zbl 1395.35048)] in which discovered the conditions for \(\alpha=1\) and \(\kappa>1\). In the proof of the main results, the authors establish a key differential inequality (see Lemma 4.12).
The proof of the theorem is delicate and seem to be right. This is a great work. The paper is well organized with a complete list of relevant references. The presentation of the paper is very clear. No typos was detected. I think it is a good paper.
However, the case is unknown whether the parabolic-parabolic Keller-Segel system with some same conditions appears blow up or not. This will arouse more interest to relevant researchers to study the above system extensively.
Reviewer: Neng Zhu (Nanchang)Random-field solutions of weakly hyperbolic stochastic partial differential equations with polynomially bounded coefficients.https://www.zbmath.org/1452.352632021-02-12T15:23:00+00:00"Ascanelli, Alessia"https://www.zbmath.org/authors/?q=ai:ascanelli.alessia"Coriasco, Sandro"https://www.zbmath.org/authors/?q=ai:coriasco.sandro"Süss, André"https://www.zbmath.org/authors/?q=ai:suss.andreThe authors consider a weakly hyperbolic stochastic equation. The involved linear partial differential equation is of SG type, cf. [\textit{C. Parenti}, Ann. Mat. Pura Appl. (4) 93, 359--389 (1972; Zbl 0291.35070)], hyperbolic with characteristic of constant multiplicity and satisfying the SG Levi condition, cf. [\textit{S. Coriasco} and \textit{L. Rodino}, Ric. Mat. 48, 25--43 (1999; Zbl 0935.35106)]. The random noise appearing in the right-hand side of the equation safisfies suitable conditions, the spectral measure being related to the multiplicity of the characteristic (the larger is the multiplicity, the smallest is the class of the allowed stochastic noises). Under such assumptions, the authors obtain the existence of a random-field solution. The proof consists of different steps. At the beginning, after factorization of the operator, the problem is reduced to an equivalent first-order system. Then, a fundamental solution is constructed for the system, by using the theory of the SG Fourier integral operators of \textit{S. Coriasco} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 44, 81--122 (1998; Zbl 0958.35160)]. Finally, one is reduced to the evaluation of stochastic and deterministic integrals, that in view of the hypotheses turn out to be well defined and provide the solution. This relevant paper gives evidence that the SG equations are a natural setting for the study of the hyperbolic stochastic problems.
Reviewer: Luigi Rodino (Torino)Boundedness in a haptotactic cross-diffusion system modeling oncolytic virotherapy.https://www.zbmath.org/1452.350772021-02-12T15:23:00+00:00"Li, Jing"https://www.zbmath.org/authors/?q=ai:li.jing.13"Wang, Yifu"https://www.zbmath.org/authors/?q=ai:wang.yifuThe authors present a cross-diffusion model describing an oncolytic virotherapy, which takes into account the influence of the extracellular matrix taxis over the tumor-oncolytic virus interaction in the form of haptotaxis of both cancer cells and oncolytic virus. Under some suitable assumptions on the system parameters, the authors succeed in proving the global boundedness of solutions to an associated spatially two-dimensional initial-boundary value problem.
Reviewer: Sofiane El-Hadi Miri (Tlemcen)Energy stable numerical schemes for the fractional-in-space Cahn-Hilliard equation.https://www.zbmath.org/1452.651512021-02-12T15:23:00+00:00"Bu, Linlin"https://www.zbmath.org/authors/?q=ai:bu.linlin"Mei, Liquan"https://www.zbmath.org/authors/?q=ai:mei.liquan"Wang, Ying"https://www.zbmath.org/authors/?q=ai:wang.ying.3|wang.ying.1|wang.ying.2|wang.ying.4|wang.ying|wang.ying.8|wang.ying.6"Hou, Yan"https://www.zbmath.org/authors/?q=ai:hou.yanSummary: In this paper, a number of energy stable numerical schemes are proposed for the fractional Cahn-Hilliard equation. We prove mass conservation, unique solvability and energy stability for three time semi-discretized schemes based on the first-order semi-implicit scheme, the Crank-Nicolson scheme and the BDF2 scheme respectively. Then we present error analysis for these numerical schemes with the Fourier spectral approximation in space. Some numerical experiments are finally carried out to confirm accuracy and effectiveness of these proposed methods.A POD-based reduced-order Crank-Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation.https://www.zbmath.org/1452.651452021-02-12T15:23:00+00:00"Abbaszadeh, Mostafa"https://www.zbmath.org/authors/?q=ai:abbaszadeh.mostafa"Dehghan, Mehdi"https://www.zbmath.org/authors/?q=ai:dehghan.mehdiSummary: This paper introduces a high-order numerical procedure to solve the two-dimensional distributed-order Riesz space-fractional diffusion equation. In the proposed technique, first, a second-order numerical integration rule is employed to estimate the integral of the distributed-order Riesz space-fractional derivative. Then, the time derivative is discretized by a second-order difference scheme. Finally, the spatial direction is approximated by a difference formulation with fourth-order accuracy. The stability of the semi-discrete scheme is analyzed. We conclude that the difference between two consecutive time steps i.e. \(U_{i, j}^n - U_{i, j}^{n - 1}\) is nearly zero when \(n \to \infty\). So, a suitable term is added to the main difference scheme as by adding this term we could derive the main ADI scheme. Furthermore, to reduce the used CPU time, we combine the fourth-order ADI formulation with the proper orthogonal decomposition method and then we gain a POD based reduced-order compact ADI finite difference plane. In the next, the convergence order of the fully discrete formulation has been investigated. The numerical results show the efficiency of new technique. It must be noted that the finite difference method is an effective and robust numerical technique for solving nonlinear equations that the ADI approach can be combined with it to improve the numerical simulations.Uniqueness and reconstruction for the fractional Calderón problem with a single measurement.https://www.zbmath.org/1452.352552021-02-12T15:23:00+00:00"Ghosh, Tuhin"https://www.zbmath.org/authors/?q=ai:ghosh.tuhin"Rüland, Angkana"https://www.zbmath.org/authors/?q=ai:ruland.angkana"Salo, Mikko"https://www.zbmath.org/authors/?q=ai:salo.mikko"Uhlmann, Gunther"https://www.zbmath.org/authors/?q=ai:uhlmann.gunther-aIn this interesting paper under review, the authors show the global uniqueness in an inverse problem for the fractional Schrödinger equation with a single measurement and provide a reconstruction algorithm.
More precisely, let \(\Omega\subset \mathbb{R}^n\) be a bounded open set and \(\Omega_e=\mathbb{R}^n\setminus \overline{\Omega}\) be the exterior domain. Let \(\Lambda_q\) be the Dirichlet-to-Neumann map given by
\[\Lambda_q: f\mapsto (-\Delta)^s u \big|_{\Omega_e},\]
where \(u\) and \(f\) are related through the fractional Schrödinger equation
\[ ((-\Delta)^s +q) u=0 \text{ in } \Omega, \quad u=f \text{ in } \Omega_e,\]
with the potential \(q\) chosen such that zero is not an eigenvalue. Assume that either \(s\in [\frac{1}{4},1)\) and \(q\in L^\infty(\Omega)\) or \(q\in C^0(\overline{\Omega})\). The authors show that, given any nonempty open sets \(W_1, W_2\subset \Omega_e\) with \(\overline{\Omega}\cap W_1=\emptyset\), and given any nonzero function \(f\) supported in \(W_1\), the potential \(q\) is uniquely determined by the single function \(f\) and the single measurement \(\Lambda_q(f)\big|_{W_2}\). Moreover, an explicit reconstruction procedure to recover the potential \(q\) from \(f\) and \(\Lambda_q(f)\) is given. Note that the previous work [\textit{T. Ghosh} et al., Anal. PDE 13, No. 2, 455--475 (2020; Zbl 1439.35530)] considers infinitely many measurements, where one knows \(\Lambda_q(f)\big|_{W_2}\) for all \(f\in C_c^\infty(W_1)\).
The proof is based on the strong approximation properties for the fractional Schrödinger equation and a unique continuation principle from sets of measure zero.
Reviewer: Wenhui Shi (Melbourne)A difference scheme for the time-fractional diffusion equation on a metric star graph.https://www.zbmath.org/1452.651712021-02-12T15:23:00+00:00"Mehandiratta, Vaibhav"https://www.zbmath.org/authors/?q=ai:mehandiratta.vaibhav"Mehra, Mani"https://www.zbmath.org/authors/?q=ai:mehra.maniSummary: In this paper, we propose an unconditionally stable numerical scheme based on finite difference for the approximation of time-fractional diffusion equation on a metric star graph. The fractional derivative is considered in Caputo sense and the so-called \(L1\) method is used for the discrete approximation of Caputo fractional derivative. The convergence and stability of the difference scheme has been proved by means of energy method. Test examples are illustrated in order to verify the feasibility of the proposed scheme.Continuity of the spectrum of quasi-periodic Schrödinger operators with finitely differentiable potentials.https://www.zbmath.org/1452.370382021-02-12T15:23:00+00:00"Zhao, Xin"https://www.zbmath.org/authors/?q=ai:zhao.xinAuthor's abstract: In this paper, we consider the spectrum of discrete quasi-periodic Schrödinger operators on \(\ell^2({\mathbb Z})\) with the potentials \(v\in C^k({\mathbb T})\). For sufficiently large \(k\), we show that the Lebesgue measure of the spectrum at irrational frequencies is the limit of the Lebesgue measure of the spectrum of its periodic approximants. This gives a partial answer to the problem proposed in [\textit{S. Jitomirskaya} and \textit{R. Mavi}, Commun. Math. Phys. 325, No. 2, 585--601 (2014; Zbl 1323.47037)]. Our results are based on a generalization of the rigidity theorem in [\textit{A. Avila} and \textit{R. Krikorian}, Ann. Math. (2) 164, No. 3, 911--940 (2006; Zbl 1138.47033)]; more precisely, we prove that in the \(C^k\) case, for almost every frequency \(\alpha \in {\mathbb R}\setminus {\mathbb Q}\) and for almost every \(E\), the corresponding quasi-periodic Schrödinger cocycles are either reducible or non-uniformly hyperbolic.
Reviewer: Meirong Zhang (Beijing)A semi-analytical approach to Caputo type time-fractional modified anomalous sub-diffusion equations.https://www.zbmath.org/1452.652752021-02-12T15:23:00+00:00"Kheybari, Samad"https://www.zbmath.org/authors/?q=ai:kheybari.samad"Darvishi, Mohammad Taghi"https://www.zbmath.org/authors/?q=ai:darvishi.mohammad-taghi"Hashemi, Mir Sajjad"https://www.zbmath.org/authors/?q=ai:hashemi.mir-sajjadSummary: This article is devoted to a new semi-analytical algorithm for solving time-fractional modified anomalous sub-diffusion equations (FMASDEs). In this method first, the main problem is reduced to a system of fractional-order ordinary differential equations (FODEs) under known initial value conditions by using the Chebyshev collocation procedure. After that, to solve this system, some auxiliary initial value problems are defined. Next, we find an optimal linear combination of some particular solutions for these problems and finally we use this linear combination to construct a semi-analytical approximate solution for the main problem. To demonstrate the convergence property of the new method, a residual error analysis is performed in details. Some test problems are investigated to show reliability and accuracy of the proposed method. Besides, convergence order's indicators are evaluated for all test problems and are compared with ones of the other methods. Moreover, a comparison between our computed numerical results and the reported results of the other numerical schemes in the literature exhibits that the proposed technique is more precise and reliable.
In summary advantages of the proposed method are: high accuracy, easy programming, high experimental convergence order, and solving another types of fractional differential equations.Numerical analysis on the mortar spectral element methods for Schrödinger eigenvalue problem with an inverse square potential.https://www.zbmath.org/1452.653162021-02-12T15:23:00+00:00"Jia, Lueling"https://www.zbmath.org/authors/?q=ai:jia.lueling"Li, Huiyuan"https://www.zbmath.org/authors/?q=ai:li.huiyuan"Zhang, Zhimin"https://www.zbmath.org/authors/?q=ai:zhang.zhiminSummary: In this paper, we present an \(hp\) analysis of the mortar spectral element method for the Schrödinger eigenvalue problem \((- \Delta + \frac{c^2}{\|x\|^2}) u = \lambda u\), and thereby justify the numerical findings in [\textit{H. Li} and \textit{Z. Zhang}, SIAM J. Sci. Comput. 39, No. 1, A114--A140 (2017; Zbl 1355.65150)], where the method was demonstrated to be efficient to handle the singularities arising from both the inverse square potential and the reentrant/obtuse corners with exponential order of convergence. Non-uniformly weighted Sobolev spaces are introduced to accommodate singularities and to measure the regularity of the eigenfunctions. Optimal error estimates for the mortar spectral element method and the lifting theorem for the eigenfunctions are established.Low regularity well-posedness for the Yang-Mills system in Fourier-Lebesgue spaces.https://www.zbmath.org/1452.351612021-02-12T15:23:00+00:00"Pecher, Hartmut"https://www.zbmath.org/authors/?q=ai:pecher.hartmutLet \(g\) be either \(so(n, \mathbb{R})\) (the algebra of trace-free skew symmetric matrices) or \(su(n,\mathbb{R})\) (the algebra of all trace free Hermitian matrices) and let \([X,Y] = XY - YX\) be the matrix commutator. Define for a given \(A_\alpha: \mathbb{R}^{1+n} \rightarrow g\) the curvature \(F=F[A]\) by
\[
F_{\alpha \beta} = \partial_\alpha A_\beta - \partial_\beta A_\alpha + [A_\alpha, A_\beta],
\]
where \(\alpha, \beta \in \{0,1,\dots,n\}\), and set \(D_\alpha := \partial_\alpha +[A_\alpha, \cdot]\). Then the Yang-Mills system is
\[
D^\alpha F_{\alpha \beta} = 0
\]
and it is a system of partial differential equations in the Minkowski space \(\mathbb{R}^{1+n}=\mathbb{R}_t \times \mathbb{R}_x^n\). For the case of \(n=3\) and Lorenz gauge (\(\partial^\alpha A_\alpha=0\)), the locally well-posedness problem associated to the initial value problem for the Yang-Mills system is studied. For this purpose, the Yang-Mills system is reduced to a nonlinear wave equation subject to initial conditions
\[
A(0)=a, \quad \partial_t A(0)=\dot a, \quad F(0)=f, \quad \partial_t F(0)=\dot f.
\]
Let \(1 < r \leq 2\) and \(\delta>0\) be fixed and set for a small \(\epsilon>0\)
\[
s= \frac{16}{7r}-\frac{2}{7} + \delta, \quad l=\frac{15}{7r} - \frac{8}{7} + \delta, \quad a = \frac{1}{r} + \epsilon, \quad b=\frac{1}{2}+\frac{1}{2r}+\epsilon.
\]
The main result of the paper states that, for given initial data \((a,\dot a)\in \widehat{H}^{s,r}\times \widehat{H}^{s-1,r}\) and \((f,\dot f)\in \widehat{H}^{l,r}\times \widehat{H}^{l-1,r}\), there exists a time \(T\) such that the Cauchy problem for the reduced nonlinear wave equation has a unique solution \(A_\mu \in X^r_{s,b,+}[0,T]+X^r_{s,b,-}[0,T]\) and \(F \in X^r_{l,a,+}[0,T]+X^r_{l,a,-}[0,T]\), where \(\widehat{H}^{s,r}\) and \(X^r_{l,a,+}[0,T]\) denote Fourier-Lebesgue spaces defined in the paper. This result implies that there exists also a solution for the Cauchy problem for the original Yang-Mills system. It is remarkable that the result holds for \(r \rightarrow 1\), which was an open problem in the field and which gives almost optimal parameters with respect to scaling.
Reviewer: Markus Holzmann (Graz)Dynamic programming principle and Hamilton-Jacobi-Bellman equations for fractional-order systems.https://www.zbmath.org/1452.490172021-02-12T15:23:00+00:00"Gomoyunov, Mikhail I."https://www.zbmath.org/authors/?q=ai:gomoyunov.mikhail-igorevichHomogenization of singular elliptic systems with nonlinear conditions on the interfaces.https://www.zbmath.org/1452.350162021-02-12T15:23:00+00:00"Amar, M."https://www.zbmath.org/authors/?q=ai:amar.micol"Riey, G."https://www.zbmath.org/authors/?q=ai:riey.giuseppeThe authors investigate the homogenization of singular elliptic systems with nonlinear conditions on the interfaces. They consider the heat conduction in a composite with two finely mixed phases having a periodic active interface and a singular source. Moreover, they assume that the heat flow across the interface is related to the temperature jump through the interface by means on of a nonlinear relation. More precisely, the fix a region \(\Omega\) in \(\mathbb{R}^N\) consisting of the two phases \(\Omega^\epsilon_1=\Omega \cap \epsilon E\) and \(\Omega^\epsilon_2=\Omega \setminus \overline{\Omega_1^\epsilon}\), separated by the interface \(\Gamma^\epsilon\), where \(E\) is a periodic open subset of \(\mathbb{R}^N\). Then they consider the problem
\[
\begin{array}{ll}
-\mathrm{div}(\lambda_1 \nabla u_\epsilon)=f/u^\theta_\epsilon & \mathrm{in}\ \Omega_1^\epsilon\, , \\
-\mathrm{div}(\lambda_2 \nabla u_\epsilon)=f/u^\theta_\epsilon & \mathrm{in}\ \Omega_2^\epsilon\, , \\
\lambda_1 \nabla u_\epsilon\cdot \nu=\lambda_2 \nabla u_\epsilon\cdot \nu & \mathrm{on}\ \Gamma^\epsilon\, , \\
\frac{1}{\epsilon^{1-k}}g\Bigg(\frac{[u_\epsilon]}{\epsilon^k}\Bigg)=\lambda_2 \nabla u_\epsilon\cdot \nu_\epsilon & \mathrm{on}\ \Gamma^\epsilon\, , \\
u_\epsilon>0 & \mathrm{in}\ \Omega\, , \\
u_\epsilon=0 & \mathrm{on}\ \partial \Omega\, ,
\end{array}
\]
where \(\theta \in (0,1)\), \(k \in \{0,1\}\), \(\lambda_1, \lambda_2 \in (0,+\infty)\), \([u_\epsilon]\) is the jump of \(u_\epsilon\) across the interface, \(\nu_\epsilon\) is the normal unit vector to \(\Gamma^\epsilon\) pointing into \(\Omega_\epsilon^2\).
The authors prove an existence and uniqueness theorem and a homogenization result via two-scale homogenization.
Reviewer: Paolo Musolino (Padova)A probabilistic look at conservative growth-fragmentation equations.https://www.zbmath.org/1452.600442021-02-12T15:23:00+00:00"Bouguet, Florian"https://www.zbmath.org/authors/?q=ai:bouguet.florianSummary: In this note, we consider general growth-fragmentation equations from a probabilistic point of view. Using Foster-Lyapunov techniques, we study the recurrence of the associated Markov process depending on the growth and fragmentation rates. We prove the existence and uniqueness of its stationary distribution, and we are able to derive precise bounds for its tails in the neighborhoods of both 0 and \(+ \infty \). This study is systematically compared to the results obtained so far in the literature for this class of integro-differential equations.
For the entire collection see [Zbl 1402.60004].Asymptotic dynamics on a chemotaxis-Navier-Stokes system with nonlinear diffusion and inhomogeneous boundary conditions.https://www.zbmath.org/1452.352282021-02-12T15:23:00+00:00"Wu, Chunyan"https://www.zbmath.org/authors/?q=ai:wu.chunyan"Xiang, Zhaoyin"https://www.zbmath.org/authors/?q=ai:xiang.zhaoyinA system describing nonlinear diffusion of cells in a viscous incompressible fluid is studied in two-dimensional domains.
Oxygen exchange through the fluid and air boundary, and oxygen influence on cells movement are modelled by chemotactic terms supplemented with nonhomogeneous boundary conditions. Sometimes, such a system is called aerotaxis-fluid model.
In the case of the porous medium like diffusion \(\Delta n^m\) with \(m>1\), the main results of the paper concern global-in-time existence of solutions and asymptotic stabilization to the unique spatial equilibrium solution.
Reviewer: Piotr Biler (Wrocław)A brief survey of FJRW theory.https://www.zbmath.org/1452.140562021-02-12T15:23:00+00:00"Francis, Amanda E."https://www.zbmath.org/authors/?q=ai:francis.amanda-e"Jarvis, Tyler J."https://www.zbmath.org/authors/?q=ai:jarvis.tyler-j"Priddis, Nathan"https://www.zbmath.org/authors/?q=ai:priddis.nathanSummary: In this paper we describe some of the constructions of FJRW theory. We also briefly describe its relation to Saito-Givental theory via Landau-Ginzburg mirror symmetry and its relation to Gromov-Witten theory via the Landau-Ginzburg/Calabi-Yau correspondence. We conclude with a discussion of some of the recent results in the field, including the gauged linear sigma model, which is expected to provide a geometric framework for unifying many of these ideas.
For the entire collection see [Zbl 1446.53004].Global radial renormalized solution to a producer-scrounger model with singular sensitivities.https://www.zbmath.org/1452.352152021-02-12T15:23:00+00:00"Cao, Xinru"https://www.zbmath.org/authors/?q=ai:cao.xinruA chemotaxis-like system with logarithmic sensitivity functions for both species is considered to model producer-scrounger competition for a consumable resource. Radially symmetric solutions of the initial-boundary value problem in the ball of \(\mathbb R^n\), \(n\ge 2\), defined globally in time are constructed as renormalized solutions trough suitable approximation process.
Reviewer: Piotr Biler (Wrocław)Nodal solutions for an elliptic equation in an annulus without the signum condition.https://www.zbmath.org/1452.340342021-02-12T15:23:00+00:00"Chen, Tianlan"https://www.zbmath.org/authors/?q=ai:chen.tianlan"Lu, Yanqiong"https://www.zbmath.org/authors/?q=ai:lu.yanqiong"Ma, Ruyun"https://www.zbmath.org/authors/?q=ai:ma.ruyunIn fact, this paper treats basically the boundary value problem
\[u''+\lambda q(t)f(t,u)=0,\;t\in(0,1),\]
\[u(0)=0=u(1),\]
where \(\lambda>0\) is a real parameter, \(q(t)\) is continuous and bounded between positive constants on \([0,1]\), and \(f:[0,1]\times \mathbb{R}\to \mathbb{R}\) is continuous.
Applying global bifurcation techniques, the authors establish the values of the parameter \(\lambda\) which ensure the existence and multiplicity of nodal solutions. The results rely on combinations of three ((A1), (A3) and (A4) or (A2), (A3) and (A4)) or four ((A1)--(A3) and (A5)) assumptions from the following ones:
\begin{itemize}
\item[(A1)] There exists a continuous and concave function \(s_1:[0,1]\to(0,+\infty)\) such that \(f(t,s_1(t))=0=f(t,0)\) and \(sf(t,s)>0\) for \(s\in\mathbb{R}\setminus\{0,s_1(t)\}\).
\item[(A2)] There exists a continuous and convex function \(s_2:[0,1]\to(-\infty,0)\) such that \(f(t,s_2(t))=0=f(t,0)\) and \(sf(t,s)>0\) for \(s\in\mathbb{R}\setminus\{0,s_2(t)\}\).
\item[(A3)] There exists a continuous function \(a:[0,1]\to(0,+\infty)\) such that \(\lim_{|s|\to0}\frac{f(t,s)}{s}=a(t)\) uniformly on \([0,1].\)
\item[(A4)] There exist positive constants \(0<\alpha<\beta<1\) such that \(\lim_{|s|\to\infty}\frac{f(t,s)}{s}=+\infty\) uniformly on \([\alpha,\beta].\)
\item[(A5)] There exists a continuous function \(b:[0,1]\to(0,\infty)\) such that \(\lim_{|s|\to\infty}\frac{f(t,s)}{s}=b(t)\) uniformly on \([0,1]\).
\end{itemize}
As the authors note, after changing the variables
\[t=-\frac{A}{r^{n-2}}+B\;\mbox{and}\;u(t)=v(r)\;\mbox{if}\;n\geq3,\;n\in\mathbb{N},\]
or
\[r=r_2(\frac{r_1}{r_2})^{^t}\mbox{and}\;u(t)=v(r)\;\mbox{if}\;n=2,\]
where \(r_2>r_1>0\), \(A=\frac{(r_1r_2)^{n-2}}{r_2^{n-2}-r_1^{n-2}}\) and \(B=\frac{r_2^{n-2}}{r_2^{n-2}-r_1^{n-2}},\)
the obtained results for the above BVP may be applied for studying the existence and multiplicity of radial nodal solutions of the semilinear elliptic problem
\[-\Delta v=\lambda h(x,v)\;\mbox{in}\;\Omega,\]
\[v=0\;\mbox{on}\;\delta\Omega,\]
where \(\Omega=\{x\in \mathbb R^n:r_1<|x|<r_2\}\), \(n\geq2\).
Reviewer: Petio S. Kelevedjiev (Sliven)Kardar-Parisi-Zhang physics in integrable rotationally symmetric dynamics on discrete space-time lattice.https://www.zbmath.org/1452.820182021-02-12T15:23:00+00:00"Krajnik, Žiga"https://www.zbmath.org/authors/?q=ai:krajnik.ziga"Prosen, Tomaž"https://www.zbmath.org/authors/?q=ai:prosen.tomazSummary: We introduce a deterministic \(\mathrm{SO}(3)\) invariant dynamics of classical spins on a discrete space-time lattice and prove its complete integrability by explicitly finding a related non-constant (baxterized) solution of the set-theoretic Yang-Baxter equation over the 2-sphere. Equipping the algebraic structure with the corresponding Lax operator we derive an infinite sequence of conserved quantities with local densities. The dynamics depend on a single continuous spectral parameter and reduce to a (lattice) Landau-Lifshitz model in the limit of a small parameter which corresponds to the continuous time limit. Using quasi-exact numerical simulations of deterministic dynamics and Monte Carlo sampling of initial conditions corresponding to a maximum entropy equilibrium state we determine spin-spin spatio-temporal (dynamical) correlation functions with relative accuracy of three orders of magnitude. We demonstrate that in the equilibrium state with a vanishing total magnetization the correlation function precisely follows Kardar-Parisi-Zhang scaling hence the spin transport belongs to the universality class with dynamical exponent \(z=3/2\), in accordance to recent related simulations in discrete and continuous time quantum Heisenberg spin 1/2 chains.Conditional regularity of solutions of the three-dimensional Navier-Stokes equations and implications for intermittency.https://www.zbmath.org/1452.760472021-02-12T15:23:00+00:00"Gibbon, J. D."https://www.zbmath.org/authors/?q=ai:gibbon.john-dSummary: Two unusual time-integral conditional regularity results are presented for the three-dimensional Navier-Stokes equations. The ideas are based on \(L^{2m}\)-norms of the vorticity, denoted by \(\Omega_m(t)\), and particularly on \(D_{m} = \big[\varpi_{0}^{-1}\Omega_{m}(t)\big]^{\alpha_{m}}\), where \(\alpha_m = 2m/(4m - 3)\) for \(m \geq 1\). The first result, more appropriate for the unforced case, can be stated simply: if there exists an \(1 \leq m < \infty\) for which the integral condition is satisfied \((Z_m = D_{m + 1}/D_m)\):
\[\int_{0}^{t}\ln \left(\frac{1 + Z_{m}}{c_{4,m}}\right)\,d\tau \geq 0,\]
then no singularity can occur on \([0, t]\). The constant \(c_{4, m}\;\searrow 2\) for large \(m\). Second, for the forced case, by imposing a critical \textit{lower} bound on \(\int_{0}^{t}D_{m}\,d\tau\), no singularity can occur in \(D_m(t)\) for large \textit{initial}\ data. Movement across this critical lower bound shows how solutions can behave intermittently, in analogy with a relaxation oscillator. Potential singularities that drive \(\int_{0}^{t}D_{m}\,d\tau\) over this critical value can be ruled out whereas other types cannot.
\copyright 2012 American Institute of PhysicsThe Fefferman-Stein decomposition for the Constantin-Lax-Majda equation: regularity criteria for inviscid fluid dynamics revisited.https://www.zbmath.org/1452.760292021-02-12T15:23:00+00:00"Ohkitani, Koji"https://www.zbmath.org/authors/?q=ai:ohkitani.kojiSummary: The celebrated Beale-Kato-Majda (BKM) criterion for the 3D Euler equations has been updated by Kozono and Taniuchi by replacing the supremum with the bounded mean oscillation norm. We consider this generalized criterion in an attempt to understand it more intuitively by giving an alternative explanation. For simplicity, we first treat the Constantin-Lax-Majda (CLM) equation \(\frac{\partial \omega }{\partial t}=H(\omega)\omega\) for the vorticity {\(\omega\)} in one-dimension and identify a mechanism underlying the update of such an estimate. We consider a Fefferman-Stein (FS) decomposition for the initial vorticity \(\omega = \omega_0 + H[\omega_1]\) and how it propagates under the dynamics of the CLM equation. In particular, we obtain a set of dynamical equations for it, which reads in its simplest case \(\frac{\partial \omega_0}{\partial t} =\omega_0 H[\omega_0]-{\omega_1} H[{\omega_1}]\) and \(\frac{\partial \omega_1}{\partial t} =\omega_0 H[\omega_1]+\omega_1 H[\omega_0]\). The equation for the second component \(\omega_1\), responsible for a possible logarithmic blow-up, is linear and homogeneous; hence it remains zero if it is so initially until a stronger blow-up takes place. This rules out a logarithmic blow-up on its own and underlies the generalized BKM criterion. Numerical results are also presented to illustrate how each component of the FS decomposition evolves in time. Higher dimensional cases are also discussed. Without knowing fully explicit FS decompositions for the 3D Euler equations, we show that the second component of the FS decomposition will not appear if it is zero initially, thereby precluding a logarithmic blow-up.(\textit{In honour of Professor Peter Constantin's 60th birthday.}){
\copyright 2012 American Institute of Physics}Invasion waves in a higher-dimensional lattice competitive system with stage structure.https://www.zbmath.org/1452.370762021-02-12T15:23:00+00:00"Li, Kun"https://www.zbmath.org/authors/?q=ai:li.kun.6|li.kun.4|li.kun.10|li.kun.3|li.kun.9|li.kun.8|li.kun.5|li.kun|li.kun.2|li.kun.7Summary: In this paper, we use Schauder's fixed point theorem to establish the existence of invasion waves in a stage-structured competitive system on higher-dimensional lattices. To illustrate our results, we construct a pair of upper and lower solutions.Extrinsic eigenvalues estimates for hypersurfaces in product spaces.https://www.zbmath.org/1452.351242021-02-12T15:23:00+00:00"Roth, Julien"https://www.zbmath.org/authors/?q=ai:roth.julienThe author provides Reilly-type upper bounds for a number of different operators on product manifolds \((\mathbb R\times N,dt^2\oplus h)\), where \((N^n,h)\) is a complete Riemannian manifold. The operators considered are: divergence-type elliptic operators, Paneitz-like operators, Steklov-Wentzell operators, and biharmonic Steklov operators. In all cases, the bounds are obtained using similar techniques, all adapted to the specific case. The last section is dedicated to a discussion of the equality cases for the new upper bounds.
Reviewer: Davide Buoso (Alessandria)Coercivity and stability results for an extended Navier-Stokes system.https://www.zbmath.org/1452.760492021-02-12T15:23:00+00:00"Iyer, Gautam"https://www.zbmath.org/authors/?q=ai:iyer.gautam"Pego, Robert L."https://www.zbmath.org/authors/?q=ai:pego.robert-l"Zarnescu, Arghir"https://www.zbmath.org/authors/?q=ai:zarnescu.arghir-daniSummary: In this paper, we study a system of equations that is known to \textit{extend} Navier-Stokes dynamics in a well-posed manner to velocity fields that are not necessarily divergence-free. Our aim is to contribute to an understanding of the role of divergence and pressure in developing energy estimates capable of both controlling the nonlinear terms, and being useful at the time-discrete level. We address questions of global existence and stability in bounded domains with no-slip boundary conditions. Through use of new \(H^1\) coercivity estimates for the linear equations, we establish a number of global existence and stability results, including results for small divergence and a time-discrete scheme. We also prove global existence in 2D for any initial data, provided sufficient divergence damping is included. (\textit{Dedicated to Peter Constantin, on the occasion of his 60th birthday.}){
\copyright 2012 American Institute of Physics}Killing spinor-valued forms and their integrability conditions.https://www.zbmath.org/1452.351172021-02-12T15:23:00+00:00"Somberg, Petr"https://www.zbmath.org/authors/?q=ai:somberg.petr"Zima, Petr"https://www.zbmath.org/authors/?q=ai:zima.petrSummary: We study invariant systems of PDEs defining Killing vector-valued forms, and then we specialize to Killing spinor-valued forms. We give a detailed treatment of their prolongation and integrability conditions by relating the pointwise values of solutions to the curvature of the underlying manifold. As an example, we completely solve the equations on model spaces of constant curvature producing brand-new solutions which do not come from the tensor product of Killing spinors and Killing-Yano forms.A new projection-based stabilized virtual element method for the Stokes problem.https://www.zbmath.org/1452.653382021-02-12T15:23:00+00:00"Guo, Jun"https://www.zbmath.org/authors/?q=ai:guo.jun"Feng, Minfu"https://www.zbmath.org/authors/?q=ai:feng.minfuSummary: We propose and analyze a stabilized virtual element method for the Stokes problem on polytopal meshes. We employ the \(C^0\) continuous arbitrary ``equal-order'' virtual element pairs to approximate both velocity and pressure, and develop a projection-based stabilization term to circumvent the discrete inf-sup condition, then we obtain the corresponding error estimates. The presented method involves neither the projection of the second derivative nor additional coupling terms, and it is parameter-free. In particularly, for the lowest-order case on triangular (tetrahedral) meshes the stabilized method introduced by \textit{P. B. Bochev} et al. [SIAM J. Numer. Anal. 44, No. 1, 82--101 (2006; Zbl 1145.76015)] is a special case of our method up to an approximation of the load term. Furthermore, numerical results are shown to confirm the theoretical predictions.Convergence analysis of the anisotropic FEM for 2D time fractional variable coefficient diffusion equations on graded meshes.https://www.zbmath.org/1452.652542021-02-12T15:23:00+00:00"Wei, Yabing"https://www.zbmath.org/authors/?q=ai:wei.yabing"Lü, Shujuan"https://www.zbmath.org/authors/?q=ai:lu.shujuan"Chen, Hu"https://www.zbmath.org/authors/?q=ai:chen.hu"Zhao, Yanmin"https://www.zbmath.org/authors/?q=ai:zhao.yanmin"Wang, Fenling"https://www.zbmath.org/authors/?q=ai:wang.fenlingSummary: In this paper, an unconditionally stable fully discrete numerical scheme for the two-dimensional (2D) time fractional variable coefficient diffusion equations with non-smooth solutions is constructed and analyzed. The \(L 2- 1_\sigma\) scheme is applied for the discretization of time fractional derivative on graded meshes and anisotropic finite element method (FEM) is employed for the spatial discretization. The unconditional stability and convergence of the proposed scheme are proved rigorously. It is shown that the order \(O ( h^2 + N^{- \min \{ r \alpha , 2 \}} )\) can be achieved, where \(h\) is the spatial step, \(N\) is the number of partition in temporal direction, \(r\) is the temporal meshes grading parameter and \(\alpha\) is the order of fractional derivative. A numerical example is provided to verify the sharpness of our error analysis.Efficient Fourier basis particle simulation.https://www.zbmath.org/1452.652882021-02-12T15:23:00+00:00"Mitchell, Matthew S."https://www.zbmath.org/authors/?q=ai:mitchell.matthew-s"Miecnikowski, Matthew T."https://www.zbmath.org/authors/?q=ai:miecnikowski.matthew-t"Beylkin, Gregory"https://www.zbmath.org/authors/?q=ai:beylkin.gregory"Parker, Scott E."https://www.zbmath.org/authors/?q=ai:parker.scott-eSummary: The standard particle-in-cell algorithm suffers from grid heating. There exists a gridless alternative which bypasses the deposition step and calculates each Fourier mode of the charge density directly from the particle positions. We show that a gridless method can be computed efficiently through the use of an Unequally Spaced Fast Fourier Transform (USFFT) algorithm. After a spectral field solve, the forces on the particles are calculated via the inverse USFFT (a rapid solution of an approximate linear system) [the third author, Appl. Comput. Harmon. Anal. 2, No. 4, 363--381 (1995; Zbl 0838.65142); \textit{A. Dutt} and \textit{V. Rokhlin}, SIAM J. Sci. Comput. 14, No. 6, 1368--1393 (1993; Zbl 0791.65108)]. We provide one and two dimensional implementations of this algorithm with an asymptotic runtime of \(O(N_p + N_m^D \log N_m^D)\) for each iteration, identical to the standard PIC algorithm (where \(N_p\) is the number of particles, \(N_m\) is the number of Fourier modes, and \(D\) is the spatial dimensionality of the problem). We demonstrate superior energy conservation and reduced noise, as well as convergence of the energy conservation at small time steps.Insights on the coercivity of the ESFR methods for elliptic problems.https://www.zbmath.org/1452.653562021-02-12T15:23:00+00:00"Quaegebeur, Samuel"https://www.zbmath.org/authors/?q=ai:quaegebeur.samuel"Nadarajah, Siva"https://www.zbmath.org/authors/?q=ai:nadarajah.siva-kSummary: The Flux Reconstruction approach is a recent high-order method which has been introduced for unsteady problems. Initial energy stability has been conducted for the advection problem, leading to the well know Energy Stable Flux Reconstruction (ESFR) scheme. Using the ESFR scheme, the energy stability proof has been extended for the advection-diffusion using the Local Discontinuous Galerkin (LDG) numerical flux. Recently, stability conditions were derived for the compact Interior Penalty (IP) and Bassi-Rebay II (BR2) numerical fluxes. Here we apply ESFR schemes to elliptic problems and derive the associated bilinear form for the Poisson equation. We show that for the compact IP and BR2 numerical fluxes, the bilinear form is independent of the auxiliary correction function. Finally, we provide some insights on the coercivity of the ESFR scheme.A partition of unity finite element method for three-dimensional transient diffusion problems with sharp gradients.https://www.zbmath.org/1452.652442021-02-12T15:23:00+00:00"Malek, Mustapha"https://www.zbmath.org/authors/?q=ai:malek.mustapha"Izem, Nouh"https://www.zbmath.org/authors/?q=ai:izem.nouh"Mohamed, M. Shadi"https://www.zbmath.org/authors/?q=ai:mohamed.m-shadi"Seaid, Mohammed"https://www.zbmath.org/authors/?q=ai:seaid.mohammed"Laghrouche, Omar"https://www.zbmath.org/authors/?q=ai:laghrouche.omarSummary: An efficient partition of unity finite element method for three-dimensional transient diffusion problems is presented. A class of multiple exponential functions independent of time variable is proposed to enrich the finite element approximations. As a consequence of this procedure, the associated matrix for the linear system is evaluated once at the first time step and the solution is obtained at subsequent time step by only updating the right-hand side of the linear system. This results in an efficient numerical solver for transient diffusion equations in three space dimensions. Compared to the conventional finite element methods with \(h\)-refinement, the proposed approach is simple, more efficient and more accurate. The performance of the proposed method is assessed using several test examples for transient diffusion in three space dimensions. We present numerical results for a transient diffusion equation with known analytical solution to quantify errors for the new method. We also solve time-dependent diffusion problems in complex geometries. We compare the results obtained using the partition of unity finite element method to those obtained using the standard finite element method. It is shown that the proposed method strongly reduces the necessary number of degrees of freedom to achieve a prescribed accuracy.A quasi-Lagrangian moving mesh discontinuous Galerkin method for hyperbolic conservation laws.https://www.zbmath.org/1452.652432021-02-12T15:23:00+00:00"Luo, Dongmi"https://www.zbmath.org/authors/?q=ai:luo.dongmi"Huang, Weizhang"https://www.zbmath.org/authors/?q=ai:huang.weizhang"Qiu, Jianxian"https://www.zbmath.org/authors/?q=ai:qiu.jianxianSummary: A moving mesh discontinuous Galerkin method is presented for the numerical solution of hyperbolic conservation laws. The method is a combination of the discontinuous Galerkin method and the mesh movement strategy which is based on the moving mesh partial differential equation approach and moves the mesh continuously in time and orderly in space. It discretizes hyperbolic conservation laws on moving meshes in the quasi-Lagrangian fashion with which the mesh movement is treated continuously and no interpolation is needed for physical variables from the old mesh to the new one. Two convection terms are induced by the mesh movement and their discretization is incorporated naturally in the DG formulation. Numerical results for a selection of one- and two-dimensional scalar and system conservation laws are presented. It is shown that the moving mesh DG method achieves the second and third order of convergence for \(P^1\) and \(P^2\) elements, respectively, for problems with smooth solutions and is able to capture shocks and concentrate mesh points in non-smooth regions. Its advantage over uniform meshes and its insensitiveness to mesh smoothness are also demonstrated.Numerical and asymptotic analysis for KdV-type equations, with nonhomogeneous \(\mathcal{C}^1\).https://www.zbmath.org/1452.351672021-02-12T15:23:00+00:00"García-Alvarado, Martín G."https://www.zbmath.org/authors/?q=ai:garcia-alvarado.martin-g"Noyola-Rodriguez, Jesús"https://www.zbmath.org/authors/?q=ai:noyola-rodriguez.jesusFor the entire collection see [Zbl 1429.53001].Rational Krylov methods for functions of matrices with applications to fractional partial differential equations.https://www.zbmath.org/1452.652012021-02-12T15:23:00+00:00"Aceto, L."https://www.zbmath.org/authors/?q=ai:aceto.lidia"Bertaccini, D."https://www.zbmath.org/authors/?q=ai:bertaccini.daniele"Durastante, F."https://www.zbmath.org/authors/?q=ai:durastante.fabio"Novati, P."https://www.zbmath.org/authors/?q=ai:novati.paoloSummary: In this paper we propose a new choice of poles to define reliable rational Krylov methods. These methods are used for approximating function of positive definite matrices. In particular, the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations. The numerical experiments on some fractional partial differential equation models confirm that the proposed approach is promising.On the wave breaking phenomena for the generalized periodic two-component Dullin-Gottwald-Holm system.https://www.zbmath.org/1452.760342021-02-12T15:23:00+00:00"Chen, Caixia"https://www.zbmath.org/authors/?q=ai:chen.caixia"Yan, Yonghua"https://www.zbmath.org/authors/?q=ai:yan.yonghuaSummary: Considered herein is the generalized two-component periodic Dullin-Gottwald-Holm system, which can be derived from the Euler equation with nonzero constant vorticity in shallow water waves moving over a linear shear flow. The precise blow-up scenarios of strong solutions and several results of blow-up solutions with certain initial profiles are described in detail. The exact blow-up rates are also determined. Finally, a sufficient condition for global solutions is established.{
\copyright 2012 American Institute of Physics}On the spectrum of a relativistic Landau Hamiltonian with a periodic electric potential.https://www.zbmath.org/1452.351212021-02-12T15:23:00+00:00"Danilov, Leonid Ivanovich"https://www.zbmath.org/authors/?q=ai:danilov.leonid-ivanovichSummary: This paper is concerned with a two-dimensional Dirac operator \(\widehat{\sigma}_1\left( -i\, \frac{\partial }{\partial x_1}\right) +\widehat{\sigma}_2\left( -i\, \frac{\partial }{\partial x_2}-Bx_1\right) +m\widehat{\sigma}_3+V\widehat{I}_2\) with a uniform magnetic field \(B\) where \(\widehat{\sigma}_j\), \(j=1,2,3\), are the Pauli matrices and \(\widehat{I}_2\) is the unit \(2\times 2\)-matrix. The function \(m\) and the electric potential \(V\) belong to the space \(L^p_{\Lambda }(\mathbb{R}^2;\mathbb{R})\) of \(\Lambda \)-periodic functions from the \(L^p_{\text{loc}}(\mathbb{R}^2;\mathbb{R})\), \(p>2\), and we suppose that for the magnetic flux \(\eta =(2\pi )^{-1}Bv(K)\in \mathbb{Q}\) where \(v(K)\) is the area of an elementary cell \(K\) of the period lattice \(\Lambda \). For any nonincreasing function \((0,1]\ni \varepsilon \mapsto\mathcal{R}(\varepsilon )\in (0,+\infty )\) for which \(\mathcal{R}(\varepsilon )\to +\infty\) as \(\varepsilon \to +0\) let \(\mathfrak{M}^p_{\Lambda}(\mathcal{R}(\cdot))\) be the set of functions \(m\in L^p_{\Lambda }(\mathbb{R}^2;\mathbb{R})\) such that for every \(\varepsilon \in (0,1]\) there exists a real-valued \(\Lambda \)-periodic trigonometric polynomial \(\mathcal{P}^{(\varepsilon )}\) such that \(\| m-\mathcal{P}^{(\varepsilon )}\|_{L^p(K)}<\varepsilon\) and for Fourier coefficients \(\mathcal{P}^{(\varepsilon )}_Y=0\) provided \(|Y|>\mathcal{R}(\varepsilon )\). It is proved that for any function \(\mathcal{R}(\cdot )\) in question there is a dense \(G_{\delta } \)-set \(\mathcal{O}\) in the Banach space \((L^p_{\Lambda }(\mathbb{R}^2;\mathbb{R}),\| \cdot \|_{L^p(K)})\) such that for every electric potential \(V\in\mathcal{O} \), for every function \(m\in\mathfrak{M}^p_{\Lambda }(\mathcal{R} (\cdot ))\), and for every uniform magnetic field \(B\) with the flux \(\eta \in \mathbb{Q}\) the spectrum of the Dirac operator is absolutely continuous.Solving the Vlasov-Maxwell equations using Hamiltonian splitting.https://www.zbmath.org/1452.653942021-02-12T15:23:00+00:00"Li, Yingzhe"https://www.zbmath.org/authors/?q=ai:li.yingzhe"He, Yang"https://www.zbmath.org/authors/?q=ai:he.yang"Sun, Yajuan"https://www.zbmath.org/authors/?q=ai:sun.yajuan"Niesen, Jitse"https://www.zbmath.org/authors/?q=ai:niesen.jitse"Qin, Hong"https://www.zbmath.org/authors/?q=ai:qin.hong.1|qin.hong|qin.hong.2"Liu, Jian"https://www.zbmath.org/authors/?q=ai:liu.jian.1|liu.jian.5|liu.jian|liu.jian.3|liu.jian.4|liu.jian.2|liu.jian.6Summary: In this paper, the numerical discretizations based on Hamiltonian splitting for solving the Vlasov-Maxwell system are constructed. We reformulate the Vlasov-Maxwell system in Morrison-Marsden-Weinstein Poisson bracket form. Then the Hamiltonian of this system is split into five parts, with which five corresponding Hamiltonian subsystems are obtained. The splitting method in time is derived by composing the solutions to these five subsystems. Combining the splitting method in time with the Fourier spectral method and finite volume method in space gives the full numerical discretizations which possess good conservation for the conserved quantities including energy, momentum, charge, etc. In numerical experiments, we simulate the Landau damping, Weibel instability and Bernstein wave to verify the numerical algorithms.A convergence analysis of generalized multiscale finite element methods.https://www.zbmath.org/1452.653192021-02-12T15:23:00+00:00"Abreu, Eduardo"https://www.zbmath.org/authors/?q=ai:abreu.eduardo"Díaz, Ciro"https://www.zbmath.org/authors/?q=ai:diaz.ciro"Galvis, Juan"https://www.zbmath.org/authors/?q=ai:galvis.juanSummary: In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be extended, without great difficulty, to more sophisticated GMsFEMs. For concreteness, the obtained error estimates generalize and simplify the convergence analysis of \textit{Y. Efendiev} et al. [ibid. 230, No. 4, 937--955 (2011; Zbl 1391.76321)]. The GMsFEM method construct basis functions that are obtained by multiplication of (approximation of) local eigenvectors by partition of unity functions. Only important eigenvectors are used in the construction. The error estimates are general and are written in terms of the eigenvalues of the eigenvectors not used in the construction. The error analysis involve local and global norms that measure the decay of the expansion of the solution in terms of local eigenvectors. Numerical experiments are carried out to verify the feasibility of the approach with respect to the convergence and stability properties of the analysis.On the numerical behavior of a chemotaxis model with linear production term.https://www.zbmath.org/1452.920082021-02-12T15:23:00+00:00"Guillén-González, F."https://www.zbmath.org/authors/?q=ai:guillen-gonzalez.francisco-m"Rodríguez-Bellido, M. A."https://www.zbmath.org/authors/?q=ai:rodriguez-bellido.maria-angeles"Rueda-Gómez, D. A."https://www.zbmath.org/authors/?q=ai:rueda-gomez.diego-aThe authors study a system of two parabolic equations modelling chemorepulsion with power type production term.
Summarizing recent reports on the numerical analysis of those systems, they discuss properties of three versions of finite element methods applicable to an initial-boundary value problem in two dimensional domains, with a particular stress on (classical) linear production term in the evolution equation for the diffusion of chemicals.
For the entire collection see [Zbl 1445.37001].
Reviewer: Piotr Biler (Wrocław)Numerical methods for construction of value functions in optimal control problems on an infinite horizon.https://www.zbmath.org/1452.490182021-02-12T15:23:00+00:00"Bagno, Aleksandr Leonidovich"https://www.zbmath.org/authors/?q=ai:bagno.aleksandr-leonidovich"Tarasyev, Aleksandr Mikhaĭlovich"https://www.zbmath.org/authors/?q=ai:tarasev.aleksandr-mikhailovichSummary: This article deals with the optimal control problem on an infinite horizon, the quality functional of which is contained in the integrand index and the discounting factor. A special feature of this formulation of the problem is the assumption of the possible unboundedness of the integrand index. The problem reduces to an equivalent optimal control problem with a stationary value function as a generalized (minimax, viscosity) solution of the Hamilton-Jacobi equation satisfying the Hölder condition and the condition of linear growth. The article describes the backward procedure on an infinite horizon. It is the method of numerical approximation of the generalized solution of the Hamilton-Jacobi equation. The main result of the article is an estimate of the accuracy of approximation of a backward procedure for solving the original problem. Problems of the analyzed type are related to modeling processes of economic growth and to problems of stabilizing dynamic systems. The results obtained can be used to construct numerical finite-difference schemes for calculating the value function of optimal control problems or differential games.Convergence of a positive nonlinear control volume finite element scheme for an anisotropic seawater intrusion model with sharp interfaces.https://www.zbmath.org/1452.652462021-02-12T15:23:00+00:00"Oulhaj, Ahmed Ait Hammou"https://www.zbmath.org/authors/?q=ai:ait-hammou-oulhaj.ahmed"Maltese, David"https://www.zbmath.org/authors/?q=ai:maltese.davidSummary: We study a sharp interface model in the context of seawater intrusion in an anisotropic unconfined aquifer. It is a degenerate parabolic system with cross-diffusion modeling the flow of fresh and saltwater. We study a nonlinear control volume finite element scheme. This scheme ensures the nonnegativity of the discrete solution without any restriction on the transmissibility coefficients. Moreover, it also provides a control on the entropy. The existence of a discrete solution and the convergence of this scheme are obtained, based on nonlinear stability results.Convergence and stability estimates in difference setting for time-fractional parabolic equations with functional delay.https://www.zbmath.org/1452.651592021-02-12T15:23:00+00:00"Hendy, Ahmed S."https://www.zbmath.org/authors/?q=ai:hendy.ahmed-s"Pimenov, Vladimir G."https://www.zbmath.org/authors/?q=ai:pimenov.vladimir-g"Macías-Díaz, Jorge E."https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardoSummary: A class of one-dimensional time-fractional parabolic differential equations with delay effects of functional type in the time component is numerically investigated in this work. To that end, a compact difference scheme is constructed for the numerical solution of those equations based on the idea of separating the current state and the prehistory function. In these terms, the prehistory function is approximated by means of an appropriate interpolation-extrapolation operator. A discrete form of the fractional Gronwall inequality is employed to provide an optimal error estimate. The existence and uniqueness of the numerical solutions, the order of approximation error for the constructed scheme, the stability and the order of convergence are mathematically investigated in this work.Stokes equations under nonlinear slip boundary conditions coupled with the heat equation: a priori error analysis.https://www.zbmath.org/1452.653312021-02-12T15:23:00+00:00"Djoko, Jules K."https://www.zbmath.org/authors/?q=ai:djoko.jules-k"Konlack, Virginie S."https://www.zbmath.org/authors/?q=ai:konlack.virginie-s"Mbehou, Mohamed"https://www.zbmath.org/authors/?q=ai:mbehou.mohamedThe authors consider the heat equation coupled with the Stokes equation under the threshold type boundary conditions, and construct a weak formulation for the continuous problem and its finite element counterpart. They next study the conditions for existence and convergence of the weak solution using Babushka-Brezzi techniques, and use a Uzawa-type iterative algorithm to determine the conditions under which the algorithm converges. The authors also present results of numerical simulations that validate their theoretical results.
Reviewer: Murli Gupta (Washington, D. C.)Analytical solutions to a class of non-Newtonian fluids with free boundaries.https://www.zbmath.org/1452.760102021-02-12T15:23:00+00:00"Fang, Li"https://www.zbmath.org/authors/?q=ai:fang.li"Guo, Zhenhua"https://www.zbmath.org/authors/?q=ai:guo.zhenhuaSummary: We study analytical solutions to a class of non-Newtonian fluids with free boundaries. For suitable non-Newtonian fluids, we construct a class of analytical solutions in \(\mathbb{R}\) with both continuous density condition and the stress free condition across the free boundaries separating the fluid from vacuum. Such solutions exhibit interesting new information such as the formation of vacuum as time tends to infinitely and explicit regularities and large time decay estimates of the velocity field.{
\copyright 2012 American Institute of Physics}Exact periodic cross-kink wave solutions for the \((2+1)\)-dimensional Korteweg-de Vries equation.https://www.zbmath.org/1452.350622021-02-12T15:23:00+00:00"Liu, Jian-Guo"https://www.zbmath.org/authors/?q=ai:liu.jian-guo|liu.jian-guo.1"Ye, Qing"https://www.zbmath.org/authors/?q=ai:ye.qingSummary: The movement of any object has a certain natural law, and the studies and solutions to many natural laws boil down to the problem of mathematical physics equations. Many important physical situations such as fluid flows, plasma physics, and solid state physics have been described by the Korteweg-de Vries (KdV)-type models. In this article, the \((2+1)\)-dimensional KdV equation is presented. By using the Hirota's bilinear form and the extended Ansätz function method, we obtain new exact periodic cross-kink wave solutions for the \((2+1)\)-dimensional KdV equation. With the aid of symbolic computation, the properties for these exact periodic cross-kink wave solutions are shown with some figures.Asymptotic expansion of the solution of the steady Stokes equation with variable viscosity in a two-dimensional tube structure.https://www.zbmath.org/1452.760432021-02-12T15:23:00+00:00"Cardone, G."https://www.zbmath.org/authors/?q=ai:cardone.giuseppe"Fares, R."https://www.zbmath.org/authors/?q=ai:fares.r"Panasenko, G. P."https://www.zbmath.org/authors/?q=ai:panasenko.grigory-pSummary: The Stokes equation with the varying viscosity is considered in a thin tube structure, i.e., in a connected union of thin rectangles with heights of order \(\varepsilon \ll 1\) and with bases of order 1 with smoothened boundary. An asymptotic expansion of the solution is constructed: it contains some Poiseuille type flows in the channels (rectangles) with some boundary layers correctors in the neighborhoods of the bifurcations of the channels. The estimates for the difference of the exact solution and its asymptotic approximation are proved.{
\copyright 2012 American Institute of Physics}A coupled grid based particle and implicit boundary integral method for two-phase flows with insoluble surfactant.https://www.zbmath.org/1452.761742021-02-12T15:23:00+00:00"Hsu, Shih-Hsuan"https://www.zbmath.org/authors/?q=ai:hsu.shih-hsuan"Chu, Jay"https://www.zbmath.org/authors/?q=ai:chu.jay"Lai, Ming-Chih"https://www.zbmath.org/authors/?q=ai:lai.mingchih"Tsai, Richard"https://www.zbmath.org/authors/?q=ai:tsai.yen-hsi-richardSummary: We develop a coupled grid based particle and implicit boundary integral method for simulation of three-dimensional interfacial flows with the presence of insoluble surfactant. The grid based particle method (GBPM, [\textit{S. Leung} and \textit{H. Zhao}, J. Comput. Phys. 228, No. 8, 2993--3024 (2009; Zbl 1161.65013)]) tracks the interface by the projection of the neighboring Eulerian grid points and does not require stitching of parameterizations nor body fitted moving meshes. Using this GBPM to represent the interface, the surfactant equation defined on the interface is discretized naturally following a new volumetric constant-along-surface-normal extension approach [\textit{J. Chu} and \textit{R. Tsai}, Res. Math. Sci. 5, No. 2, Paper No. 19, 38 p. (2018; Zbl 1431.65131)]. We first examine the proposed scheme to solve the convection-diffusion equation for the problems with available analytical solutions. The numerical results demonstrate second-order accuracy of the scheme. We then perform a series of simulations for interfacial flows with insoluble surfactant. The numerical results agree well with the theory, and are comparable with other numerical works in literature.Algorithms for the partitioned solution of weakly coupled fluid models for cardiovascular flows.https://www.zbmath.org/1452.762872021-02-12T15:23:00+00:00"Malossi, A. Cristiano I."https://www.zbmath.org/authors/?q=ai:malossi.a-cristiano-i"Blanco, Pablo J."https://www.zbmath.org/authors/?q=ai:blanco.pablo-javier"Deparis, Simone"https://www.zbmath.org/authors/?q=ai:deparis.simone"Quarteroni, Alfio"https://www.zbmath.org/authors/?q=ai:quarteroni.alfio-mSummary: The main goal of this work is to devise robust iterative strategies to partition the solution of the Navier -- Stokes equations in a three-dimensional (3D) domain, into non-overlapping 3D subdomains, which communicate through the exchange of averaged/integrated quantities across the interfaces. The novel aspect of the present approach is that at coupling boundaries, the conservation of flow rates and of the associated dual variables is implicitly imposed, entailing a weak physical coupling. For the solution of the non-linear interface problem two strategies are compared: relaxed fixed-point and Newton iterations. The algorithm is tested in several configurations for problems ranging from academic ones to some related to the computational haemodynamics field, which involve more than two components at each coupling interface. In some cases, it is shown that relaxed fixed-point methods are not convergent, whereas the Newton method leads in all tested cases to convergent schemes. One of the appealing aspects of the strategy proposed here is the flexibility in the setting of boundary conditions at the interfaces, where no hierarchy is established a priori (unlike Gauss -- Seidel methods). The usefulness of this methodology is also discussed in the context of dimensionally heterogeneous coupling and preconditioning for domain decomposition methods.A simple pseudospectral method for the computation of the time-dependent Dirac equation with perfectly matched layers.https://www.zbmath.org/1452.652672021-02-12T15:23:00+00:00"Antoine, Xavier"https://www.zbmath.org/authors/?q=ai:antoine.xavier"Lorin, Emmanuel"https://www.zbmath.org/authors/?q=ai:lorin.emmanuelSummary: A simple time-splitting pseudospectral method for the computation of the Dirac equation with Perfectly Matched Layers is proposed. Within this approach, basic and widely used FFT-based solvers can be adapted without much effort to compute Initial Boundary Value Problems for the time-dependent Dirac equation with absorbing boundary layers. Some numerical examples from laser-physics are proposed to illustrate the method.Approximating solutions of linear elliptic PDE's on a smooth manifold using local kernel.https://www.zbmath.org/1452.653122021-02-12T15:23:00+00:00"Gilani, Faheem"https://www.zbmath.org/authors/?q=ai:gilani.faheem"Harlim, John"https://www.zbmath.org/authors/?q=ai:harlim.johnSummary: A mesh-free numerical method for solving linear elliptic PDE's using the local kernel theory that was developed for manifold learning is proposed. In particular, this novel approach exploits the local kernel theory which allows one to approximate the Kolmogorov operator associated with Itô diffusion processes on compact Riemannian manifolds without boundary or with Neumann boundary conditions using an integral operator. Theoretical justification for the convergence of this numerical technique is provided under the standard conditions for the existence of the weak solutions of the PDEs. Numerical results on various instructive examples, ranging from PDE's defined on flat and non-flat manifolds with known and unknown embedding functions show accurate approximation with error on the order of the kernel bandwidth parameter.Non-isothermal Navier-Stokes system with mixed boundary conditions and friction law: uniqueness and regularity properties.https://www.zbmath.org/1452.760422021-02-12T15:23:00+00:00"Boukrouche, Mahdi"https://www.zbmath.org/authors/?q=ai:boukrouche.mahdi"Boussetouan, Imane"https://www.zbmath.org/authors/?q=ai:boussetouan.imane"Paoli, Laetitia"https://www.zbmath.org/authors/?q=ai:paoli.laetitia-aSummary: We consider an unsteady non-isothermal fluid flow subjected to non-homogeneous Dirichlet conditions on a part of the boundary and Tresca's friction law on the other part. For this problem an existence result has been proved recently in our work [Q. Appl. Math. 78, No. 3, 525--543 (2020; Zbl 1435.76021)] but uniqueness has been left as an open question. Starting from the approximation of the problem based on a regularization of the free boundary condition due to friction combined with a special penalty method, we establish some sharp a priori estimates leading to better regularity properties for the velocity field and to the uniqueness of the solutions. Finally we study the regularity of the pressure field and of the stress tensor.Spatiotemporal dynamics in a diffusive bacterial and viral diseases propagation model with chemotaxis.https://www.zbmath.org/1452.350242021-02-12T15:23:00+00:00"Tang, Xiaosong"https://www.zbmath.org/authors/?q=ai:tang.xiaosong"Ouyang, Peichang"https://www.zbmath.org/authors/?q=ai:ouyang.peichangSummary: In this article, we study the effect of chemotaxis on the dynamics of a diffusive bacterial and viral diseases propagation model. From three aspects: \(\chi >0\), \(\chi =0\) and \(\chi <0\), we investigate the existence of Turing bifurcations and stability of positive equilibrium under Neumann boundary conditions. We find that Turing bifurcations can induced by chemotaxis, which does not occur in the original model. Moreover, for the model with diffusion and chemotaxis, we need explore the new expression of the normal form on Turing bifurcation. By the newly obtained normal form, we can determine the properties of Turing bifurcation. Finally, we perform some numerical simulations to verify the theoretical analysis and obtain stable steady state solutions, spots pattern and spots-strip pattern, which also expand the main results in this article.Hardy-Poincaré type inequalities related to \(k\)-Hessian operator.https://www.zbmath.org/1452.350082021-02-12T15:23:00+00:00"Jin, Yongyang"https://www.zbmath.org/authors/?q=ai:jin.yongyang"Chen, Haiting"https://www.zbmath.org/authors/?q=ai:chen.haiting"Shen, Shoufeng"https://www.zbmath.org/authors/?q=ai:shen.shoufeng"Wu, Yurong"https://www.zbmath.org/authors/?q=ai:wu.yurongSummary: In this paper, we obtain some improved Hardy inequalities for Hessian integrals \(I_{p,k}[u,\Omega]\) by symmetrization method under conditions that \(\Omega\) is a bounded \((k-1)\)-convex starshaped domain of \(\mathbb{R}^n\) with \(1<p<n-k+1\), and \(u\in A_{k-1}(\Omega)\) whose sub-level set \(\Omega_t=\{x\in \Omega \mid u(x)<t\}\) is \((k-1)\)-convex starshaped, where \(A_{k-1}(\Omega)\) is a particular class of function space whose sub-level domains satisfy some monotonicity property. Especially in the case of \(p=2\), the best contant for the remainder term is given.Sharp estimate of the mean exit time of a bounded domain in the zero white noise limit.https://www.zbmath.org/1452.310172021-02-12T15:23:00+00:00"Nectoux, Boris"https://www.zbmath.org/authors/?q=ai:nectoux.borisSummary: We prove a sharp asymptotic formula for the mean exit time from a bounded domain \(D\subset\mathbb{R}^d\) for the overdamped Langevin dynamics \[dX_t=-\nabla f(X_t)dt+\sqrt{2\varepsilon}\,dB_t\] when \(\varepsilon\to 0\) and in the case when \(D\) contains a unique non degenerate minimum of \(f\) and \(\partial_nf>0\) on \(\partial D\), where \(\mathbf{n}\) is the unit outward normal vector to \(D\). This formula was actually first derived in \textit{B. J. Matkowsky} and \textit{Z. Schuss} [SIAM J. Appl. Math. 33, 365--382 (1977; Zbl 0369.60071)] using formal computations and we thus provide, in the reversible case, the first proof of it. As a direct consequence, we obtain when \(\varepsilon\to 0\), a sharp asymptotic estimate of the smallest eigenvalue of the operator \[L_\varepsilon=-\varepsilon\Delta+\nabla f\cdot\nabla\] associated with Dirichlet boundary conditions on \(\partial D\). The approach does not require \(f|_{\partial D}\) to be a Morse function. The proof is based on results from \textit{M. Day} [SIAM J. Math. Anal. 13, 532--540 (1982; Zbl 0513.60077); Stochastics 8, 297--323 (1983; Zbl 0504.60032)] and a formula for the mean exit time from \(D\) introduced in \textit{A. Bovier} et al. [J. Eur. Math. Soc. (JEMS) 6, No. 4, 399--424 (2004; Zbl 1076.82045); \textit{A. Bovier} et al., J. Eur. Math. Soc. (JEMS) 7, No. 1, 69--99 (2005; Zbl 1105.82025)].Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes. I: Volume integration.https://www.zbmath.org/1452.652582021-02-12T15:23:00+00:00"You, Hojun"https://www.zbmath.org/authors/?q=ai:you.hojun"Kim, Chongam"https://www.zbmath.org/authors/?q=ai:kim.chongamSummary: This work deals with the development of the direct reconstruction method (DRM) and its application to the volume integration of the discontinuous Galerkin (DG) method on multi-dimensional high-order mixed-curved meshes. The conventional quadrature-based DG methods require the humongous computational cost on high-order curved elements due to their non-linear shape functions. To overcome this issue, the flux function is directly reconstructed in the physical domain using nodal polynomials on a target space in a quadrature-free manner. Regarding the target space and distribution of the nodal points, DRM has two variations: the brute force points (BFP) and shape function points (SFP) methods. In both methods, one nodal point corresponds to one nodal basis function of the target space. The DRM-BFP method uses a set of points that empirically minimizes a condition number of the generalized Vandermonde matrix. In the DRM-SFP method, the conventional nodal points are used to span an enlarged target space of the flux function. It requires a larger number of reconstruction points than DRM-BFP but offers easy extendability to the higher-degree polynomial space and a better de-aliasing effect. A robust way to compute orthonormal polynomials is provided to achieve lower round-off errors. The proposed methods are validated by the 2-D/3-D Navier-Stokes equations on high-order mixed-curved meshes. The numerical results confirm that the DRM volume integration greatly reduces the computational cost and memory overhead of the conventional quadrature-based DG methods on high-order curved meshes while maintaining an optimal order-of-accuracy as well as resolving the flow physics accurately.Hyperbolic stochastic Galerkin formulation for the \(p\)-system.https://www.zbmath.org/1452.650172021-02-12T15:23:00+00:00"Gerster, Stephan"https://www.zbmath.org/authors/?q=ai:gerster.stephan"Herty, Michael"https://www.zbmath.org/authors/?q=ai:herty.michael-matthias"Sikstel, Aleksey"https://www.zbmath.org/authors/?q=ai:sikstel.alekseySummary: We analyze properties of stochastic hyperbolic systems using a Galerkin formulation, which reformulates the stochastic system as a deterministic one that describes the evolution of polynomial chaos modes. We investigate conditions such that the resulting systems are hyperbolic. We state the eigendecompositions in closed form. A Roe flux is presented and theoretical results are illustrated numerically.Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions.https://www.zbmath.org/1452.653932021-02-12T15:23:00+00:00"Cai, Wenjun"https://www.zbmath.org/authors/?q=ai:cai.wenjun"Jiang, Chaolong"https://www.zbmath.org/authors/?q=ai:jiang.chaolong"Wang, Yushun"https://www.zbmath.org/authors/?q=ai:wang.yushun"Song, Yongzhong"https://www.zbmath.org/authors/?q=ai:song.yongzhongSummary: This paper presents two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation, while most existing structure-preserving algorithms are only valid for zero or periodic boundary conditions. The first strategy is based on the conventional second-order central difference quotient but with a cell-centered grid, while the other is established on the regular grid but incorporated with summation by parts (SBP) operators. Both the methodologies can provide conservative semi-discretizations with different forms of Hamiltonian structures and the discrete energy. However, utilizing the existing SBP formulas, schemes obtained by the second strategy can directly achieve higher-order accuracy while it is not obvious for schemes based on the cell-centered grid to make accuracy improved easily. Further combining the implicit midpoint method and the scalar auxiliary variable (SAV) approach, we construct symplectic integrators and linearly implicit energy-preserving schemes for the two-dimensional sine-Gordon equation, respectively. Extensive numerical experiments demonstrate their effectiveness with the homogeneous Neumann boundary conditions.Multi-symplectic quasi-interpolation method for Hamiltonian partial differential equations.https://www.zbmath.org/1452.653972021-02-12T15:23:00+00:00"Sun, Zhengjie"https://www.zbmath.org/authors/?q=ai:sun.zhengjieSummary: In this paper, we propose a multi-symplectic quasi-interpolation method for solving multi-symplectic Hamiltonian partial differential equations. Based on the method of lines, we first discretize the multi-symplectic PDEs using quasi-interpolation method and then employ appropriate time integrators to obtain the full-discrete system. The local conservation properties including multi-symplectic conservation laws, energy conservation laws and momentum conservation laws are discussed in detail. For illustration, we provide two concrete examples: the nonlinear wave equation and the nonlinear Schrödinger equation. The salient feature of our multi-symplectic quasi-interpolation method is that it is valid both on uniform grids and nonuniform grids. The numerical results show the good accuracy and excellent conservation properties of the proposed method.On the asymptotic behavior of the solutions to parabolic variational inequalities.https://www.zbmath.org/1452.350332021-02-12T15:23:00+00:00"Colombo, Maria"https://www.zbmath.org/authors/?q=ai:colombo.maria"Spolaor, Luca"https://www.zbmath.org/authors/?q=ai:spolaor.luca"Velichkov, Bozhidar"https://www.zbmath.org/authors/?q=ai:velichkov.bozhidarThe asymptotic behavior of the solutions to a parabolic variational inequality is studied in connection with a Łojasjiewicz-type inequality. The results are applied to parabolic obstacle and thin-obstacle problems.
Reviewer: Dumitru Motreanu (Perpignan)A fully Lagrangian meshfree framework for PDEs on evolving surfaces.https://www.zbmath.org/1452.651782021-02-12T15:23:00+00:00"Suchde, Pratik"https://www.zbmath.org/authors/?q=ai:suchde.pratik"Kuhnert, Jörg"https://www.zbmath.org/authors/?q=ai:kuhnert.jorgSummary: We propose a novel framework to solve PDEs on moving manifolds, where the evolving surface is represented by a moving point cloud. This has the advantage of avoiding the need to discretize the bulk volume around the surface, while also avoiding the need to have a global mesh. Distortions in the point cloud as a result of the movement are fixed by local adaptation. We first establish a comprehensive Lagrangian framework for arbitrary movement of curves and surfaces given by point clouds. Collision detection algorithms between point cloud surfaces are introduced, which also allow the handling of evolving manifolds with topological changes. We then couple this Lagrangian framework with a meshfree Generalized Finite Difference Method (GFDM) to approximate surface differential operators, which together give a method to solve PDEs on evolving manifolds. The applicability of this method is illustrated with a range of numerical examples, which include advection-diffusion equations with large deformations of the surface, curvature dependent geometric motion, and wave equations on evolving surfaces.A continuation principle to the Cauchy problem of two-dimensional compressible Navier-Stokes equations with variable viscosity.https://www.zbmath.org/1452.351352021-02-12T15:23:00+00:00"Zhong, Xin"https://www.zbmath.org/authors/?q=ai:zhong.xinSummary: The formation of singularity of strong solutions to the two-dimensional (2D) Cauchy problem of the compressible Navier-Stokes equations with variable viscosity is considered. It is shown that for the initial density allowing vacuum, the strong solution exists globally if the density is bounded from above. Some weighted estimates play a crucial role in the proof.A computational wavelet method for variable-order fractional model of dual phase lag bioheat equation.https://www.zbmath.org/1452.651962021-02-12T15:23:00+00:00"Hosseininia, M."https://www.zbmath.org/authors/?q=ai:hosseininia.m"Heydari, M. H."https://www.zbmath.org/authors/?q=ai:heydari.mohammad-hossien"Roohi, R."https://www.zbmath.org/authors/?q=ai:roohi.r"Avazzadeh, Z."https://www.zbmath.org/authors/?q=ai:avazzadeh.zakiehSummary: In this study, we focus on the mathematical model of hyperthermia treatment as one the most constructive and effective procedures. Considering the sophisticated nature of involving phenomena in bioheat transfer inside a living tissue, several models with different levels of simplifications have been proposed. One of the general forms of the bioheat transfer equation which is introduced and studied in this paper for the first time, is the 2D-transient, dual phase lag (DPL), variable-order fractional energy equation. For finding the numerical solution of this general case, we propose an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs). Precisely, the variable-order fractional derivatives of the model are discretized in the first stage, and then the response of the model is expanded by the 2D LWs. Consequently, the main problem is transformed into an equivalent system of algebraic equations, which can be simply tackled. The stability of the proposed method is examined theoretically and experimentally. Also, the procedure is described for one example to examine the computational efficiency of method. The experimental results show the stability and spectral accuracy of the proposed method. According to the achieved results, increasing the fractional order from 0.1 to 1.0, leads to increment of maximum tissue temperatures by about 29\% near the center of the targeted region.A higher-order error estimation framework for finite-volume CFD.https://www.zbmath.org/1452.761362021-02-12T15:23:00+00:00"Tyson, William C."https://www.zbmath.org/authors/?q=ai:tyson.william-c"Roy, Christopher J."https://www.zbmath.org/authors/?q=ai:roy.christopher-jSummary: Computational fluid dynamics is an invaluable tool for both the design and analysis of aerospace vehicles. Reliable error estimation techniques are needed to ensure that simulation results are accurate enough to be used in engineering decision-making processes. In this work, a framework for estimating error and improving solution accuracy is presented. A linearized error transport equation (ETE) is used to estimate local discretization errors. A truncation error estimation technique is proposed which combines aspects of higher-order residual methods and continuous residual methods. The equivalence between adjoint and ETE methods for functional error estimation is demonstrated. Using adjoint/ETE equivalence, the higher-order properties of adjoint methods are extended to ETE methods. Consequently, ETE error estimates are shown to converge to the true discretization error at a higher-order rate. ETE error estimates are then used to correct the entire primal solution, and by extension, all output functionals, to higher order. The computational advantages of this ETE approach are discussed. Results are presented for 1D and 2D inviscid and viscous flow problems on grids with smoothly varying and non-smoothly varying grid metrics.The generalized Fourier series method. Bending of elastic plates.https://www.zbmath.org/1452.740012021-02-12T15:23:00+00:00"Constanda, Christian"https://www.zbmath.org/authors/?q=ai:constanda.christian"Doty, Dale"https://www.zbmath.org/authors/?q=ai:doty.dalePublisher's description: This book explains in detail the generalized Fourier series technique for the approximate solution of a mathematical model governed by a linear elliptic partial differential equation or system with constant coefficients. The power, sophistication, and adaptability of the method are illustrated in application to the theory of plates with transverse shear deformation, chosen because of its complexity and special features. In a clear and accessible style, the authors show how the building blocks of the method are developed, and comment on the advantages of this procedure over other numerical approaches. An extensive discussion of the computational algorithms is presented, which encompasses their structure, operation, and accuracy in relation to several appropriately selected examples of classical boundary value problems in both finite and infinite domains. The systematic description of the technique, complemented by explanations of the use of the underlying software, will help the readers create their own codes to find approximate solutions to other similar models. The work is aimed at a diverse readership, including advanced undergraduates, graduate students, general scientific researchers, and engineers.
The book strikes a good balance between the theoretical results and the use of appropriate numerical applications. The first chapter gives a detailed presentation of the differential equations of the mathematical model, and of the associated boundary value problems with Dirichlet, Neumann, and Robin conditions. The second chapter presents the fundamentals of generalized Fourier series, and some appropriate techniques for orthonormalizing a complete set of functions in a Hilbert space. Each of the remaining six chapters deals with one of the combinations of domain-type (interior or exterior) and nature of the prescribed conditions on the boundary. The appendices are designed to give insight into some of the computational issues that arise from the use of the numerical methods described in the book.Tykhonov well-posedness of a viscoplastic contact problem.https://www.zbmath.org/1452.740862021-02-12T15:23:00+00:00"Sofonea, Mircea"https://www.zbmath.org/authors/?q=ai:sofonea.mircea"Xiao, Yi-bin"https://www.zbmath.org/authors/?q=ai:xiao.yibinSummary: We consider an initial and boundary value problem \(\mathcal{P}\) which describes the frictionless contact of a viscoplastic body with an obstacle made of a rigid body covered by a layer of elastic material. The process is quasistatic and the time of interest is \(\mathbb{R}_+ = [0,+\infty) \). We list the assumptions on the data and derive a variational formulation \(\mathcal{P}_V\) of the problem, in a form of a system coupling an implicit differential equation with a time-dependent variational-hemivariational inequality, which has a unique solution. We introduce the concept of Tykhonov triple \(\mathcal{T} = (I,\Omega, \mathcal{C}) \) where \(I \) is set of parameters, \( \Omega \) represents a family of approximating sets and \(\mathcal{C} \) is a set of sequences, then we define the well-posedness of Problem \(\mathcal{P}_V \) with respect to \(\mathcal{T} \). Our main result is Theorem 3.4, which provides sufficient conditions guaranteeing the well-posedness of \(\mathcal{P}_V \) with respect to a specific Tykhonov triple. We use this theorem in order to provide the continuous dependence of the solution with respect to the data. Finally, we state and prove additional convergence results which show that the weak solution to problem \(\mathcal{P}\) can be approached by the weak solutions of different contact problems. Moreover, we provide the mechanical interpretation of these convergence results.MFS-fading regularization method for inverse BVPs in anisotropic heat conduction.https://www.zbmath.org/1452.652122021-02-12T15:23:00+00:00"Marin, Liviu"https://www.zbmath.org/authors/?q=ai:marin.liviuSummary: We investigate the application of the fading regularization method, in conjunction with the method of fundamental solutions, to the Cauchy problem in 2D anisotropic heat conduction. More precisely, we present a numerical reconstruction of the missing data on an inaccessible part of the boundary from the knowledge of over-prescribed noisy data taken on the remaining accessible boundary part. The accuracy, convergence, stability and robustness of the proposed numerical algorithm, as well as its capability to deblur noisy data, are validated by considering a test example in a 2D simply connected domain.
For the entire collection see [Zbl 1445.65001].Optimal stabilization and time step constraints for the forward Euler-local discontinuous Galerkin method applied to fractional diffusion equations.https://www.zbmath.org/1452.652282021-02-12T15:23:00+00:00"Castillo, Paul"https://www.zbmath.org/authors/?q=ai:castillo.paul-e"Gómez, Sergio"https://www.zbmath.org/authors/?q=ai:gomez.sergio-alejandroSummary: A time dependent model problem with the Riesz or the Riemann-Liouville fractional differential operator of order \(1 < \alpha < 2\) is considered. By penalyzing the primary variable of the minimal dissipation Local Discontinuous Galerkin (mdLDG) method with a term of order \(h^{1 - \alpha}\) and using a von Neumann analysis, stability conditions proportional to \(h^\alpha\) are derived for the forward Euler method and both fractional operators in one dimensional domains. The CFL condition is numerically studied with respect to the approximation degree and the stabilization parameter. Our analysis and computations carried out using explicit high order strong stability preserving Runge-Kutta schemes reveal that the proposed penalization term is suitable for high order approximations and explicit time advancing schemes when \(\alpha\) is close to one. A series of numerical experiments in 1D and 2D problems are presented to validate our theoretical results and those not covered by the theory.The Monte Carlo Markov chain method for solving the modified anomalous fractional sub-diffusion equation.https://www.zbmath.org/1452.651822021-02-12T15:23:00+00:00"Yan, Zhi-Zhong"https://www.zbmath.org/authors/?q=ai:yan.zhizhong"Zheng, Cheng-Feng"https://www.zbmath.org/authors/?q=ai:zheng.cheng-feng"Zhang, Chuanzeng"https://www.zbmath.org/authors/?q=ai:zhang.chuanzengSummary: In this paper, the Monte Carlo Markov chain method for solving the modified anomalous fractional sub-diffusion equation is studied. Most of the previous methods are low in temporal and spatial accuracy order. Based on the idea of Monte Carlo Markov chain method and compact finite difference schemes, a probability model for solving the modified anomalous fractional sub-diffusion equation is established. Numerical examples are given to show the feasibility of the proposed scheme. Compared with the compact finite difference method, the present method is truly meshless and is easy to be implemented with high temporal and spatial accuracy order. And it is also applied to solve partial differential equation in irregular domains.Matrix-free multigrid block-preconditioners for higher order discontinuous Galerkin discretisations.https://www.zbmath.org/1452.653222021-02-12T15:23:00+00:00"Bastian, Peter"https://www.zbmath.org/authors/?q=ai:bastian.peter"Müller, Eike Hermann"https://www.zbmath.org/authors/?q=ai:muller.eike-hermann"Müthing, Steffen"https://www.zbmath.org/authors/?q=ai:muthing.steffen"Piatkowski, Marian"https://www.zbmath.org/authors/?q=ai:piatkowski.marianSummary: Efficient and suitably preconditioned iterative solvers for elliptic partial differential equations (PDEs) of the convection-diffusion type are used in all fields of science and engineering, including for example computational fluid dynamics, nuclear reactor simulations and combustion models. To achieve optimal performance, solvers have to exhibit high arithmetic intensity and need to exploit every form of parallelism available in modern manycore CPUs. This includes both distributed- or shared memory parallelisation between processors and vectorisation on individual cores. The computationally most expensive components of the solver are the repeatedapplications of the linear operator and the preconditioner. For discretisations based on higher-order Discontinuous Galerkin methods, sum-factorisation results in a dramatic reduction of the computational complexity of the operator application while, at the same time, the matrix-free implementation can run at a significant fraction of the theoretical peak floating point performance. Multigrid methods for high order methods often rely on block-smoothers to reduce high-frequency error components within one grid cell. Traditionally, this requires the assembly and expensive dense matrix solve in each grid cell, which counteracts any improvements achieved in the fast matrix-free operator application. To overcome this issue, we present a new matrix-free implementation of block-smoothers. Inverting the block matrices iteratively avoids storage and factorisation of the matrix and makes it is possible to harness the full power of the CPU. We implemented a hybrid multigrid algorithm with matrix-free block-smoothers in the high order Discontinuous Galerkin (DG) space combined with a low order coarse grid correction using algebraic multigrid where only low order components are explicitly assembled. The effectiveness of this approach is demonstrated by solving a set of representative elliptic PDEs of increasing complexity, including a convection dominated problem and the stationary SPE10 benchmark.Lagrangian controllability of the 1-dimensional Korteweg-de Vries equation.https://www.zbmath.org/1452.351662021-02-12T15:23:00+00:00"Gagnon, Ludovick"https://www.zbmath.org/authors/?q=ai:gagnon.ludovickOptimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems.https://www.zbmath.org/1452.652302021-02-12T15:23:00+00:00"Fu, Guosheng"https://www.zbmath.org/authors/?q=ai:fu.guosheng"Shu, Chi-Wang"https://www.zbmath.org/authors/?q=ai:shu.chi-wangSummary: We propose energy-conserving discontinuous Galerkin (DG) methods for symmetric linear hyperbolic systems on general unstructured meshes. Optimal a priori error estimates of order \(k + 1\) are obtained for the semi-discrete scheme in one dimension, and in multi-dimensions on Cartesian meshes when tensor-product polynomials of degree \(k\) are used. A high-order energy-conserving Lax-Wendroff time discretization is also presented.
Extensive numerical results in one dimension, and two dimensions on both rectangular and triangular meshes are presented to support the theoretical findings and to assess the new methods. One particular method (with the doubling of unknowns) is found to be optimally convergent on triangular meshes for all the examples considered in this paper. The method is also compared with the classical (dissipative) upwinding DG method and (conservative) DG method with a central flux. It is numerically observed for the new method to have a superior performance for long time simulations.Well-posedness of the plasma-vacuum interface problem.https://www.zbmath.org/1452.762742021-02-12T15:23:00+00:00"Secchi, Paolo"https://www.zbmath.org/authors/?q=ai:secchi.paolo"Trakhinin, Yuri"https://www.zbmath.org/authors/?q=ai:trakhinin.yuri-lAsymptotically complexity diminishing schemes (ACDS) for kinetic equations in the diffusive scaling.https://www.zbmath.org/1452.650042021-02-12T15:23:00+00:00"Crestetto, Anaïs"https://www.zbmath.org/authors/?q=ai:crestetto.anais"Crouseilles, Nicolas"https://www.zbmath.org/authors/?q=ai:crouseilles.nicolas"Dimarco, Giacomo"https://www.zbmath.org/authors/?q=ai:dimarco.giacomo"Lemou, Mohammed"https://www.zbmath.org/authors/?q=ai:lemou.mohammedSummary: In this work, we develop a new class of numerical schemes for collisional kinetic equations in the diffusive regime. The first step consists in reformulating the problem by decomposing the solution in the time evolution of an equilibrium state plus a perturbation. Then, the scheme combines a Monte Carlo solver for the perturbation with an Eulerian method for the equilibrium part, and is designed in such a way to be uniformly stable with respect to the diffusive scaling and to be consistent with the asymptotic diffusion equation. Moreover, since particles are only used to describe the perturbation part of the solution, the scheme becomes computationally less expensive -- and is thus an asymptotically complexity diminishing scheme (ACDS) -- as the solution approaches the equilibrium state due to the fact that the number of particles diminishes accordingly. This contrasts with standard methods for kinetic equations where the computational cost increases (or at least does not decrease) with the number of interactions. At the same time, the statistical error due to the Monte Carlo part of the solution decreases as the system approaches the equilibrium state: the method automatically degenerates to a solution of the macroscopic diffusion equation in the limit of infinite number of interactions. After a detailed description of the method, we perform several numerical tests and compare this new approach with classical numerical methods on various problems up to the full three dimensional case.A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations.https://www.zbmath.org/1452.652842021-02-12T15:23:00+00:00"Zeng, Fanhai"https://www.zbmath.org/authors/?q=ai:zeng.fanhai"Turner, Ian"https://www.zbmath.org/authors/?q=ai:turner.ian-william"Burrage, Kevin"https://www.zbmath.org/authors/?q=ai:burrage.kevin"Wright, Stephen J."https://www.zbmath.org/authors/?q=ai:wright.stephen-jSummary: In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain.On the management fourth-order Schrödinger-Hartree equation.https://www.zbmath.org/1452.351782021-02-12T15:23:00+00:00"Banquet, Carlos"https://www.zbmath.org/authors/?q=ai:banquet-brango.carlos"Villamizar-Roa, Élder J."https://www.zbmath.org/authors/?q=ai:villamizar-roa.elder-jesusSummary: We consider the Cauchy problem associated to the fourth-order nonlinear Schrödinger-Hartree equation with variable dispersion coefficients. The variable dispersion coefficients are assumed to be continuous or periodic and piecewise constant in time functions. We prove local and global well-posedness results for initial data in \(H^s\)-spaces. We also analyze the scaling limit of the fast dispersion management and the convergence to a model with averaged dispersions.Adversarial uncertainty quantification in physics-informed neural networks.https://www.zbmath.org/1452.681712021-02-12T15:23:00+00:00"Yang, Yibo"https://www.zbmath.org/authors/?q=ai:yang.yibo"Perdikaris, Paris"https://www.zbmath.org/authors/?q=ai:perdikaris.paris-gSummary: We present a deep learning framework for quantifying and propagating uncertainty in systems governed by non-linear differential equations using physics-informed neural networks. Specifically, we employ latent variable models to construct probabilistic representations for the system states, and put forth an adversarial inference procedure for training them on data, while constraining their predictions to satisfy given physical laws expressed by partial differential equations. Such physics-informed constraints provide a regularization mechanism for effectively training deep generative models as surrogates of physical systems in which the cost of data acquisition is high, and training data-sets are typically small. This provides a flexible framework for characterizing uncertainty in the outputs of physical systems due to randomness in their inputs or noise in their observations that entirely bypasses the need for repeatedly sampling expensive experiments or numerical simulators. We demonstrate the effectiveness of our approach through a series of examples involving uncertainty propagation in non-linear conservation laws, and the discovery of constitutive laws for flow through porous media directly from noisy data.Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data.https://www.zbmath.org/1452.681722021-02-12T15:23:00+00:00"Zhu, Yinhao"https://www.zbmath.org/authors/?q=ai:zhu.yinhao"Zabaras, Nicholas"https://www.zbmath.org/authors/?q=ai:zabaras.nicholas-j"Koutsourelakis, Phaedon-Stelios"https://www.zbmath.org/authors/?q=ai:koutsourelakis.phaedon-stelios"Perdikaris, Paris"https://www.zbmath.org/authors/?q=ai:perdikaris.paris-gSummary: Surrogate modeling and uncertainty quantification tasks for PDE systems are most often considered as supervised learning problems where input and output data pairs are used for training. The construction of such emulators is by definition a small data problem which poses challenges to deep learning approaches that have been developed to operate in the big data regime. Even in cases where such models have been shown to have good predictive capability in high dimensions, they fail to address constraints in the data implied by the PDE model. This paper provides a methodology that incorporates the governing equations of the physical model in the loss/likelihood functions. The resulting physics-constrained, deep learning models are trained without any labeled data (e.g. employing only input data) and provide comparable predictive responses with data-driven models while obeying the constraints of the problem at hand. This work employs a convolutional encoder-decoder neural network approach as well as a conditional flow-based generative model for the solution of PDEs, surrogate model construction, and uncertainty quantification tasks. The methodology is posed as a minimization problem of the reverse Kullback-Leibler (KL) divergence between the model predictive density and the reference conditional density, where the later is defined as the Boltzmann-Gibbs distribution at a given inverse temperature with the underlying potential relating to the PDE system of interest. The generalization capability of these models to out-of-distribution input is considered. Quantification and interpretation of the predictive uncertainty is provided for a number of problems.Path integral solutions of the governing equation of SDEs excited by Lévy white noise.https://www.zbmath.org/1452.650212021-02-12T15:23:00+00:00"Xu, Yong"https://www.zbmath.org/authors/?q=ai:xu.yong.1"Zan, Wanrong"https://www.zbmath.org/authors/?q=ai:zan.wanrong"Jia, Wantao"https://www.zbmath.org/authors/?q=ai:jia.wantao"Kurths, Jürgen"https://www.zbmath.org/authors/?q=ai:kurths.jurgenSummary: In this paper, the probability density functions (PDFs) of scalar stochastic differential equations (SDEs) subject to \(\alpha\)-stable Lévy white noise are investigated. The path integral (PI) method is extended to solve one-dimensional space fractional Fokker-Planck-Kolmogorov (FPK) equations, which are the governing equations corresponded to scalar SDEs excited by \(\alpha\)-stable Lévy white noise. First, we derive a short time solution of the one-dimensional space fractional FPK equation, which is used in the Chapman-Kolmogorov-Smoluchowski (CKS) equation to obtain the PI solution. Then, the accuracy of the PI solution is analyzed theoretically in terms of its characteristic function. Our results demonstrate that the PI method has a higher accuracy than the first order finite difference method for one step iteration in time. Finally, several illustrative examples are carried out in detail to verify the feasibility and effectiveness of the PI method for solving one-dimensional space fractional FPK equations. We find that the PI solution agrees well with the exact solution or the Monte Carlo one.Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data.https://www.zbmath.org/1452.351332021-02-12T15:23:00+00:00"Larios, Adam"https://www.zbmath.org/authors/?q=ai:larios.adam"Pei, Yuan"https://www.zbmath.org/authors/?q=ai:pei.yuanSummary: We propose a data assimilation algorithm for the 2D Navier-Stokes equations, based on the Azouani, Olson, and Titi (AOT) algorithm, but applied to the 2D Navier-Stokes-Voigt equations. Adapting the AOT algorithm to regularized versions of Navier-Stokes has been done before, but the innovation of this work is to drive the assimilation equation with observational data, rather than data from a regularized system. We first prove that this new system is globally well-posed. Moreover, we prove that for any admissible initial data, the \(L^2\) and \(H^1\) norms of error are bounded by a constant times a power of the Voigt-regularization parameter \(\alpha>0\), plus a term which decays exponentially fast in time. In particular, the large-time error goes to zero algebraically as \(\alpha\) goes to zero. Assuming more smoothness on the initial data and forcing, we also prove similar results for the \(H^2\) norm.Remarks on the damped nonlinear Schrödinger equation.https://www.zbmath.org/1452.351942021-02-12T15:23:00+00:00"Saanouni, Tarek"https://www.zbmath.org/authors/?q=ai:saanouni.tarekSummary: It is the purpose of this note to investigate the initial value problem for a focusing semi-linear damped Schrödinger equation. Indeed, in the energy sub-critical regime, one obtains global well-posedness and scattering in the energy space, depending on the order of the fractional dissipation.Pointwise control of the linearized Gear-Grimshaw system.https://www.zbmath.org/1452.351652021-02-12T15:23:00+00:00"Capistrano-Filho, Roberto de A."https://www.zbmath.org/authors/?q=ai:capistrano-filho.roberto-de-a"Komornik, Vilmos"https://www.zbmath.org/authors/?q=ai:komornik.vilmos"Pazoto, Ademir F."https://www.zbmath.org/authors/?q=ai:pazoto.ademir-fernandoSummary: In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability by using two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise controllability when only one control function acts internally. In both cases we can find, under some assumptions on the coefficients of the system, the sharp time of the controllability.Infinite time blow-up for the fractional heat equation with critical exponent.https://www.zbmath.org/1452.352442021-02-12T15:23:00+00:00"Musso, Monica"https://www.zbmath.org/authors/?q=ai:musso.monica"Sire, Yannick"https://www.zbmath.org/authors/?q=ai:sire.yannick"Wei, Juncheng"https://www.zbmath.org/authors/?q=ai:wei.juncheng"Zheng, Youquan"https://www.zbmath.org/authors/?q=ai:zheng.youquan"Zhou, Yifu"https://www.zbmath.org/authors/?q=ai:zhou.yifuAuthors' abstract: ``We consider positive solutions for the fractional heat equation with critical exponent
\[
\begin{cases}
u_t = -(-\Delta)^s + u^{\frac{n+2s}{n-2s}} &\text{ in } \Omega \times(0,\infty),\\
u = 0 &\text{ on } (\mathbb R^n \setminus \Omega) \times(0,\infty),\\
u(\cdot,0) = u_0 &\text{ in } \mathbb R^n,\\
\end{cases}
\]
where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^n\), \(n>4s\), \(s\in (0, 1)\), \(u:\mathbb{R}^n\times [0, \infty)\to \mathbb{R}\) and \(u_0\) is a positive smooth initial datum with \(u_0|_{\mathbb{R}^n\setminus \Omega} = 0\). We prove the existence of \(u_0\) such that the solution blows up precisely at prescribed distinct points \(q_1,\dots,q_k\) in \(\Omega\) as \(t \to \infty\). The main ingredient of the proofs is a new inner-outer gluing scheme for the fractional parabolic problems.''
Recently, several researchers have been interested in exploring both classical and modern definitions of fractional derivatives and integrals in order to have a good understanding to some phenomena in physics and engineering that can be modelled better by the fractional-order derivatives than the integer ones. It is well-known that fractional calculus is a very powerful tool in modelling many phenomena that exhibit memory effect which is one of the main advantages of applying fractional derivatives to the models in natural sciences and engineering. However, some systems formulated in the sense of fractional derivatives cannot be solved analytical, and if they can be solved, the analytical solution will be very complicated to obtain for such systems. Generally, any new mathematical definition comes with advantages and disadvantages. Therefore, pure and applied mathematicians have dedicated their efforts to overcome some challenges that are associated with the existence of solutions to some certain fractional differential equations by proposing new approximate-analytical approaches, numerical techniques, or investigating some topics of the mathematical analysis of some interesting fractional differential equations. For more examples about some recent studies related to fractional differential equations, we refer the reader to [\textit{N. Cusimano} et al., ESAIM, Math. Model. Numer. Anal. 54, No. 3, 751--774 (2020; Zbl 1452.35237); \textit{S. D. Taliaferro}, J. Math. Pures Appl. (9) 133, 287--328 (2020; Zbl 1437.35697); \textit{M. K. A. Kaabar} et al., ``New approximate-analytical solutions for the nonlinear fractional Schrödinger equation with second-order spatio-temporal dispersion via double Laplace transform method'', Preprint, \url{arXiv:2010.10977}; \textit{A. Ghanmi} and \textit{Z. Zhang}, Bull. Korean Math. Soc. 56, No. 5, 1297--1314 (2019; Zbl 1432.34012); \textit{L. Zhang} et al., Topol. Methods Nonlinear Anal. 54, No. 2A, 537--566 (2019; Zbl 1445.47039); \textit{T. Ghosh} et al., Anal. PDE 13, No. 2, 455--475 (2020; Zbl 1439.35530); \textit{F. Camilli } and \textit{A. Goffi}, Nonlinear Differ. Equ. Appl. 27, No. 22, 1--37 (2020; Zbl 1452.35234); \textit{K. Ryszewska}, J. Math. Anal. Appl. 483, No. 2, Article ID 123654, 17 p. (2020; Zbl 1436.35323); \textit{M. Kaabar}, ``Novel methods for solving the conformable wave equations'', J. New Theory 2020, No. 31, 56--85 (2020); \textit{F. Martínez} et al., ``New results on complex conformable integral'', AIMS Mathematics. 5, No. 6, 7695--7710 (2020); \textit{F. Martínez} et al., ``Note on the conformable boundary value problems: Sturm's theorems and Green's function'', Preprint (2020; \url{doi: 10.20944/preprints202009.0440.v1}); \textit{Y. Gholami} and \textit{K. Ghanbari}, S\(\vec{\text{e}}\)MA J. 75, No. 2, 305--333 (2018; Zbl 1400.26012); \textit{H. Dong} and \textit{D. Kim}, J. Funct. Anal. 278, No. 3, Article ID 108338, 66 p. (2020; Zbl 1427.35316); \textit{F. M. Gaafar}, J. Egypt. Math. Soc. 26, 469--482 (2018; Zbl 1441.34007); \textit{F. Martínez} et al., ``Some new results on conformable fractional power series'', Asia Pac. J. Math. 7, No. 31, 1--14 (2020); \textit{L. C. F. Ferreira} et al., Bull. Sci. Math. 153, 86--117 (2019; Zbl 1433.35185); \textit{F. Martínez} et al., ``Note on the conformable fractional derivatives and integrals of complex-valued functions of a real variable'', IAENG Int. J. Appl. Math. 50, No. 3, 609--615 (2020)], and [\textit{K. Hidano } and \textit{C. Wang}, Sel. Math., New Ser. 25, No. 1, 1--28 (2019; Zbl 1428.35662)]. In addition, for more research-based information on the topic of nonlocal modeling, analysis, and computation, the reader is also highly recommended to refer to recently published book by Qiang Du about this topic of research where this book offers all needed tools for young researchers to conduct research studies on this topic [\textit{Q. Du}, Nonlocal modeling, analysis, and computation. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2019; Zbl 1423.00007)]. In this paper, the authors have investigated the infinite time (\(t \rightarrow \infty\)) blow-up for the fractional heat equation with critical exponent by proposing a new technique called the inner-outer gluing scheme which is helpful in solving fractional parabolic problems. This scheme when it is used with a parabolic equation, the solution is needed to be found for the linearized form of the studied equation around the bubble with sufficiently fast decay as the authors mentioned in their research paper. According to the authors of this research paper, the fractional part of the problem makes working with this scheme a real challenge; therefore, the blow-up argument is applied depending on the nondegeneracy of bubbles and with the help of the removable singular property for the limit equations in this paper. The proposed problem in this paper can be written as follows:
Given a function \(u: \Re^{n}\times[0,\infty)\rightarrow \Re\); \(\Omega\) is assumed to be smooth bounded domain in \(\Re^{n}\) where \(n\geq1\); There is a smooth, positive initial datum, denoted by \(u_{0}\) such that \(u_{0}|_{\Re^{n} \backslash \Omega}\); For \(0<s<1\) where \(n>4s\) such that \(n\geq1\), any point \(x\in \Re^{n}\), and a positive normalizing constant, denoted by \(C(n,s)\), the fractional Laplacian, denoted by
\((-\Delta)^{s}u(x)\) is expressed in this paper as: \((-\Delta)^{s}u(x):= C(n,s)P.V.\int_{\Re}^{n} \frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\). Then, the proposed fractional heat equation with critical exponent in this paper
can be written as follows:
\(u_{t}=-(-\Delta)^{s}u+u^{\frac{n+2s}{n-2s}}\) in \(\Omega \times (0,\infty)\), subject to \(u=0\) on \((\Re^{n}\backslash \Omega)\times (0,\infty)\), and \(u(\cdot,0)=u_{0}\) in \(\Re^{n}\).
This paper is well-organized, and it has 64 pages. Although this research paper is very long, this paper is written in a simplified way so the reader can read and understand every single page because it is worthy and contains valuable results and information. This research work consists of six major parts: the constructed approximate form of solutions, the procedure of the inner-outer gluing scheme, the outer problem, the linear theory for the formulated nonlinear nonlocal problem (see equation 1.12 in the research paper), investigating the solvability condition for equation 1.12, and the inner problem (gluing part which is the proof of theorem 1.1 (the main objective of this paper)) for solving the formulated problem in equation 1.12 via the method of the linear theory of the proposed linear parabolic problem (see equation 1.13 in the research paper) and with the help of the Contraction Mapping Theorem. Finally, the authors have done a great job by proving the existence of positive smooth initial datum, denoted by \(u_{0}\), where the obtained solution blows up precisely at distinct points in the assumed smooth bounded domain in \(\Re^{n}\) where \(n\geq1\) as \( t \rightarrow \infty \). All in all, this interesting result will surely motivate other interested researchers and mathematicians to work on improving or developing this scheme in future research works.
Reviewer: Mohammed Kaabar (Gelugor)On the solution of 3D problems in physics: from the geometry definition in CAD to the solution by a meshless method.https://www.zbmath.org/1452.653722021-02-12T15:23:00+00:00"Mirfatah, S. M."https://www.zbmath.org/authors/?q=ai:mirfatah.s-m"Boroomand, B."https://www.zbmath.org/authors/?q=ai:boroomand.bijan"Soleimanifar, E."https://www.zbmath.org/authors/?q=ai:soleimanifar.eSummary: This paper presents a simple meshfree approach, from the grid generation to the final solution, for the simulation of 3D problems geometrically defined by CAD. First, the domain grid for 3D problems is generated through a discrete searching algorithm. Non-Uniform Rational B-Splines (NURBS) are employed, just as a choice among similar tools, to define the boundaries through geometrical control points obtained by the CAD program. A predefined regular grid of nodes, embedding the whole geometry, is then trimmed to follow the constructed boundaries. The spatial solution is performed by construction of shape functions satisfying the governing differential equation, through using exponential basis functions (EBFs). A straightforward strategy is proposed for choosing appropriate EBFs via their shape-parameters making the method efficient for solution of 3D problems. Several 3D Laplace and Helmholtz problems are solved and the results are compared with those of commercial programs to show the efficiency of the method.A fast compact time integrator method for a family of general order semilinear evolution equations.https://www.zbmath.org/1452.651612021-02-12T15:23:00+00:00"Huang, Jianguo"https://www.zbmath.org/authors/?q=ai:huang.jianguo"Ju, Lili"https://www.zbmath.org/authors/?q=ai:ju.lili"Wu, Bo"https://www.zbmath.org/authors/?q=ai:wu.boSummary: In this paper we develop a fast compact time integrator method for numerically solving a family of general order semilinear evolution equations in regular domains. The spatial discretization is carried out by a fourth-order accurate compact difference scheme in which fast Fourier transform can be utilized for efficient implementation. The resulting semi-discretized problem consists of a system of ordinary differential equations whose solution can be explicitly expressed in term of time integrators, and a desired numerical method is then obtained by further adopting multistep approximations of the nonlinear terms based on the solution formula. Linear stability analysis is performed for the method for second-order in time evolution equations. Extensive numerical experiments with applications are also presented to demonstrate efficiency, accuracy, and stability of the proposed method in practice.Supplementary variable method for structure-preserving approximations to partial differential equations with deduced equations.https://www.zbmath.org/1452.651602021-02-12T15:23:00+00:00"Hong, Qi"https://www.zbmath.org/authors/?q=ai:hong.qi"Li, Jun"https://www.zbmath.org/authors/?q=ai:li.jun.8|li.jun.10|li.jun.3|li.jun.14|li.jun|li.jun.12|li.jun.1|li.jun.11|li.jun.2|li.jun.13|li.jun.7|li.jun.6"Wang, Qi"https://www.zbmath.org/authors/?q=ai:wang.qi.3|wang.qi.1|wang.qi.6|wang.qi.2|wang.qi.4|wang.qi|wang.qi.5Summary: We present a supplementary variable method (SVM) for developing structure-preserving numerical approximations to a partial differential equation system with deduced equations. The PDE system with deduced equations constitutes an over-determined, yet consistent and structurally unstable system of equations. We augment a proper set of supplementary variables to the over-determined system to make it well-determined with a stable structure. We then discretize the modified system to arrive at a structure-preserving numerical approximation to the over-determined PDE system. We illustrate the idea using a dissipative network generating partial differential equation model by developing an energy-dissipation-rate preserving scheme. We then simulate the network generating phenomenon using the numerical scheme. This numerical method is so general that it applies literally to any PDE systems with deduced equations.Spectrally-accurate numerical method for acoustic scattering from doubly-periodic 3D multilayered media.https://www.zbmath.org/1452.653882021-02-12T15:23:00+00:00"Cho, Min Hyung"https://www.zbmath.org/authors/?q=ai:cho.minhyungSummary: A periodizing scheme and the method of fundamental solutions are used to solve acoustic wave scattering from doubly-periodic three-dimensional multilayered media. A scattered wave in a unit cell is represented by the sum of the near and distant contribution. The near contribution uses the free-space Green's function and its eight immediate neighbors. The contribution from the distant sources is expressed using proxy source points over a sphere surrounding the unit cell and its neighbors. The Rayleigh-Bloch radiation condition is applied to the top and bottom layers. Extra unknowns produced by the periodizing scheme in the linear system are eliminated using a Schur complement. The proposed numerical method avoids using singular quadratures and the quasi-periodic Green's function or complicated lattice sum techniques. Therefore, the proposed scheme is robust at all scattering parameters including Wood anomalies. The algorithm is also applicable to electromagnetic problems by using the dyadic Green's function. Numerical examples with 10-digit accuracy are provided. Finally, reflection and transmission spectra are computed over a wide range of incident angles for device characterization.Solution of the boundary-value problem of heat conduction with periodic boundary conditions.https://www.zbmath.org/1452.652102021-02-12T15:23:00+00:00"Kanca, F."https://www.zbmath.org/authors/?q=ai:kanca.fatma"Baglan, I."https://www.zbmath.org/authors/?q=ai:baglan.irem-sakincSummary: We investigate the solution of the inverse problem for a linear two-dimensional parabolic equation with periodic boundary and integral overdetermination conditions. Under certain natural regularity and consistency conditions imposed on the input data, we establish the existence, uniqueness of the solution, and its continuous dependence on the data by using the generalized Fourier method. In addition, an iterative algorithm is constructed for the numerical solution of the problem.Forced \((2+1)\)-dimensional discrete three-wave equation.https://www.zbmath.org/1452.351642021-02-12T15:23:00+00:00"Zhu, Junyi"https://www.zbmath.org/authors/?q=ai:zhu.junyi"Zhou, Sishou"https://www.zbmath.org/authors/?q=ai:zhou.sishou"Qiao, Zhijun"https://www.zbmath.org/authors/?q=ai:qiao.zhijunA critical fractional Laplace equation in the resonant case.https://www.zbmath.org/1452.490052021-02-12T15:23:00+00:00"Servadei, Raffaella"https://www.zbmath.org/authors/?q=ai:servadei.raffaellaSummary: In this paper we complete the study of the following non-local fractional equation involving critical nonlinearities
\[
\begin{cases}
(-\Delta)^s u-\lambda u=|u|^{2^*-2}u & {\text{in }} \Omega,\\
u=0 & {\text{in }} \mathbb{R}^n\setminus \Omega,
\end{cases}
\]
started in the recent papers [\textit{R. Servadei}, Adv. Nonlinear Anal. 2, No. 3, 235--270 (2013; Zbl 1273.49011); \textit{R. Servadei} and \textit{E. Valdinoci}, Trans. Am. Math. Soc. 367, No. 1, 67--102 (2015; Zbl 1323.35202); \textit{R. Servadei} and \textit{E. Valdinoci}, Rev. Mat. Complut. 28, No. 3, 655--676 (2015; Zbl 1338.35481)].
Here \(s\in (0,1)\) is a fixed parameter, \((-\Delta )^s\) is the fractional Laplace operator, \(\lambda\) is a positive constant, \(2^*=2n/(n-2s)\) is the fractional critical Sobolev exponent and \(\Omega\) is an open bounded subset of \(\mathbb R^n\), \(n> 2s\), with Lipschitz boundary. Aim of this paper is to study this critical problem in the special case when \(n\not=4s\) and \(\lambda\) is an eigenvalue of the operator \((-\Delta)^s\) with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a non-trivial solution, so that with the results obtained in [Servadei, loc. cit.; \textit{R. Servadei} and \textit{E. Valdinoci}, Trans. Am. Math. Soc. 367, No. 1, 67--102 (2015; Zbl 1323.35202); \textit{R. Servadei} and \textit{E. Valdinoci}, Rev. Mat. Complut. 28, No. 3, 655--676 (2015; Zbl 1338.35481)], we are able to show that this critical problem admits a nontrivial solution provided
\begin{itemize}
\item \(n> 4s\) and \(\lambda> 0\),
\item \(n=4s\) and \(\lambda> 0\) is different from the eigenvalues of \((-\Delta)^s\),
\item \(2s< n< 4s\) and \(\lambda> 0\) is sufficiently large.
\end{itemize}
In this way we extend completely the famous result of Brezis and Nirenberg (see [\textit{H. Brézis} and \textit{L. Nirenberg}, Commun. Pure Appl. Math. 36, 437--477 (1983; Zbl 0541.35029); \textit{A. Capozzi} et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2, No. 6, 463--470 (1985; Zbl 0612.35053); \textit{F. Gazzola} and \textit{B. Ruf}, Adv. Differ. Equ. 2, No. 4, 555--572 (1997; Zbl 1023.35508); \textit{D. Zhang}, Nonlinear Anal., Theory Methods Appl. 13, No. 4, 353--372 (1989; Zbl 0704.35053)]) for the critical Laplace equation to the non-local setting of the fractional Laplace equation.On the convergence of the local discontinuous Galerkin method applied to a stationary one dimensional fractional diffusion problem.https://www.zbmath.org/1452.653252021-02-12T15:23:00+00:00"Castillo, Paul."https://www.zbmath.org/authors/?q=ai:castillo.paul-e"Gómez, Sergio Alejandro"https://www.zbmath.org/authors/?q=ai:gomez.sergio-alejandroSummary: The mixed formulation of the Local Discontinuous Galerkin (LDG) method is presented for a two boundary value problem that involves the Riesz operator with fractional order \(1<\alpha <2\). Well posedness of the stabilized and non stabilized LDG method is proved. Using a penalty term of order \(\mathcal{O}(h^{1-\alpha})\) a sharp error estimate in a mesh dependent energy semi-norm is developed for sufficiently smooth solutions. Error estimates in the \(L^2\)-norm are obtained for two auxiliary variables which characterize the LDG formulation. Our analysis indicates that the non stabilized version of the method achieves higher order of convergence for all fractional orders. A numerical study suggests a less restrictive, \(\mathcal{O}(h^{-\alpha})\), spectral condition number of the stiffness matrix by using the proposed penalty term compared to the \(\mathcal{O}(h^{-2})\) growth obtained when the traditional \(\mathcal{O}(h^{-1})\) penalization term is chosen. The sharpness of our error estimates is numerically validated with a series of numerical experiments. The present work is the first attempt to elucidate the main differences between both versions of the method.Asymptotically compatible schemes for stochastic homogenization.https://www.zbmath.org/1452.350222021-02-12T15:23:00+00:00"Sun, Qi"https://www.zbmath.org/authors/?q=ai:sun.qi"Du, Qiang"https://www.zbmath.org/authors/?q=ai:du.qiang"Ming, Ju"https://www.zbmath.org/authors/?q=ai:ming.juNumerical algorithms of the two-dimensional Feynman-Kac equation for reaction and diffusion processes.https://www.zbmath.org/1452.651732021-02-12T15:23:00+00:00"Nie, Daxin"https://www.zbmath.org/authors/?q=ai:nie.daxin"Sun, Jing"https://www.zbmath.org/authors/?q=ai:sun.jing"Deng, Weihua"https://www.zbmath.org/authors/?q=ai:deng.weihuaThis paper considers the numerical solution of a backward Feynman-Kac equation which governs the distribution of functionals of the path for a particle undergoing both reaction and diffusion. The method is based on the first-order and second-order schemes for discretizing the time tempered fractional substantial derivative and the finite difference method to approximate the two-dimensional tempered fractional Laplacian. Error estimates of the schemes are proved, which depend only on the regularity of the solution on \(\Omega\) rather than on the whole space. Numerical examples are included.
Reviewer: Zhiming Chen (Beijing)A truly forward semi-Lagrangian WENO scheme for the Vlasov-Poisson system.https://www.zbmath.org/1452.761622021-02-12T15:23:00+00:00"Sirajuddin, David"https://www.zbmath.org/authors/?q=ai:sirajuddin.david"Hitchon, William N. G."https://www.zbmath.org/authors/?q=ai:hitchon.william-n-gSummary: A class of modular high order forward semi-Lagrangian (FSL) schemes with a number of advantages (outlined below) for linear advection equations is extended to have high resolution, in the sense that sharp gradients can be captured without loss of accuracy, by means of weighted essentially non-oscillatory derivative (DWENO) calculations based on finite differences. Principally, our approach (the \textit{convected scheme}, CS) differs from conventional high order FSL schemes in the means by which it reaches higher order accuracy. The high order CS progressed in this work reserves use of a low order histogram representation of the solution and a mass-conservative volume-weighted projection operator which ordinarily produces a second order accurate solution. A higher order version is obtained by compensating for the amount by which the CS solution and the exact solution disagree at every order in a Taylor series sense. Instead of including these error terms algebraically, which changes the projection operator itself, the novel idea of the CS is to incorporate this information geometrically as a commensurate adjustment to the volume-weighted proportion assigned to each cell. We leverage DWENO approximations to compute these Taylor series error terms more sensitively, and show the corresponding \textit{WENO convected schemes} (WENO-CS) achieve the desired dual high order and high resolution behavior. Fifth, seventh, and ninth order WENO-CS base solvers are presented for the 1D constant speed advection equation prototype, and are employed in a Strang splitting approach for the numerical solution of linear and nonlinear two-dimensional hyperbolic equations. Results are showcased for 2D linear transport and rigid body rotation, as well as for classic benchmark problems for the \((1 + 1)\)-dimensional Vlasov-Poisson system including Landau damping, and the two-stream instability. The schemes enjoy the usual benefits of semi-Lagrangian schemes (e.g. no CFL time step restriction) while further claiming distinct advantages afforded to forward schemes including automatic mass conservation, compact support, and therefore straightforward parallelization.Some hyperbolic conservation laws on \(\mathbb{R}^n\).https://www.zbmath.org/1452.350962021-02-12T15:23:00+00:00"Bedida, Nabila"https://www.zbmath.org/authors/?q=ai:bedida.nabila"Hermas, Nadji"https://www.zbmath.org/authors/?q=ai:hermas.nadjiSummary: In this paper, we prove the existence and the uniqueness of maximum classical solutions in the temporal variable for some quasi-linear hyperbolic systems.A direct approach to imaging in a waveguide with perturbed geometry.https://www.zbmath.org/1452.653092021-02-12T15:23:00+00:00"Borcea, Liliana"https://www.zbmath.org/authors/?q=ai:borcea.liliana"Cakoni, Fioralba"https://www.zbmath.org/authors/?q=ai:cakoni.fioralba"Meng, Shixu"https://www.zbmath.org/authors/?q=ai:meng.shixuSummary: We introduce a direct, linear sampling approach to imaging in an acoustic waveguide with sound hard walls. The waveguide terminates at one end and has unknown geometry due to compactly supported wall deformations. The goal of imaging is to determine these deformations and to identify localized scatterers in the waveguide, using a remote array of sensors that emits time harmonic probing waves and records the echoes. We present a theoretical analysis of the imaging approach and illustrate its performance with numerical simulations.Level set methods for stochastic discontinuity detection in nonlinear problems.https://www.zbmath.org/1452.650202021-02-12T15:23:00+00:00"Pettersson, Per"https://www.zbmath.org/authors/?q=ai:pettersson.per"Doostan, Alireza"https://www.zbmath.org/authors/?q=ai:doostan.alireza"Nordström, Jan"https://www.zbmath.org/authors/?q=ai:nordstrom.janSummary: Stochastic problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zeros of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods.A regularity criterion at one scale without pressure for suitable weak solutions to the Navier-Stokes equations.https://www.zbmath.org/1452.760392021-02-12T15:23:00+00:00"Wang, Yanqing"https://www.zbmath.org/authors/?q=ai:wang.yanqing"Wu, Gang"https://www.zbmath.org/authors/?q=ai:wu.gang"Zhou, Daoguo"https://www.zbmath.org/authors/?q=ai:zhou.daoguoSummary: In this paper, we continue our work in [\textit{Q. Jiu} et al., J. Math. Fluid Mech. 21, No. 2, Paper No. 22, 16 p. (2019; Zbl 1411.76022)] to derive \({\varepsilon}\)-regularity criteria at one scale without pressure for suitable weak solutions to the Navier-Stokes equations. We establish an \({\varepsilon}\)-regularity criterion below of suitable weak solutions, for any \[\delta > 0,\iint\limits_{Q(1)} | u |^{\frac{5}{2} + \delta} d x d t \leq \varepsilon .\] As an application, we extend the previous corresponding results concerning the improvement of the classical Caffarelli-Kohn-Nirenberg theorem by a logarithmic factor.Local existence of strong solutions of a fluid-structure interaction model.https://www.zbmath.org/1452.351542021-02-12T15:23:00+00:00"Mitra, Sourav"https://www.zbmath.org/authors/?q=ai:mitra.souravSummary: We are interested in studying a system coupling the compressible Navier-Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler-Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system.Reconstruction of generalized impedance functions for 3D acoustic scattering.https://www.zbmath.org/1452.653102021-02-12T15:23:00+00:00"Ivanyshyn Yaman, Olha"https://www.zbmath.org/authors/?q=ai:ivanyshyn.olhaSummary: We consider the inverse obstacle scattering problem of determining both of the surface impedance functions from far field measurements for a few incident plane waves at a fixed frequency. The reconstruction algorithm we propose is based on an iteratively regularized Newton-type method and nonlinear integral equations. The mathematical foundation of the method is presented and the feasibility is illustrated by numerical examples.Weak discrete maximum principle of finite element methods in convex polyhedra.https://www.zbmath.org/1452.653482021-02-12T15:23:00+00:00"Leykekhman, Dmitriy"https://www.zbmath.org/authors/?q=ai:leykekhman.dmitriy"Li, Buyang"https://www.zbmath.org/authors/?q=ai:li.buyangSummary: We prove that the Galerkin finite element solution \(u_h\) of the Laplace equation in a convex polyhedron \(\varOmega \), with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree \(r\geqslant 1\), satisfies the following weak maximum principle:
\[
\| u_h\|_{L^{\infty}(\varOmega)} \leqslant C\left \Vert u_h\right \Vert_{L^{\infty}(\partial \varOmega)} ,
\]
with a constant \(C\) independent of the mesh size \(h\). By using this result, we show that the Ritz projection operator \(R_h\) is stable in \(L^\infty\) norm uniformly in \(h\) for \(r\geq 2\), i.e.,
\[
\Vert R_hu\Vert_{L^{\infty}(\varOmega)} \leqslant C\Vert u\Vert_{L^{\infty}(\varOmega)} .
\]
Thus we remove a logarithmic factor appearing in the previous results for convex polyhedral domains.Novel multilevel techniques for convergence acceleration in the solution of systems of equations arising from RBF-FD meshless discretizations.https://www.zbmath.org/1452.653782021-02-12T15:23:00+00:00"Zamolo, Riccardo"https://www.zbmath.org/authors/?q=ai:zamolo.riccardo"Nobile, Enrico"https://www.zbmath.org/authors/?q=ai:nobile.enrico"Šarler, Božidar"https://www.zbmath.org/authors/?q=ai:sarler.bozidarSummary: The present paper develops two new techniques, namely additive correction multicloud (ACMC) and smoothed restriction multicloud (SRMC), for the efficient solution of systems of equations arising from Radial Basis Function-generated Finite Difference (RBF-FD) meshless discretizations of partial differential equations (PDEs). RBF-FD meshless methods employ arbitrary distributed nodes, without the need to generate a mesh, for the numerical solution of PDEs. The proposed techniques are specifically designed for the RBF-FD data structure and employ simple restriction and interpolation strategies in order to obtain a hierarchy of coarse-level node distributions and the corresponding correction equations. Both techniques are kept as simple as possible in terms of code implementation, which is an important feature of meshless methods. The techniques are verified on 2D and 3D Poisson equations, defined on non-trivial domains, showing very high benefits in terms of both time consumption and work to convergence when comparing the present techniques to the most common solver approaches. These benefits make the RBF-FD approach competitive with standard grid-based approaches when the number of nodes is very high, allowing large size problems to be tackled by the RBF-FD method.Blow-up for generalized Boussinesq equation with double damping terms.https://www.zbmath.org/1452.350452021-02-12T15:23:00+00:00"Hao, Jianghao"https://www.zbmath.org/authors/?q=ai:hao.jianghao"Gao, Aiyuan"https://www.zbmath.org/authors/?q=ai:gao.aiyuanSummary: In this paper, we consider the Cauchy problem for a generalized Boussinesq equation with double damping terms. By using improved convexity method combined with potential well method and Fourier transform, we show the finite time blow-up of the solution with arbitrarily high initial energy while many similar results require the corresponding energy to be less than some certain numbers.Fluctuation splitting Riemann solver for a non-conservative modeling of shear shallow water flow.https://www.zbmath.org/1452.651852021-02-12T15:23:00+00:00"Bhole, Ashish"https://www.zbmath.org/authors/?q=ai:bhole.ashish"Nkonga, Boniface"https://www.zbmath.org/authors/?q=ai:nkonga.boniface"Gavrilyuk, Sergey"https://www.zbmath.org/authors/?q=ai:gavrilyuk.sergey-l"Ivanova, Kseniya"https://www.zbmath.org/authors/?q=ai:ivanova.kseniyaSummary: In this paper we propose a fluctuation splitting finite volume scheme for a non-conservative modeling of shear shallow water flow (SSWF). This model was originally proposed by \textit{V. M. Teshukov} [Prikl. Mekh. Tekh. Fiz. 48, No. 3, 8--15 (2007; Zbl 1150.76335); translation in J. Appl. Mech. Tech. Phys. 48, No. 3, 303--309 (2007)] and was extended to include modeling of friction by \textit{S. Gavrilyuk} et al. [J. Comput. Phys. 366, 252--280 (2018; Zbl 1406.65068)]. The directional splitting scheme proposed by Gavrilyuk et al. [loc. cit.] is tricky to apply on unstructured grids. Our scheme is based on the physical splitting in which we separate the characteristic waves of the model to form two different hyperbolic sub-systems. The fluctuations associated with each sub-systems are computed by developing Riemann solvers for these sub-systems in a local coordinate system. These fluctuations enable us to develop a Godunov-type scheme that can be easily applied on mixed/unstructured grids. While the equation of energy conservation is solved along with the SSWF model in [loc. cit.], in this paper we solve only SSWF model equations. We develop a cell-centered finite volume code to validate the proposed scheme with the help of some numerical tests. As expected, the scheme shows first order convergence. The numerical simulation of 1D roll waves shows a good agreement with the experimental results. The numerical simulations of 2D roll waves show similar transverse wave structures as observed in [loc. cit.].Global hypoelliptic vector fields in ultradifferentiable classes and normal forms.https://www.zbmath.org/1452.350682021-02-12T15:23:00+00:00"Albanese, Angela A."https://www.zbmath.org/authors/?q=ai:albanese.angela-annaThe problem of the global hypoellipticy on the torus attracted attention in the last years. The first contribution for the 2-torus [\textit{S. J. Greenfield} and \textit{N. R. Wallach}, Proc. Am. Math. Soc. 31, 112--114 (1972; Zbl 0229.35023)], where the authors proved the global hypoellipticity of the vector fields with a leading irrational non-Liouville constant coefficient. The result was later generalized to higher-order operators, see, for example, [\textit{T. Gramchev} et al., Rend. Semin. Mat., Torino 51, No. 2, 145--172 (1993; Zbl 0824.35027)]. The main reference for the present new contribution is [\textit{W. Chen} and \textit{M. Y. Chi}, Commun. Partial Differ. Equations 25, No. 1--2, 337--354 (2000; Zbl 0945.35007)], where the authors proved that a vector field with real variable coefficients on the \(n\)-dimensional torus is hypoelliptic if and only if it can be reduced by a diffeomorphism to the case of the constant cofficients satisfying a Diophantine condition, generalization of the non-Liouville assumption. In the present paper, the same result is proved to be valid in the frame of the Gevrey regularity and, more generally, in the setting of the ultradifferentiable classes of [\textit{R. W. Braun} et al., Result. Math. 17, No. 3--4, 206--237 (1990; Zbl 0735.46022)], under suitable Diophantine conditions. The arguments involve a precise analysis of diffeomorphisms of ultradifferentiable classes and a new Paley-Wiener-type theorem for ultradifferentiable function spaces.
Reviewer: Luigi Rodino (Torino)An improved WLS-ENO method for solving hyperbolic conservation laws.https://www.zbmath.org/1452.761502021-02-12T15:23:00+00:00"Chen, Li Li"https://www.zbmath.org/authors/?q=ai:chen.lili"Huang, Cong"https://www.zbmath.org/authors/?q=ai:huang.congSummary: Liu and Jiao proposed a WLS-ENO method (Weighted-Least-Squares based Essentially Non-Oscillatory). Comparing to the WENO method (Weighted ENO), the WLS-ENO method is more flexible for solving the hyperbolic conservation laws. Notice that, a WLS problem is generated for each WLS-ENO procedure, although the QR factorization with column pivoting is employed, the computational cost of solving WLS problem is still heavy. So in this paper, we will propose an improved WLS-ENO method for improving the efficiency. The basic idea of new method is that, let the reconstructed polynomial approximate the cell averages of some smooth cells in conservation fashion, and approximate other cell averages in WLS fashion, so that the resulted WLS problem only has one unknown. Because the resulted WLS problem can be solved by using the normal equation directly, the whole computational cost is small, thus the new method has better efficiency.A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes.https://www.zbmath.org/1452.761432021-02-12T15:23:00+00:00"Zhu, Jun"https://www.zbmath.org/authors/?q=ai:zhu.jun"Shu, Chi-Wang"https://www.zbmath.org/authors/?q=ai:shu.chi-wangSummary: In this paper, we continue our work in [J. Comput. Phys. 375, 659--683 (2018; Zbl 1416.65286)] and propose a new type of high-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. Although termed ``multi-resolution WENO schemes'', we only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. We construct new third-order, fourth-order, and fifth-order WENO schemes using three or four unequal-sized central spatial stencils, different from the classical WENO procedure using equal-sized biased/central spatial stencils for the spatial reconstruction. The new WENO schemes could obtain the optimal order of accuracy in smooth regions, and could degrade gradually to first-order of accuracy so as to suppress spurious oscillations near strong discontinuities. This is the first time that only a series of unequal-sized hierarchical central spatial stencils are used in designing arbitrary high-order finite volume WENO schemes on triangular meshes. The main advantages of these schemes are their compactness, robustness, and their ability to maintain good convergence property for steady-state computation. The linear weights of such WENO schemes can be any positive numbers on the condition that they sum to one. Extensive numerical results are provided to illustrate the good performance of these new finite volume WENO schemes.GPU-accelerated particle methods for evaluation of sparse observations for inverse problems constrained by diffusion PDEs.https://www.zbmath.org/1452.652052021-02-12T15:23:00+00:00"Borggaard, Jeff"https://www.zbmath.org/authors/?q=ai:borggaard.jeff-t"Glatt-Holtz, Nathan"https://www.zbmath.org/authors/?q=ai:glatt-holtz.nathan-e"Krometis, Justin"https://www.zbmath.org/authors/?q=ai:krometis.justinSummary: We consider the inverse problem of estimating parameters of a driven diffusion (e.g., the underlying fluid flow, diffusion coefficient, or source terms) from point measurements of a passive scalar (e.g., the concentration of a pollutant). We present two particle methods that leverage the structure of the inverse problem to enable efficient computation of the forward map, one for time evolution problems and one for Dirichlet boundary-value problems. The methods scale in a natural fashion to modern computational architectures, enabling substantial speedup for applications involving sparse observations and high-dimensional unknowns. Numerical examples of applications to Bayesian inference and numerical optimization are provided.Finite time blow-up for wave equations with strong damping in an exterior domain.https://www.zbmath.org/1452.350442021-02-12T15:23:00+00:00"Fino, Ahmad Z."https://www.zbmath.org/authors/?q=ai:fino.ahmad-zSummary: We consider the initial boundary value problem in exterior domain for strongly damped wave equations with power-type nonlinearity \(|u|^p\). We will establish blow-up results under some conditions on the initial data and the exponent \(p\), using the method of test function with an appropriate harmonic functions. We also study the existence of mild solution and its relation with the weak formulation.Towards perfectly matched layers for time-dependent space fractional PDEs.https://www.zbmath.org/1452.652662021-02-12T15:23:00+00:00"Antoine, Xavier"https://www.zbmath.org/authors/?q=ai:antoine.xavier"Lorin, Emmanuel"https://www.zbmath.org/authors/?q=ai:lorin.emmanuelSummary: Perfectly Matched Layers (PML) are proposed for time-dependent space fractional PDEs. Within this approach, widely used powerful Fourier solvers based on FFTs can be adapted without much effort to compute Initial Boundary Value Problems (IBVP) for well-posed fractional equations with absorbing boundary layers. We analyze mathematically the method and propose some illustrating numerical experiments.A parallel unified gas kinetic scheme for three-dimensional multi-group neutron transport.https://www.zbmath.org/1452.820382021-02-12T15:23:00+00:00"Shuang, Tan"https://www.zbmath.org/authors/?q=ai:shuang.tan"Wenjun, Sun"https://www.zbmath.org/authors/?q=ai:wenjun.sun"Junxia, Wei"https://www.zbmath.org/authors/?q=ai:junxia.wei"Guoxi, Ni"https://www.zbmath.org/authors/?q=ai:guoxi.niSummary: In this paper, a parallel unified gas kinetic scheme (UGKS) is developed to simulate 3D multi-group neutron transport. The physical processes of neutron absorption, scattering, fission are included in the scheme. In order to design a multiscale method for capturing the physics in different regimes, the scheme is composed of solving coupled equations of microscopic neutron transport and macroscopic moment equation. As a result, a time accurate numerical fluxes for the neutron transport can be uniquely formulated and used in the construction of the multiscale method. The isotropic scattering neutron transport is studied in this paper, and it can be proved mathematically that this extended UGKS for multi-group neutron transport equation is an asymptotic preserving scheme. The scheme can be used effectively to capture the diffusion limit even with the mesh size being much larger than the neutron's transport characteristic scale, i.e., the so-called mean free path. Thus, the computation efficiency is greatly improved in comparison with the conventional single-scale scheme. In order to further speed up the computation, a parallelized UGKS has been developed for 3D neutron transport simulation as well. The newly developed scheme is verified through many numerical tests and shows high accuracy and strong robustness in the simulations under a large variation of transport condition.Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity.https://www.zbmath.org/1452.652602021-02-12T15:23:00+00:00"Yu, Hui"https://www.zbmath.org/authors/?q=ai:yu.hui"Liu, Hailiang"https://www.zbmath.org/authors/?q=ai:liu.hailiangSummary: For a class of convection-diffusion equations with variable diffusivity, we construct third order accurate discontinuous Galerkin (DG) schemes on both one and two dimensional rectangular meshes. The DG method with an explicit time stepping can well be applied to nonlinear convection-diffusion equations. It is shown that under suitable time step restrictions, the scaling limiter proposed in [\textit{H. Liu} and \textit{H. Yu}, SIAM J. Sci. Comput. 36, No. 5, A2296--A2325 (2014; Zbl 1341.65038)] when coupled with the present DG schemes preserves the solution bounds indicated by the initial data, i.e., the maximum principle, while maintaining uniform third order accuracy. These schemes can be extended to rectangular meshes in three dimension. The crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions. This is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages. Numerical results are presented to validate the numerical methods and demonstrate their effectiveness.Volume penalization for inhomogeneous Neumann boundary conditions modeling scalar flux in complicated geometry.https://www.zbmath.org/1452.653012021-02-12T15:23:00+00:00"Sakurai, Teluo"https://www.zbmath.org/authors/?q=ai:sakurai.teluo"Yoshimatsu, Katsunori"https://www.zbmath.org/authors/?q=ai:yoshimatsu.katsunori"Okamoto, Naoya"https://www.zbmath.org/authors/?q=ai:okamoto.naoya"Schneider, Kai"https://www.zbmath.org/authors/?q=ai:schneider.kaiSummary: We develop a volume penalization method for inhomogeneous Neumann boundary conditions, generalizing the flux-based volume penalization method for homogeneous Neumann boundary condition proposed by \textit{B. Kadoch} et al. [J. Comput. Phys. 231, No. 12, 4365--4383 (2012; Zbl 1244.76074)]. The generalized method allows us to model scalar flux through walls in geometries of complex shape using simple, e.g. Cartesian, domains for solving the governing equations. We examine the properties of the method, by considering a one-dimensional Poisson equation with different Neumann boundary conditions. The penalized Laplace operator is discretized by second order central finite-differences and interpolation. The discretization and penalization errors are thus assessed for several test problems. Convergence properties of the discretized operator and the solution of the penalized equation are analyzed. The generalized method is then applied to an advection-diffusion equation coupled with the Navier-Stokes equations in an annular domain which is immersed in a square domain. The application is verified by numerical simulation of steady free convection in a concentric annulus heated through the inner cylinder surface using an extended square domain.A mesh-free pseudospectral approach to estimating the fractional Laplacian via radial basis functions.https://www.zbmath.org/1452.653742021-02-12T15:23:00+00:00"Rosenfeld, Joel A."https://www.zbmath.org/authors/?q=ai:rosenfeld.joel-a"Rosenfeld, Spencer A."https://www.zbmath.org/authors/?q=ai:rosenfeld.spencer-a"Dixon, Warren E."https://www.zbmath.org/authors/?q=ai:dixon.warren-eSummary: This paper investigates the use of radial basis function (RBF) interpolants to estimate a function's fractional Laplacian of a given order through a mesh-free pseudospectral method. The mesh-free approach yields an algorithm that can be implemented in high dimensional settings without adjustment. Moreover, the fractional Laplacian is defined in terms of the Fourier transform, and the symmetry of RBFs can be exploited to simplify the estimation problem. Convergence rates are established for RBFs when the function whose fractional Laplacian to be estimated is compactly supported. Further results demonstrate convergence when a function is in the native space for a Wendland RBF (i.e. a Sobolev space) and satisfies a certain \(L^1\) condition. Numerical experiments demonstrate the developed method by estimating the fractional Laplacian of several functions and by solving a fractional Poisson equation with extended Dirichlet condition in one and two dimensions.The uniqueness of the exact solution of the Riemann problem for the shallow water equations with discontinuous bottom.https://www.zbmath.org/1452.760282021-02-12T15:23:00+00:00"Aleksyuk, Andrey I."https://www.zbmath.org/authors/?q=ai:aleksyuk.andrey-i"Belikov, Vitaly V."https://www.zbmath.org/authors/?q=ai:belikov.vitaly-vSummary: The Riemann problem for the shallow water equations with discontinuous topography is considered. In a general case the exact solution of this problem is not unique, which complicates the application of an exact Riemann solver in numerical methods, since it is not clear which solution should be chosen. In the present work it is shown that involving an additional physical assumption makes it possible to prove the existence and uniqueness of the solution. The assumption is that the discharge at the bottom discontinuity should continuously depend on the initial conditions. The proven uniqueness opens up a possibility to use an exact Riemann solver for a numerical solution of the shallow water equations with complex discontinuous topography.Riesz transforms associated with Schrödinger operator on vanishing generalized Morrey spaces.https://www.zbmath.org/1452.420132021-02-12T15:23:00+00:00"Guliyev, V. S."https://www.zbmath.org/authors/?q=ai:guliyev.vagif-sabir"Guliyev, R. V."https://www.zbmath.org/authors/?q=ai:guliyev.ramin-v"Omarova, M. N."https://www.zbmath.org/authors/?q=ai:omarova.mehriban-nIn this paper the authors study the boundedness of the dual Riesz transform \(\mathfrak{R}^\ast=\mathcal{L}^{-\frac{1}{2}}\nabla\) and their commutator on generalized Morrey spaces \(M^{\alpha, V}_{p, \varphi}\) associated with Schrödinger operator \(\mathcal{L}= -\Delta +V(x)\) on \(\mathbb{R}^n\) and vanishing generalized Morrey spaces \(VM^{\alpha, V}_{p, \varphi}\) associated with Schrödinger operator.
We assume that the potential \(V\) is non-negative, \(V \neq 0\), and belongs to a reverse Hölder class \(RH_{q}\) for some \(q \geq n/2\) with the reverse Hölder index \(q_{0}=\sup\{q: V\in RH_q\}\) and \(1/p_0=1/q_0-1/n\).
The function \(\rho\) on \(\mathbb{R}^n\) is defined by
\[
\rho(x)=\sup\{r >0: \frac{1}{r^{n-2}}\int_{B(x,r)}V(y)\ dy \leq 1 \},
\]
where \(B(x,r)\) is a ball in \(\mathbb{R}^n\).
Let \(\varphi(x,t)\) be a positive measurable function on \(\mathbb{R}^n \times (0, \infty), \ 1 \leq p < \infty, \ \alpha \geq 0\) and \(V \in RH_q,\ 1 \leq q\). Then the generalized Morrey space \(M^{\alpha, V}_{p, \varphi}\) associated with Schrödinger operator is defined as the space of all functions \(f \in L^p_{loc}(\mathbb{R}^n)\) with
\[
\|f\|_{M^{\alpha, V}_{p, \varphi}}=\sup_{x \in \mathbb{R}^n,\ r>0}
\mathfrak{U}^{\alpha,V}_{p, \varphi}(f: x,r) < \infty,
\]
where
\[
\mathfrak{U}^{\alpha,V}_{p, \varphi}(f: x, r)=(1+\frac{r}{\rho(x)})^{\alpha}
r^{-n/p}\varphi(x,r)^{-1}\|f\|_{L^p(B(x,r))}.
\]
The vanishing generalized Morrey space \(VM^{\alpha, V}_{p, \varphi}\) associated with Schrödinger operator is defined as the space of functions \(f \in M^{\alpha, V}_{p, \varphi}\) such that
\[
\lim_{r \rightarrow 0}\sup_{x \in \mathbb{R}^n}\mathfrak{U}^{\alpha,V}_{p, \varphi}(f: x, r)=0.
\]
The \(BMO\) type space \(BMO_{\theta}(\rho)\) with \(\theta \geq 0\) is defined as a set of locally integrable functions \(b\) such that
\[
\frac{1}{|B(x,r)|}\int_{B(x,r)}|b(y)-b_{B}|\ dy\leq C(1+\frac{r}{\rho(x)})^{\theta},
\]
where \(x \in \mathbb{R}^n,\ r > 0,\ b_{B}=\frac{1}{|B|}\int_{B}b(y)\ dy\) with the norm \([b]_{\theta}\) as the infimum of the constants \(C\) in the inequality above.
We denote by \(\Omega^{\alpha, V}_{p}\) the set of all positive measurable functions \(\varphi\) on \(\mathbb{R}^n \times (0, \infty)\) such that for all \(t > 0\)
\[
\sup_{x \in \mathbb{R}^n}\|(1+\frac{r}{\rho(x)})^{\alpha}\frac{r^{-\frac{n}{p}}}{\varphi(x,r)}\|_{L^{\infty}(t, \infty)}< \infty
\]
and
\[
\sup_{x \in \mathbb{R}^n}\|(1+\frac{r}{\rho(x)})^{\alpha}\varphi(x,r)^{-1}\|_{L^{\infty}(0,t)}< \infty.
\]
The authors prove the following theorems:
Theorem 1.
Let \(V \in RH_{q}\) with \(n/2 \leq q < n,\ \alpha \geq 0,\ p_{0}^\prime < p < \infty\) and \(\varphi_{1}, \varphi_{2} \in \Omega^{\alpha, V}_p\) satisfy the condition
\[
\int_{r}^{\infty}\mathrm{ess\ sup}_{t< s< \infty}\varphi_{1}(x,s)s^{\frac{n}{p}} t^{-\frac{n}{p}}\frac{dt}{t} \leq c_{0}\varphi_{2}(x,r).
\]
Then the operator \(\mathcal{R}^\ast\) is bounded from \(M^{\alpha, V}_{p, \varphi_{1}}\) to \(M^{\alpha, V}_{p, \varphi_{2}}\), and
\[
\|\mathcal{R}^\ast f\|_{M^{\alpha, V}_{p, \varphi_{2}}} \leq C\|f\|_{M^{\alpha, V}_{p, \varphi_{1}}}.
\]
Theorem 2.
Let \(V \in RH_{q}\) with \(n/2 \leq q < n,\ \alpha \geq 0,\ p_{0}^\prime< p < \infty,\ b \in BMO_{\theta}(\rho)\) and \(\varphi_{1}, \varphi_{2} \in \Omega^{\alpha, V}_p\) satisfy the condition
\[
\int_{r}^{\infty}(1+\ln \frac{t}{r})\ \mathrm{ess\ inf}_{t< s< \infty}\varphi_{1}(x,s)s^{\frac{n}{p}} t^{-\frac{n}{p}}\frac{dt}{t} \leq c_{0}\varphi_{2}(x,r).
\]
Then the commutator \([b,\mathcal{R}^\ast]\) is bounded from \(M^{\alpha, V}_{p, \varphi_{1}}\) to \(M^{\alpha, V}_{p, \varphi_{2}}\), and
\[
\|[b,\mathcal{R}^\ast]f\|_{M^{\alpha, V}_{p, \varphi_{2}}} \leq C[b]_{\theta}\|f\|_{M^{\alpha, V}_{p, \varphi_{1}}}.
\]
The authors also prove the boundedness of the Riesz transform and their commutator on the vanishing generalized Morrey space \(VM^{\alpha, V}_{p, \varphi}\) associated with Schrödinger operator.
Reviewer: Koichi Saka (Akita)Discontinuous Galerkin discretizations of the Boltzmann-BGK equations for nearly incompressible flows: semi-analytic time stepping and absorbing boundary layers.https://www.zbmath.org/1452.652362021-02-12T15:23:00+00:00"Karakus, A."https://www.zbmath.org/authors/?q=ai:karakus.abdullah-harun|karakus.ali"Chalmers, N."https://www.zbmath.org/authors/?q=ai:chalmers.noel"Hesthaven, J. S."https://www.zbmath.org/authors/?q=ai:hesthaven.jan-s"Warburton, T."https://www.zbmath.org/authors/?q=ai:warburton.timothySummary: We present an efficient nodal discontinuous Galerkin method for approximating nearly incompressible flows using the Boltzmann equations. The equations are discretized with Hermite polynomials in velocity space yielding a first order conservation law. A stabilized unsplit perfectly matching layer (PML) formulation is introduced for the resulting nonlinear flow equations. The proposed PML equations exponentially absorb the difference between the nonlinear fluctuation and the prescribed mean flow. We introduce semi-analytic time discretization methods to improve the time step restrictions in small relaxation times. We also introduce a multirate semi-analytic Adams-Bashforth method which preserves efficiency in stiff regimes. Accuracy and performance of the method are tested using distinct cases including isothermal vortex, flow around square cylinder, and wall mounted square cylinder test cases.A stable second order of accuracy difference scheme for a fractional Schrödinger differential equation.https://www.zbmath.org/1452.651472021-02-12T15:23:00+00:00"Ashyralyev, A."https://www.zbmath.org/authors/?q=ai:ashyralyev.allaberen"Hicdurmaz, B."https://www.zbmath.org/authors/?q=ai:hicdurmaz.betulSummary: In the present paper, we present and analyze a second order of accuracy difference scheme for solving a fractional Schrödinger differential equation with the fractional derivative in the Riemann Liouville sense. A stability analysis is performed on the presented difference scheme. Numerical results confirm the expected convergence rates and illustrate the effectiveness of the method.Normal oscillations of hydrosystein of two viscoelastic fluids in stationary container (a model problem).https://www.zbmath.org/1452.760142021-02-12T15:23:00+00:00"Kopachevsky, N. D."https://www.zbmath.org/authors/?q=ai:kopachevsky.nikolay-dmitrievichSummary: In the paper, we consider a problem on small motions and normal oscillations of two viscoelastic fluids in a stationary container. One of models of such fluids is Oldroid's model. It is described, for example, in the book \textit{F. R. Eirich} (ed.) [Rheology. Theory and applications. Vol. I. New York: Academic Press Inc. (1956; Zbl 0072.19401)]. It is important to notice that the present paper is devoted to the study of the scalar model problem. Also it should be noted that the present paper based on the previous author's works together with Azizov, T. Ya., Orlova L. D., Krein, S. G. Namely, problem on small movements of one or two viscoelastic fluid for generalized Oldroid's model and normal oscillations of a viscoelastic fluid in an open container were investigated in these papers. The aim of this paper is to use an operator approach of mentioned works, to prove the theorem on correct solvability for the scalar model initial boundary-value problem generated by a problem of small motions of two viscoelastic fluids in a stationary container and to get properties of eigenvalues and eigenelements of corresponding spectral problem. This paper is organized as follows. In section 1 we describe a model of viscoelastic fluid, formulate mathematical statement of the problem: linearized equations of movements, stickiness condition, kinematic and dynamic conditions. Further, in this section we receive the law of full energy balance and choose the functional spaces generated by the problem. For applying of method of orthogonal projection we need to get orthogonal projector on corresponding space. The law of action of this projector we receive in this section. In section 2 we make transition to operator equation by using orthogonal projector received in section 1. Further, we solve some auxiliary problems and obtain the Cauchy problem for the system of integro-differential equation in some Hilbert space. Then we make transition to a system of differential equation. This system can be rewrite as operator differential equation in the sum of Hilbert spaces. Properties of main operator of this problem are studied in this section. The existence and uniqueness theorems for final operator differential equation as for original initial-boundary-value problem based on factorization, closure and accretivity property of operator matrix. Finally, in this section we consider the spectral problem on normal oscillations corresponding to the evolution problem. This means that external forces equal to zero and dependence by time for the unknown function has the form \(e^{-\lambda t}\). Here we obtain the spectral problem for operator pencil and study main properties of it. Section 3 is devoted to investigating of model spectral problem in rectangular domain. The more detailed properties of eigenvalues are obtained here.Fast Fourier solvers for the tensor product high-order FEM for a Poisson type equation.https://www.zbmath.org/1452.653662021-02-12T15:23:00+00:00"Zlotnik, A. A."https://www.zbmath.org/authors/?q=ai:zlotnik.alexander-a"Zlotnik, I. A."https://www.zbmath.org/authors/?q=ai:zlotnik.i-aSummary: Logarithmically optimal in theory and fast in practice, direct algorithms for implementing a tensor product finite element method (FEM) based on tensor products of 1D high-order FEM spaces on multi-dimensional rectangular parallelepipeds are proposed for solving the \(N\)-dimensional Poisson-type equation \(- \Delta u + \alpha u = f\) \((N \geqslant 2)\) with Dirichlet boundary conditions. The algorithms are based on well-known Fourier approaches. The key new points are a detailed description of the eigenpairs of the 1D eigenvalue problems for the high-order FEM, as well as fast direct and inverse eigenvector expansion algorithms that simultaneously employ several versions of the fast Fourier transform. Results of numerical experiments in the 2D and 3D cases are presented. The algorithms can be used in numerous applications, in particular, to implement tensor product high-order finite element methods for various time-dependent partial differential equations, including the multidimensional heat, wave, and Schrödinger ones.Bifurcation in mean phase portraits for stochastic dynamical systems with multiplicative Gaussian noise.https://www.zbmath.org/1452.370592021-02-12T15:23:00+00:00"Wang, Hui"https://www.zbmath.org/authors/?q=ai:wang.hui.6"Tsiairis, Athanasios"https://www.zbmath.org/authors/?q=ai:tsiairis.athanasios"Duan, Jinqiao"https://www.zbmath.org/authors/?q=ai:duan.jinqiaoStationary waves for nonlinear Schrödinger equations with potentials.https://www.zbmath.org/1452.350712021-02-12T15:23:00+00:00"Ambrosetti, Antonio"https://www.zbmath.org/authors/?q=ai:ambrosetti.antonio(no abstract)Nonlinear self-organized population dynamics induced by external selective nonlocal processes.https://www.zbmath.org/1452.352262021-02-12T15:23:00+00:00"Tumbarell Aranda, Orestes"https://www.zbmath.org/authors/?q=ai:tumbarell-aranda.orestes"Penna, André L. A."https://www.zbmath.org/authors/?q=ai:penna.andre-l-a"Oliveira, Fernando A."https://www.zbmath.org/authors/?q=ai:oliveira.fernando-albuquerqueSummary: Self-organization evolution of a population is studied considering generalized reaction-diffusion equations. We proposed a model based on non-local operators that has several of the equations traditionally used in research on population dynamics as particular cases. Then, employing a relatively simple functional form of the non-local kernel, we determined the conditions under which the analyzed population develops spatial patterns, as well as their main characteristics. Finally, we established a relationship between the developed model and real systems by making simulations of bacterial populations subjected to non-homogeneous lighting conditions. Our proposal reproduces some of the experimental results that other approaches considered previously had not been able to obtain.The limit \(\alpha \to 0\) of the \(\alpha \)-Euler equations in the half-plane with no-slip boundary conditions and vortex sheet initial data.https://www.zbmath.org/1452.351452021-02-12T15:23:00+00:00"Busuioc, Adriana V."https://www.zbmath.org/authors/?q=ai:busuioc.adriana-valentina"Iftimie, Dragos"https://www.zbmath.org/authors/?q=ai:iftimie.dragos"Lopes Filho, Milton D."https://www.zbmath.org/authors/?q=ai:lopes-filho.milton-da-costa"Nussenzveig Lopes, Helena J."https://www.zbmath.org/authors/?q=ai:nussenzveig-lopes.helena-jLet \(\mathbb H = \{ x \in \mathbb R^2: x_2 > 0\}\) denote the half-plane. The authors study the initial-value problem with no-slip boundary conditions for the \(\alpha\)-Euler system \(\partial_t(u-\alpha \Delta u) + u \cdot \nabla (u-\alpha \Delta u) + \sum_j(u-\alpha \Delta u)_j\nabla u_j = - \nabla p\), \(\operatorname{div} u=0\) in \(\mathbb H\) and the limit of its solutions as \(\alpha \rightarrow 0\). The existence of subsequences converging to a weak solution of the incompressible Euler equations is established under the condition of the nonnegative initial vorticities in the space of bounded Radon measures in \(H^{-1}\).
Reviewer: Vladimir Mityushev (Kraków)Generalized fractional Cauchy-Riemann operator associated with the fractional Cauchy-Riemann operator.https://www.zbmath.org/1452.352352021-02-12T15:23:00+00:00"Ceballos, Johan"https://www.zbmath.org/authors/?q=ai:ceballos.johan"Coloma, Nicolás"https://www.zbmath.org/authors/?q=ai:coloma.nicolas"Di Teodoro, Antonio"https://www.zbmath.org/authors/?q=ai:di-teodoro.antonio-nicola|teodoro.antonio-di"Ochoa-Tocachi, Diego"https://www.zbmath.org/authors/?q=ai:ochoa-tocachi.diegoSummary: In this paper, we present a characterization of all linear fractional order partial differential operators with complex-valued coefficients that are associated to the generalized fractional Cauchy-Riemann operator in the Riemann-Liouville sense. To achieve our goal, we make use of the technique of an associated differential operator applied to the fractional case.On the improved regularity criterion of the solutions to the Navier-Stokes equations.https://www.zbmath.org/1452.351322021-02-12T15:23:00+00:00"Gala, Sadek"https://www.zbmath.org/authors/?q=ai:gala.sadekSummary: This note deals with the question of the regularity of (Leray) weak solutions of the Navier-Stokes equations in terms of the pressure. This criterion improves on the existing results.Global existence and asymptotic behavior of a plate equation with a constant delay term and logarithmic nonlinearities.https://www.zbmath.org/1452.350922021-02-12T15:23:00+00:00"Remil, Melouka"https://www.zbmath.org/authors/?q=ai:remil.meloukaSummary: In this paper, we investigate the viscoelastic plate equation with a constant delay term and logarithmic nonlinearities. Under some conditions, we will prove the global existence. Furthermore, we use weighted spaces to establish a general decay rate of solution.A discontinuous Galerkin method for an elliptic hemivariational inequality for semipermeable media.https://www.zbmath.org/1452.653612021-02-12T15:23:00+00:00"Wang, Fei"https://www.zbmath.org/authors/?q=ai:wang.fei.2|wang.fei.1"Qi, Haoran"https://www.zbmath.org/authors/?q=ai:qi.haoranSummary: In this paper, we study a discontinuous Galerkin (DG) method for solving an elliptic hemivariational inequality for semipermeable media. A priori error analysis is established, and we show that the DG scheme with the linear element reaches optimal convergence order under appropriate solution regularity assumptions. One numerical example is presented to support the theoretical analysis.A stabilizer free weak Galerkin method for the biharmonic equation on polytopal meshes.https://www.zbmath.org/1452.653622021-02-12T15:23:00+00:00"Ye, Xiu"https://www.zbmath.org/authors/?q=ai:ye.xiu"Zhang, Shangyou"https://www.zbmath.org/authors/?q=ai:zhang.shangyouA boundary value problem for a degenerate moisture transfer equation with a condition of the third kind.https://www.zbmath.org/1452.651502021-02-12T15:23:00+00:00"Beshtokov, Murat Khamidbievich"https://www.zbmath.org/authors/?q=ai:beshtokov.murat-khamidbievich"Kanchukoyev, Vladimir Zedunovich"https://www.zbmath.org/authors/?q=ai:kanchukoyev.vladimir-zedunovich"Èrzhibova, Farida Aleksandrovna"https://www.zbmath.org/authors/?q=ai:erzhibova.farida-aleksandrovnaSummary: In this work, we study the pseudoparabolic equation in the three dimensional space. The equation of this form implies the presence of cylindrical or spherical symmetry that enables one to move from a three-dimensional problem to one-dimensional problem, but with degeneration. In this regard, we study the solvability and stability of solutions to boundary value problems for degenerate pseudoparabolic equation of the third order of general form with variable coefficients and third kind condition, as well as difference schemes approximating this problem on uniform grids. The main result consists in proving a priori estimates for a solution to both the differential and difference problems by means of the method of energy inequalities. The obtained inequalities imply stability of the solution relative to initial data and right side. Because of the linearity of the considered problems these inequalities allow us to state the convergence of the approximate solution to the exact solution of the considered differential problem under the assumption of the existence of the solutions in the class of sufficiently smooth functions. On the test examples the numerical experiments are performed confirming the theoretical results obtained in the work.Error analysis of the SAV-MAC scheme for the Navier-Stokes equations.https://www.zbmath.org/1452.651672021-02-12T15:23:00+00:00"Li, Xiaoli"https://www.zbmath.org/authors/?q=ai:li.xiaoli"Shen, Jie"https://www.zbmath.org/authors/?q=ai:shen.jie.2|shen.jie.1|shen.jie|shen.jie.3|shen.jie.4|shen.jie.5Convergence of Dziuk's linearly implicit parametric finite element method for curve shortening flow.https://www.zbmath.org/1452.652402021-02-12T15:23:00+00:00"Li, Buyang"https://www.zbmath.org/authors/?q=ai:li.buyangA fourth-order exponential wave integrator Fourier pseudo-spectral method for the Klein-Gordon equation.https://www.zbmath.org/1452.653692021-02-12T15:23:00+00:00"Ji, Bingquan"https://www.zbmath.org/authors/?q=ai:ji.bingquan"Zhang, Luming"https://www.zbmath.org/authors/?q=ai:zhang.lumingSummary: This paper is concerned with a fourth-order exponential wave integrator Fourier pseudo-spectral method for solving the Klein-Gordon equation. We suggest a new numerical integration formula to approximate the time integral term in the phase space. The error estimate shows the suggested numerical method is fourth-order accurate in time and spectral accurate in space. The theoretical findings are confirmed by numerical experiments.Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition.https://www.zbmath.org/1452.350252021-02-12T15:23:00+00:00"Zheng, Jiayue"https://www.zbmath.org/authors/?q=ai:zheng.jiayue"Cui, Shangbin"https://www.zbmath.org/authors/?q=ai:cui.shangbinSummary: In this paper we study existence of nonradial stationary solutions of a free boundary problem modeling the growth of nonnecrotic tumors. Unlike the models studied in existing literatures on this topic where boundary value condition for the nutrient concentration \(\sigma \) is linear, in this model this is a nonlinear boundary condition. By using the bifurcation method, we prove that nonradial stationary solutions do exist when the surface tension coefficient \(\gamma\) takes values in small neighborhoods of certain eigenvalues of the linearized problem at the radial stationary solution.Fractional approximations of abstract semilinear parabolic problems.https://www.zbmath.org/1452.350852021-02-12T15:23:00+00:00"Bezerra, Flank D. M."https://www.zbmath.org/authors/?q=ai:bezerra.flank-david-morais"Carvalho, Alexandre N."https://www.zbmath.org/authors/?q=ai:nolasco-de-carvalho.alexandre"Nascimento, Marcelo J. D."https://www.zbmath.org/authors/?q=ai:nascimento.marcelo-jose-diasSummary: In this paper we study the abstract semilinear parabolic problem of the form \[\frac{du}{dt}+Au = f(u),\] as the limit of the corresponding fractional approximations \[ \frac{du}{dt} + A^{\alpha}u = f(u), \] in a Banach space \(X\), where the operator \(A:D(A) \subset X \to X\) is a sectorial operator in the sense of \textit{D. Henry} [Geometric theory of semilinear parabolic equations. Springer, Cham (1981; Zbl 0456.35001)]. Under suitable assumptions on nonlinearities \(f:X^\alpha\to X\) (\( X^\alpha: = D(A^\alpha \)), we prove the continuity with rate (with respect to the parameter \( \alpha \)) for the global attractors (as seen in [\textit{A. V. Babin} and \textit{M. I. Vishik}, Attractors of evolution equations. Transl. from the Russian by A. V. Babin. Amsterdam etc.: North-Holland (1992; Zbl 0778.58002)], Chapter 8, Theorem 2.1]). As an application of our analysis we consider a fractional approximation of the strongly damped wave equations and we study the convergence with rate of solutions of such approximations.A multi-D model for Raman amplification.https://www.zbmath.org/1452.762822021-02-12T15:23:00+00:00"Colin, Mathieu"https://www.zbmath.org/authors/?q=ai:colin.mathieu"Colin, Thierry"https://www.zbmath.org/authors/?q=ai:colin.thierrySummary: In this paper, we continue the study of the Raman amplification in plasmas that we initiated in [the authors, Differ. Integral Equ. 17, No. 3--4, 297--330 (2004; Zbl 1174.35528); J. Comput. Appl. Math. 193, No. 2, 535--562 (2006; Zbl 1092.35101)]. We point out that the Raman instability gives rise to three components. The first one is collinear to the incident laser pulse and counter propagates. In 2-D, the two other ones make a non-zero angle with the initial pulse and propagate forward. Furthermore they are symmetric with respect to the direction of propagation of the incident pulse. We construct a non-linear system taking into account all these components and perform some 2-D numerical simulations.A fast algorithm for quadrature by expansion in three dimensions.https://www.zbmath.org/1452.654132021-02-12T15:23:00+00:00"Wala, Matt"https://www.zbmath.org/authors/?q=ai:wala.matt"Klöckner, Andreas"https://www.zbmath.org/authors/?q=ai:klockner.andreasSummary: This paper presents an accelerated quadrature scheme for the evaluation of layer potentials in three dimensions. Our scheme combines a generic, high order quadrature method for singular kernels called Quadrature by Expansion (QBX) with a modified version of the Fast Multipole Method (FMM). Our scheme extends a recently developed formulation of the FMM for QBX in two dimensions, which, in that setting, achieves mathematically rigorous error and running time bounds. In addition to generalization to three dimensions, we highlight some algorithmic and mathematical opportunities for improved performance and stability. Lastly, we give numerical evidence supporting the accuracy, performance, and scalability of the algorithm through a series of experiments involving the Laplace and Helmholtz equations.Wigner measures and effective mass theorems.https://www.zbmath.org/1452.350182021-02-12T15:23:00+00:00"Chabu, Victor"https://www.zbmath.org/authors/?q=ai:chabu.victor"Fermanian Kammerer, Clotilde"https://www.zbmath.org/authors/?q=ai:fermanian-kammerer.clotilde"Macià, Fabricio"https://www.zbmath.org/authors/?q=ai:macia.fabricioSummary: We study a Schrödinger equation which describes the dynamics of an electron in a crystal in the presence of impurities. We consider the regime of small wave-lengths comparable to the characteristic scale of the crystal. It is well-known that under suitable assumptions on the initial data and for highly oscillating potential, the wave function can be approximated by the solution of a simpler equation, the effective mass equation. Using Floquet-Bloch decomposition, as it is classical in this subject, we establish effective mass equations in a rather general setting. In particular, Bloch bands are allowed to have degenerate critical points, as may occur in dimension strictly larger than one. Our analysis leads to a new type of effective mass equations which are operator-valued and of Heisenberg form and relies on Wigner measure theory and, more precisely, to its applications to the analysis of dispersion effects.Detection of multiple impedance obstacles by non-iterative topological gradient based methods.https://www.zbmath.org/1452.650412021-02-12T15:23:00+00:00"Le Louër, F."https://www.zbmath.org/authors/?q=ai:le-louer.frederique"Rapún, M.-L."https://www.zbmath.org/authors/?q=ai:rapun.maria-luisaSummary: We investigate a fast, one-step imaging method of multiple 2D and 3D acoustic obstacles fully-coated by a complex surface impedance with either monochromatic or multi-frequency noisy data. Introducing the topological gradient of the misfit functional as a limit of shape derivatives, closed-form expressions of the obstacle indicator are derived using Fourier and Mie series expansions of the radiating solution. We provide a wide variety of numerical experiments that assesses the performance and limitations of the one step single and multi-frequency imaging strategies when dealing both with full and limited aperture measurements.Solving the Riemann problem for realistic astrophysical fluids.https://www.zbmath.org/1452.850052021-02-12T15:23:00+00:00"Chen, Zhuo"https://www.zbmath.org/authors/?q=ai:chen.zhuo"Coleman, Matthew S. B."https://www.zbmath.org/authors/?q=ai:coleman.matthew-s-b"Blackman, Eric G."https://www.zbmath.org/authors/?q=ai:blackman.eric-g"Frank, Adam"https://www.zbmath.org/authors/?q=ai:frank.adamSummary: We present new methods to solve the Riemann problem both exactly and approximately for general equations of state (EoS) to facilitate realistic modeling and understanding of astrophysical flows. The existence and uniqueness of the new exact general EoS Riemann solution can be guaranteed if the EoS is monotone regardless of the physical validity of the EoS. We confirm that: (1) the solution of the new exact general EoS Riemann solver and the solution of the original exact Riemann solver match when calculating perfect gas Euler equations; (2) the solution of the new Harten-Lax-van Leer-Contact (HLLC) general EoS Riemann solver and the solution of the original HLLC Riemann solver match when working with perfect gas EoS; and (3) the solution of the new HLLC general EoS Riemann solver approaches the new exact solution. We solve the EoS with two methods, one is to interpolate 2D EoS tables by the bi-linear interpolation method, and the other is to analytically calculate thermodynamic variables at run-time. The interpolation method is more general as it can work with other monotone and realistic EoS while the analytic EoS solver introduced here works with a relatively idealized EoS. Numerical results confirm that the accuracy of the two EoS solvers is similar. We study the efficiency of these two methods with the HLLC general EoS Riemann solver and find that analytic EoS solver is faster in the test problems. However, we point out that a combination of the two EoS solvers may become favorable in some specific problems. Throughout this research, we assume local thermal equilibrium.A multigroup moment-accelerated deterministic particle solver for 1-D time-dependent thermal radiative transfer problems.https://www.zbmath.org/1452.652892021-02-12T15:23:00+00:00"Park, H."https://www.zbmath.org/authors/?q=ai:park.hyeongkae"Chacón, L."https://www.zbmath.org/authors/?q=ai:chacon.luis"Matsekh, A."https://www.zbmath.org/authors/?q=ai:matsekh.anna-m"Chen, G."https://www.zbmath.org/authors/?q=ai:chen.guangyeSummary: We propose an efficient, robust, Lagrangian (characteristic-based) transport solver for 1-D time-dependent thermal radiative transfer (TRT) applications within the context of a moment-accelerated (High-Order/Low-Order, HOLO) algorithm. This novel transport algorithm inherits the best features of both particle methods (e.g., time accuracy, phase-space adaptivity, positivity) and deterministic, grid-based methods (e.g., no stochastic noise). Particles are evolved by the method of characteristics, with a time-dependent weight that accounts for stiff absorption and reemission process self-consistently. As a result, this approach is able to obtain accurate results while employing large time steps and a moderate number of particles (compared to Implicit Monte Carlo).The inner-boundary value problem for a mixed type equation with a nonsmooth parabolic degeneration line.https://www.zbmath.org/1452.351152021-02-12T15:23:00+00:00"Vodakhova, V. A."https://www.zbmath.org/authors/?q=ai:vodakhova.v-a.1"Nakhusheva, F. M."https://www.zbmath.org/authors/?q=ai:nakhusheva.f-m"Ezaova, A. G."https://www.zbmath.org/authors/?q=ai:ezaova.alena-georgievna"Kanukoeva, L. V."https://www.zbmath.org/authors/?q=ai:kanukoeva.l-vSummary: In this paper, equations of mixed elliptic-hyperbolic type with two degeneration lines parabolic investigated the question of unique solvability of a nonlocal problem, when on the elliptic part of the boundary region is set to a Dirichlet condition, and on the hyperbolic parts of the border, conditions are defined, point-by-point linking the values of the fractional derivative of the solution on the characteristics with the values of the solution at the parabolic lines of degeneracy within the region. Under certain restrictions preventing type on the specified functions and fractional order derivatives in boundary conditions is the method of energy integrals is proved the uniqueness of the solution of the problem. The question of existence of the solution are equivalent is reduced to the question of solvability of system of singular integral equations of the second order Cauchy kernel with respect to derivatives of the traces of the sought solution on the lines of degeneration. Discharged conditions which guarantee the existence of regularizers, leading to singular integral equations to the Fredholm equation of the second kind, unconditional solvability of which follows from the uniqueness of the problem solution. Investigated differential properties of the solution. The influence on the posedness of the problem of orders of fractional derivatives in boundary conditions and their connection with the order of degeneracy of the equation.Efficient parallel solution of the 3D stationary Boltzmann transport equation for diffusive problems.https://www.zbmath.org/1452.820352021-02-12T15:23:00+00:00"Moustafa, Salli"https://www.zbmath.org/authors/?q=ai:moustafa.salli"Févotte, François"https://www.zbmath.org/authors/?q=ai:fevotte.francois"Faverge, Mathieu"https://www.zbmath.org/authors/?q=ai:faverge.mathieu"Plagne, Laurent"https://www.zbmath.org/authors/?q=ai:plagne.laurent"Ramet, Pierre"https://www.zbmath.org/authors/?q=ai:ramet.pierreSummary: This paper presents an efficient parallel method for the deterministic solution of the 3D stationary Boltzmann transport equation applied to diffusive problems such as nuclear core criticality computations. Based on standard MultiGroup-Sn-DD discretization schemes, our approach combines a highly efficient nested parallelization strategy with the PDSA parallel acceleration technique applied for the first time to 3D transport problems. These two key ingredients enable us to solve extremely large neutronic problems involving up to \(10^{12}\) degrees of freedom in less than an hour using 64 super-computer nodes.A finite volume method for two-dimensional Riemann-Liouville space-fractional diffusion equation and its efficient implementation.https://www.zbmath.org/1452.651892021-02-12T15:23:00+00:00"Fu, Hongfei"https://www.zbmath.org/authors/?q=ai:fu.hongfei"Liu, Huan"https://www.zbmath.org/authors/?q=ai:liu.huan"Wang, Hong"https://www.zbmath.org/authors/?q=ai:wang.hong.1Summary: We develop a finite volume method based on Crank-Nicolson time discretization for the two-dimensional nonsymmetric Riemann-Liouville space-fractional diffusion equation. Stability and convergence are then carefully discussed. We prove that the finite volume scheme is unconditionally stable and convergent with second-order accuracy in time and \(\min \{1 + \alpha, 1 + \beta \}\) order in space with respect to a weighted discrete norm. Here \(0 < \alpha, \beta < 1\) are the space-fractional order indexes in \(x\) and \(y\) directions, respectively. Furthermore, we rewrite the finite volume scheme into a matrix form and develop a matrix-free preconditioned fast Krylov subspace iterative method, which only requires storage of \(\mathcal{O}(N)\) and computational cost of \(\mathcal{O}(N \log N)\) per iteration without losing any accuracy compared to the direct Gaussian elimination method. Here \(N\) is the total number of spatial unknowns. Consequently, the fast finite volume method is particularly suitable for large-scale modeling and simulation. Numerical experiments verify the theoretical results and show strong potential of the fast method.Equivalent formulations and numerical schemes for a class of pseudo-parabolic equations.https://www.zbmath.org/1452.350822021-02-12T15:23:00+00:00"Fan, Y."https://www.zbmath.org/authors/?q=ai:fan.yetian|fan.yuyao|fan.yuling|fan.yimin|fan.yitong|fan.yekun|fan.yunsheng|fan.yanni|fan.yibo|fan.yawen|fan.yuqing|fan.ya|fan.yanglong|fan.yaoying|fan.yayun|fan.yaochi|fan.yize|fan.yonhong|fan.yaping|fan.yangin|fan.yonghu|fan.yuguang|fan.yongsheng|fan.yifan|fan.yabo|fan.yutao|fan.yongbin|fan.yanping|fan.ye|fan.yuanyuan|fan.yuzheng|fan.yizhong|fan.yu|fan.yougao|fan.yongqing|fan.yushun|fan.yong|fan.yajing|fan.yali|fan.yilin|fan.yuqin|fan.yanhong|fan.yingle|fan.yuwei|fan.yangyu|fan.yunxia|fan.yubo|fan.yi|fan.yijun|fan.yingying|fan.yongfeng|fan.youping|fan.yuanze|fan.yizheng|fan.yuliang|fan.yulian|fan.yanming|fan.yingli|fan.yuehui|fan.yueqian|fan.yushuang|fan.yanfei|fan.yunyun|fan.yehua|fan.yanpeng|fan.yue|fan.yongliang|fan.yiqun|fan.yung|fan.yonghong|fan.yunzheng|fan.yeli|fan.yan|fan.yunlan|fan.yin|fan.yadong|fan.yaqing|fan.yachun|fan.yuan|fan.yeming|fan.yun|fan.yongquan|fan.yingmei|fan.yanan|fan.yongbing|fan.yindi|fan.yinhai|fan.yingnan|fan.yugang|fan.yurun|fan.yongyan|fan.yihong|fan.yingg|fan.yuren|fan.yinghui|fan.yonghui|fan.yueyue|fan.yinshui|fan.yang|fan.yigang|fan.ying|fan.yingsong|fan.yanqin|fan.yale|fan.yuchen|fan.yuchao|fan.yongchen|fan.yihan|fan.yanhuan|fan.yuming|fan.yijia|fan.yumei|fan.yuhua|fan.yingzhe|fan.yingfei|fan.yuying|fan.yulei|fan.yamin|fan.yuqi|fan.yunpeng|fan.yunfeng|fan.yongqiang|fan.yiren|fan.yabin|fan.yingjie|fan.yanjun|fan.yupeng|fan.yuhang|fan.youzhe"Pop, I. S."https://www.zbmath.org/authors/?q=ai:pop.iuliu-sorinSummary: This paper investigates three different formulations for a class of pseudo-parabolic equations. Such equations are encountered, for example, as a model for two-phase porous media flows when dynamic effects in the capillary pressure are included. We first show the equivalence of the three different formulations for the original equation. On the basis of this, we further investigate the corresponding discretization in time and give some numerical examples.Inverse scattering and soliton solutions of nonlocal complex reverse-spacetime mKdV equations.https://www.zbmath.org/1452.370692021-02-12T15:23:00+00:00"Ma, Wen-Xiu"https://www.zbmath.org/authors/?q=ai:ma.wen-xiuThe paper is devoted to a nonlocal modification of the Korteweg-de Vries equation. The nonlocality of the model manifests in the presence, in the corresponding nonlinear differential equation, of a term depending on inverted space-time coordinates. In addition, the author considers even larger, multicomponent class of mKdV equations. It turns out that such nonlolcal generalization possesses a Lax pair formulation which results in the complete integrability of the model.
Then the author applies the inverse scattering method by means of the corresponding Riemann-Hilbert problem, which is one of the equivalent approaches to finding the explicit solutions. The investigation is done along the lines of classical works on the inverse scattering method. The author routinely formulates the related spectral problem, finds the time evolution of the scattering data, and finds the corresponding Gelfand-Levitan-Marchenko integral equations, which encode the soliton solutions. The presentation of the paper is quite complete, and contains many technical details and explicit calculations.
Reviewer: Arsen Melikyan (Brasília)On Mellin transforms of solutions of differential equation \(\chi^{(n)}(x)+\gamma_nx\chi (x)=0\).https://www.zbmath.org/1452.330022021-02-12T15:23:00+00:00"Askari, Hassan"https://www.zbmath.org/authors/?q=ai:askari.hassan-randjbar"Ansari, Alireza"https://www.zbmath.org/authors/?q=ai:ansari.alirezaSummary: In this paper, for \(n=2,3,\dots\), we consider the differential equation
\[
\chi^{(n)}(x)+\gamma_nx\chi (x)=0,\quad
\begin{cases}
\gamma_n=(-1)^k,\quad &n=2k,\\
\gamma_n=-1,\quad &n=2k+1,
\end{cases}
\]
and find the linear independent solutions in terms of the higher-order Airy functions \((n=2k)\) and the higher-order Lévy stable functions \((n=2k+1)\). The integral representations of solutions are presented and their Mellin transforms are also given.An unconditionally energy stable scheme for simulating wrinkling phenomena of elastic thin films on a compliant substrate.https://www.zbmath.org/1452.741172021-02-12T15:23:00+00:00"Huang, Qiong-Ao"https://www.zbmath.org/authors/?q=ai:huang.qiongao"Jiang, Wei"https://www.zbmath.org/authors/?q=ai:jiang.wei.1|jiang.wei.4|jiang.wei.2|jiang.wei.5|jiang.wei|jiang.wei.3"Yang, Jerry Zhijian"https://www.zbmath.org/authors/?q=ai:yang.jerry-zhijianSummary: By introducing a scalar auxiliary variable (SAV), we propose a class of semi-implicit time-stepping, unconditionally energy stable numerical schemes for simulating wrinkling phenomena of an elastic thin films on a compliant substrate. These SAV schemes only need to solve, twice, a decoupled linear, fourth-order integro-differential equations with constant coefficients at each time step, and these linear equations can be efficiently implemented by using Fourier spectral method to discretize spatial derivatives. Numerical results have demonstrated that these SAV schemes (including first-order SAV/BDF1, second-order SAV/BDF2 and SAV/CN2) are highly efficient and accurate. Furthermore, many interesting wrinkling phenomena, such as pattern formations (e.g., stripes, checkerboards, labyrinths, herringbones) and pattern transitions under different loading processes, are investigated by using the proposed schemes.Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations.https://www.zbmath.org/1452.652412021-02-12T15:23:00+00:00"Liu, Yuan"https://www.zbmath.org/authors/?q=ai:liu.yuan"Cheng, Yingda"https://www.zbmath.org/authors/?q=ai:cheng.yingda"Chen, Shanqin"https://www.zbmath.org/authors/?q=ai:chen.shanqin"Zhang, Yong-Tao"https://www.zbmath.org/authors/?q=ai:zhang.yongtaoSummary: Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained.The asymptotic analysis of a Darcy-Stokes system coupled through a curved interface.https://www.zbmath.org/1452.351552021-02-12T15:23:00+00:00"Morales, Fernando A."https://www.zbmath.org/authors/?q=ai:morales.fernando-aSummary: We present the asymptotic analysis of a Darcy-Stokes coupled system, modeling the fluid exchange between a narrow channel (Stokes flow) and a porous medium (Darcy flow), coupled through a \(C^2\) curved interface. The channel is a cylindrical domain between the interface \((\Gamma)\) and a parallel translation of itself (\(\Gamma+\varepsilon \widehat{e}_N\), \(\varepsilon>0\)). The introduction of a change variable (to fix the domain geometry) and the introduction of two systems of coordinates: the Cartesian and a local one (consistent with the geometry of the surface), permit to find the limiting form of the system when the width of the channel tends to zero \((\varepsilon\to 0)\). The limit problem is a coupled system with Darcy flow in the porous medium and Brinkman flow on the curved interface \((\Gamma)\).On wave splitting, source separation and echo removal with absorbing boundary conditions.https://www.zbmath.org/1452.651482021-02-12T15:23:00+00:00"Baffet, Daniel"https://www.zbmath.org/authors/?q=ai:baffet.daniel-h"Grote, Marcus J."https://www.zbmath.org/authors/?q=ai:grote.marcus-jSummary: Starting from classical absorbing boundary conditions (ABC), we propose a method for the separation of time-dependent wave fields given measurements of the total wave field. The method is local in space and time, deterministic, and makes no prior assumptions on the frequency spectrum and the location of sources or physical boundaries. By using increasingly higher order ABC, the method can be made arbitrarily accurate and is, in that sense, exact. Numerical examples illustrate the usefulness for source separation and echo removal.Numerical solution of fractional diffusion-reaction problems based on BURA.https://www.zbmath.org/1452.650602021-02-12T15:23:00+00:00"Harizanov, Stanislav"https://www.zbmath.org/authors/?q=ai:harizanov.stanislav"Lazarov, Raytcho"https://www.zbmath.org/authors/?q=ai:lazarov.raytcho-d"Margenov, Svetozar"https://www.zbmath.org/authors/?q=ai:margenov.svetozar-d"Marinov, Pencho"https://www.zbmath.org/authors/?q=ai:marinov.pencho-g|marinov.pencho-asSummary: The paper is devoted to the numerical solution of algebraic systems of the type \((\mathbb{A}^\alpha+q\mathbb{I})\mathbf{u}=\mathbf{f}\), \(0<\alpha<1\), \(q>0\), \(\mathbf{u},\mathbf{f}\in\mathbb{R}^N\), where \(\mathbb{A}\) is a symmetric and positive definite matrix. We assume that \(\mathbb{A}\) is obtained by finite difference approximation of a second order diffusion problem in \(\Omega\subset\mathbb{R}^d\), \(d=1,2\) so that \(\mathbb{A}^\alpha+q\mathbb{I}\) approximates the related fractional diffusion-reaction operator or could be a result of a time-stepping procedure in solving time-dependent sub-diffusion problems. We also assume that a method of optimal complexity for solving linear systems with matrices \(\mathbb{A}+c\mathbb{I}\), \(c\geq 0\) is available. We analyze and study numerically a class of solution methods based on the best uniform rational approximation (BURA) of a certain scalar function in the unit interval.
The first such method, originally proposed in [\textit{S. Harizanov} et al., Numer. Linear Algebra Appl. 25, No. 5, e2167, 24 p. (2018; Zbl 06986996)] for numerical solution of fractional-in-space diffusion problems, was based on the BURA \(r_\alpha(\xi)\) of \(\xi^{1-\alpha}\) in \([0,1]\) through scaling of the matrix \(\mathbb{A}\) by its largest eigenvalue. Then the BURA of \(t^{-\alpha}\) in \([1,\infty)\) is given by \(t^{-1}r_\alpha(t)\) and correspondingly, \(\mathbb{A}^{-1}r_\alpha(\mathbb{A})\) is used as an approximation of \(\mathbb{A}^{-\alpha}\). Further, this method was improved in [\textit{S. Harizanov} et al., ``Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation'', J. Comput. Phys. 408, Article ID 109285, 21 p. (2019; \url{doi:10.1016/j.jcp.2020.109285}), \url{arXiv:1905.08155}] using the same concept but by scaling the matrix \(\mathbb{A}\) by its smallest eigenvalue. In this paper we consider the BURA \(r_\alpha(\xi)\) of \(1/(\xi^{-\alpha}+q)\) for \(\xi\in(0,1]\). Then we define the approximation of \((\mathbb{A}^\alpha+q\mathbb{I})^{-1}\) as \(r_\alpha(\mathbb{A}^{-\alpha})\). We also propose an alternative method that uses BURA of \(\xi^\alpha\) to produce certain uniform rational approximation (URA) of \(1/(\xi^{-\alpha}+q)\). Comprehensive numerical experiments are used to demonstrate the computational efficiency and robustness of the new BURA and URA methods.Uniform stabilization of Boussinesq systems in critical \(\mathbf{L}^q \)-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls.https://www.zbmath.org/1452.352292021-02-12T15:23:00+00:00"Lasiecka, Irena"https://www.zbmath.org/authors/?q=ai:lasiecka.irena"Priyasad, Buddhika"https://www.zbmath.org/authors/?q=ai:priyasad.buddhika"Triggiani, Roberto"https://www.zbmath.org/authors/?q=ai:triggiani.robertoSummary: We consider the \(d\)-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, with homogeneous boundary conditions, and subject to external sources, assumed to cause instability. The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, feedback controls, which are localized on an arbitrarily small interior subdomain. In addition, they will be minimal in number, and of reduced dimension: more precisely, they will be of dimension \((d-1) \) for the fluid component and of dimension 1 for the heat component. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to \(\mathbf{L}^3(\Omega)\) for \(d = 3\)) and the space \( L^q(\Omega)\) for the thermal component, \( q > d \). Thus, this paper may be viewed as an extension of \textit{I. Lasiecka, B. Priyasad} and \textit{R. Triggiani} [``Uniform stabilization of Navier-Stokes equations in critical \(L^q\)-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls'', Appl. Math. Optim. (2019; \url{doi.org/10.1007/s00245-019-09607-9})], where the same interior localized uniform stabilization outcome was achieved by use of finite dimensional feedback controls for the Navier-Stokes equations, in the same Besov setting.Hexagonal spike clusters for some PDE's in 2D.https://www.zbmath.org/1452.350152021-02-12T15:23:00+00:00"Kolokolnikov, Theodore"https://www.zbmath.org/authors/?q=ai:kolokolnikov.theodore"Wei, Juncheng"https://www.zbmath.org/authors/?q=ai:wei.junchengSummary: We study hexagonal spike cluster patterns for Gierer-Meinhardt reaction-diffusion system with a precursor on all of \(\mathbb{R}^2 \). These clusters consist of \(N\) spikes which form a nearly hexagonal lattice of a finite size. The lattice density is locally nearly constant, but globally non-uniform. We also characterize a similar hexagonal spike cluster steady state for a simple elliptic PDE \(0 = \Delta u - u +u^2 + \varepsilon |x|^2 \) with a small ``confinement well'' \( \varepsilon |x|^2 \). The key idea is to explicitly exploit the local hexagonality structure to asymptotically approximate the solution using certain lattice sums. In the limit of many spikes, we derive the effective spike density as well as the cluster radius. This effective density is a solution to a certain separable first-order ODE coupled to an integral boundary condition.Accurate particle time integration for solving Vlasov-Fokker-Planck equations with specified electromagnetic fields.https://www.zbmath.org/1452.762842021-02-12T15:23:00+00:00"Jenny, Patrick"https://www.zbmath.org/authors/?q=ai:jenny.patrick"Gorji, Hossein"https://www.zbmath.org/authors/?q=ai:gorji.hosseinSummary: The Vlasov-Fokker-Planck equation (together with Maxwell's equations) provides the basis for plasma flow calculations. While the terms accounting for long range forces are established, different drift and diffusion terms are used to describe Coulomb collisions. Here, linear drift and a constant diffusion coefficient are considered and the electromagnetic fields are imposed, i.e., plasma frequency is not addressed. The solution algorithm is based on evolving computational particles of a large ensemble according to a Langevin equation, whereas the time step size is typically limited by plasma frequency, Coulomb collision frequency and cyclotron frequency. To overcome the latter two time step size constraints, a novel time integration scheme for the particle evolution is presented. It only requires that gradients of mean velocity, bath temperature, magnetic field and electric field have to be resolved along the trajectories. In fact, if these gradients are zero, then the new integration scheme is statistically exact; no matter how large the time step is chosen. Obviously, this is a computational advantage compared to classical integration schemes, which is demonstrated with numerical experiments of isolated charged particle trajectories under the influence of constant magnetic- and electric fields. Besides single ion trajectories, also plasma flow in spatially varying electromagnetic fields was investigated, that is, the influence of time step size and grid resolution on the final solution was studied.Algorithm for overcoming the curse of dimensionality for state-dependent Hamilton-Jacobi equations.https://www.zbmath.org/1452.490162021-02-12T15:23:00+00:00"Chow, Yat Tin"https://www.zbmath.org/authors/?q=ai:chow.yat-tin"Darbon, Jérôme"https://www.zbmath.org/authors/?q=ai:darbon.jerome"Osher, Stanley"https://www.zbmath.org/authors/?q=ai:osher.stanley-j"Yin, Wotao"https://www.zbmath.org/authors/?q=ai:yin.wotaoSummary: In this paper, we develop algorithms to overcome the curse of dimensionality in non-convex state-dependent Hamilton-Jacobi partial differential equations (HJ PDEs) arising from optimal control and differential game problems. The subproblems are independent and they can be implemented in an embarrassingly parallel fashion. This is ideal for perfect scaling in parallel computing. The algorithm is proposed to overcome the curse of dimensionality when solving HJ PDE. The major contribution of the paper is to change either the solving of a PDE problem or an optimization problem over a space of curves to an optimization problem of a single vector, which goes beyond the work of [\textit{B. Rimoldi}, IEEE Trans. Inf. Theory 47, No. 6, 2432--2442 (2001; Zbl 1021.94516)]. We extend the method in [\textit{Y. T. Chow} et al., Ann. Math. Sci. Appl. 3, No. 2, 369--403 (2018; Zbl 1415.35087); \textit{Y. T. Chow} et al., J. Sci. Comput. 73, No. 2--3, 617--643 (2017; Zbl 1381.65048); \textit{J. Darbon} and \textit{S. Osher}, ``Algorithms for overcoming the curse of dimensionality for certain Hamilton-Jacobi equations arising in control theory and elsewhere'', UCLA CAM report 15-50 (2015)], and \textit{conjecture} a (Lax-type) minimization principle to solve \textit{state-dependent} HJ PDE when the Hamiltonian is convex, as well as a (Hopf-type) maximization principle to solve \textit{state-dependent} HJ PDE when the Hamiltonian is \textit{non-convex}, as a generalization of the well-known Hopf formula in [\textit{L. C. Evans}, Partial differential equations. 2nd ed. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1194.35001); \textit{E. Hopf}, J. Math. Mech. 14, 951--973 (1965; Zbl 0168.35101); \textit{I. V. Rublev}, Comput. Math. Model. 11, No. 4, 391--400 (2000; Zbl 1020.49023); translation from Prikl. Mat. Inf. 3, 81--89 (1999)]. We showed the validity of the formula under restricted assumption for the sake of completeness, and would like to bring our readers to [\textit{I. Yegorov} and \textit{P. Dower}, ``Perspectives on characteristics based curse-of-dimensionality-free numerical approaches for solving Hamilton-Jacobi equations'', Appl. Math. Optim. (to appear)] which validates our conjectures in a more general setting. We conjectured the weakest assumption of our formula to hold is a pseudoconvexity assumption similar to one stated in [\textit{I. V. Rublev}, Comput. Math. Model. 11, No. 4, 391--400 (2000; Zbl 1020.49023); translation from Prikl. Mat. Inf. 3, 81--89 (1999)]. Our method is expected to have application in control theory, differential game problems and elsewhere.Estimating the division rate from indirect measurements of single cells.https://www.zbmath.org/1452.352182021-02-12T15:23:00+00:00"Doumic, Marie"https://www.zbmath.org/authors/?q=ai:doumic.marie"Olivier, Adélaïde"https://www.zbmath.org/authors/?q=ai:olivier.adelaide"Robert, Lydia"https://www.zbmath.org/authors/?q=ai:robert.lydiaSummary: Is it possible to estimate the dependence of a growing and dividing population on a given trait in the case where this trait is not directly accessible by experimental measurements, but making use of measurements of another variable? This article adresses this general question for a very recent and popular model describing bacterial growth, the so-called incremental or adder model. In this model, the division rate depends on the increment of size between birth and division, whereas the most accessible trait is the size itself. We prove that estimating the division rate from size measurements is possible, we state a reconstruction formula in a deterministic and then in a statistical setting, and solve numerically the problem on simulated and experimental data. Though this represents a severely ill-posed inverse problem, our numerical results prove to be satisfactory.On the 1D modeling of fluid flowing through a junction.https://www.zbmath.org/1452.351382021-02-12T15:23:00+00:00"Colombo, Rinaldo M."https://www.zbmath.org/authors/?q=ai:colombo.rinaldo-m"Garavello, Mauro"https://www.zbmath.org/authors/?q=ai:garavello.mauroSummary: A compressible fluid flows through a junction between two different pipes. Its evolution is described by the 2D or 3D Euler equations, whose analytical theory is far from complete and whose numerical treatment may be rather costly. This note compares different 1D approaches to this phenomenon.A finite element method of the self-consistent field theory on general curved surfaces.https://www.zbmath.org/1452.651142021-02-12T15:23:00+00:00"Wei, Huayi"https://www.zbmath.org/authors/?q=ai:wei.huayi"Xu, Ming"https://www.zbmath.org/authors/?q=ai:xu.ming"Si, Wei"https://www.zbmath.org/authors/?q=ai:si.wei"Jiang, Kai"https://www.zbmath.org/authors/?q=ai:jiang.kaiSummary: Block copolymers provide a wonderful platform in the study of soft condensed matter systems. Many fascinating ordered structures have been discovered in bulk and confined systems. Among various theories, the self-consistent field theory (SCFT) has been proven to be a powerful tool for studying the equilibrium ordered structures. Many numerical methods have been developed to solve the SCFT model. However, most of these focus on the bulk systems, and little work on the confined systems, especially on general curved surfaces. In this work, we developed a linear surface finite element method, which has a rigorous mathematical theory to guarantee numerical precision, to study the self-assembled phases of block copolymers on general curved surfaces based on the SCFT. Furthermore, to capture the consistent surface for a given self-assembled pattern, an adaptive approach to optimize the size of the general curved surface has been proposed. To demonstrate the power of this approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces, including five closed surfaces and an unclosed surface. Numerical results illustrate the efficiency of the proposed method. The obtained ordered structures are consistent with the previous results on standard surfaces, such as sphere and torus. More significantly, the proposed numerical framework can be applied to study the phase behaviors of block copolymers on general surfaces accurately.Generalized solutions to models of inviscid fluids.https://www.zbmath.org/1452.351362021-02-12T15:23:00+00:00"Breit, Dominic"https://www.zbmath.org/authors/?q=ai:breit.dominic"Feireisl, Eduard"https://www.zbmath.org/authors/?q=ai:feireisl.eduard"Hofmanová, Martina"https://www.zbmath.org/authors/?q=ai:hofmanova.martinaSummary: We discuss several approaches to generalized solutions of problems describing the motion of inviscid fluids. We propose a new concept of dissipative solution to the compressible Euler system based on a careful analysis of possible oscillations and/or concentrations in the associated generating sequence. Unlike the conventional measure-valued solutions or rather their expected values, the dissipative solutions comply with a natural compatibility condition -- they are classical solutions as long as they enjoy a certain degree of smoothness.Adiabatic limit in Ginzburg-Landau and Seiberg-Witten equations.https://www.zbmath.org/1452.351992021-02-12T15:23:00+00:00"Sergeev, A. G."https://www.zbmath.org/authors/?q=ai:sergeev.armen-glebovichThe author reviews results on adiabatic theorems for Ginzburg-Landau and Seiber-Witten equations, which have originally appeared in [\textit{A. G. Sergeev}, Proc. Steklov Inst. Math. 289, 227--285 (2015; Zbl 1351.35203); translation from Tr. Mat. Inst. Steklova 289, 242--303 (2015)] and [\textit{C. H. Taubes}, J. Am. Math. Soc. 9, No. 3, 845--918 (1996; Zbl 0867.53025)], respectively. The adiabatic principle appearing in [Sergeev, loc. cit.] has been proved rigorously in [\textit{R. V. Pal'velev}, Trans. Mosc. Math. Soc. 2011, 219-244 (2011; Zbl 1246.82125); translation from Tr. Mosk. Mat. O.-va 2011, No. 1, 26 p. (2011)] (see also [\textit{R. V. Palvelev} and \textit{A. G. Sergeev}, Proc. Steklov Inst. Math. 277, 191--205 (2012; Zbl 1311.35304); translation from Tr. Mat. Inst. Steklova 277, 199--214 (2012)]). The exposition is very clearly written and well-suited to get a good first understanding of the results in the reviewed articles. Topics covered in the chapter on the Ginzburg-Landau equations are vortex solutions of the time-independent Ginzburg-Landau equations, the Taubes theorem characterizing equivalence classes of such solutions when taking a quotient w.r.t. gauge transformations, the time-dependent Ginzburg-Landau equations including an adiabatic theorem for them. The adiabatic theorem states that the relevant adiabatic trajectories are the geodesics of a Riemannian metric induced by the kinetic energy functional related to the time-dependent Ginzburg-Landau equations. In the chapter on the Seiberg-Witten equations, the author briefly discusses the geometric structures that are necessary to introduce the Seiberg-Witten equations on four-dimensional symplectic manifolds. Afterwards, an appropriate adiabatic limit is introduced and a Taubes theorem is discussed.
Reviewer: Andreas Deuchert (Zürich)Global existence for a one-dimensional non-relativistic Euler model with relaxation.https://www.zbmath.org/1452.351402021-02-12T15:23:00+00:00"Xiang, Shuyang"https://www.zbmath.org/authors/?q=ai:xiang.shuyang"Cao, Yangyang"https://www.zbmath.org/authors/?q=ai:cao.yangyangSummary: We study the initial value problem for a kind of Euler equation with a source term. Our main result is the existence of a globally-in-time weak solution whose total variation is bounded on the domain of definition, allowing the existence of shock waves. Our proof relies on a well-balanced random choice method called Glimm method which preserves the fluid equilibria and we construct a sequence of approximate weak solutions which converges to the exact weak solution of the initial value problem, based on the construction of exact solutions of the generalized Riemann problem associated with initially piecewise steady state solutions.Convergence of a full discrete finite element method for the Korteweg-de Vries equation.https://www.zbmath.org/1452.652342021-02-12T15:23:00+00:00"Huang, Pengzhan"https://www.zbmath.org/authors/?q=ai:huang.pengzhanSummary: In this paper, we put emphasis on discussing a full discrete finite element scheme for the Korteweg-de Vries equation, where nonlinear term is dealt with a semi-implicit scheme and temporal term is discreted by the Euler scheme. Theoretical analysis is based on error splitting technique, i.e., error function is split as temporal error function plus spatial error function, and then unconditionally optimal error estimates of the considered full discrete scheme are obtained. Numerical results are provided to confirm our theoretical analysis, which show that no time-step condition is needed.Families of interior penalty hybridizable discontinuous Galerkin methods for second order elliptic problems.https://www.zbmath.org/1452.653362021-02-12T15:23:00+00:00"Fabien, Maurice S."https://www.zbmath.org/authors/?q=ai:fabien.maurice-s"Knepley, Matthew G."https://www.zbmath.org/authors/?q=ai:knepley.matthew-g"Riviere, Beatrice M."https://www.zbmath.org/authors/?q=ai:riviere.beatrice-mSummary: The focus of this paper is the analysis of families of hybridizable interior penalty discontinuous Galerkin methods for second order elliptic problems. We derive \textit{a priori} error estimates in the energy norm that are optimal with respect to the mesh size. Suboptimal \(L^2\)-norm error estimates are proven. These results are valid in two and three dimensions. Numerical results support our theoretical findings, and we illustrate the computational cost of the method.Reduced basis approximations of the solutions to spectral fractional diffusion problems.https://www.zbmath.org/1452.653242021-02-12T15:23:00+00:00"Bonito, Andrea"https://www.zbmath.org/authors/?q=ai:bonito.andrea"Guignard, Diane"https://www.zbmath.org/authors/?q=ai:guignard.diane"Zhang, Ashley R."https://www.zbmath.org/authors/?q=ai:zhang.ashley-rSummary: We consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction-diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction-diffusion problems. The reduced basis does not depend on the fractional power \(s\) for \(0 < s_{min} \leqslant s \leqslant s_{max} < 1\). It is built \textit{offline} once for all and used \textit{online} irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.A two-dimensional method for a family of dispersive shallow water models.https://www.zbmath.org/1452.652212021-02-12T15:23:00+00:00"Aïssiouene, Nora"https://www.zbmath.org/authors/?q=ai:aissiouene.nora"Bristeau, Marie-Odile"https://www.zbmath.org/authors/?q=ai:bristeau.marie-odile"Godlewski, Edwige"https://www.zbmath.org/authors/?q=ai:godlewski.edwige"Mangeney, Anne"https://www.zbmath.org/authors/?q=ai:mangeney.anne"Parés Madroñal, Carlos"https://www.zbmath.org/authors/?q=ai:pares-madronal.carlos"Sainte-Marie, Jacques"https://www.zbmath.org/authors/?q=ai:sainte-marie.jacquesThis article presents a numerical method for a family of two-dimensional dispersive shallow water systems with topography. The approach relies on shallow water approximations without the hydrostatic assumption of the incompressible Euler system with free surface. Numerical experiments are also included to support the theoretical findings.
Reviewer: Marius Ghergu (Dublin)Novel simulations to the time-fractional Fisher's equation.https://www.zbmath.org/1452.652962021-02-12T15:23:00+00:00"Veeresha, P."https://www.zbmath.org/authors/?q=ai:veeresha.pundikala"Prakasha, D. G."https://www.zbmath.org/authors/?q=ai:prakasha.doddabhadrappla-gowda"Baskonus, Haci Mehmet"https://www.zbmath.org/authors/?q=ai:baskonus.haci-mehmetSummary: In the present work, an efficient numerical technique, called \(q\)-homotopy analysis transform method (briefly, \(q\)-HATM), is applied to nonlinear Fisher's equation of fractional order. The homotopy polynomials are employed, in order to handle the nonlinear terms. Numerical examples are illustrated to examine the efficiency of the proposed technique. The suggested algorithm provides the auxiliary parameters \(\hbar\) and \(n\), which help us to control and adjust the convergence region of the series solution. The outcomes of the study reveal that the \(q\)-HATM is computationally very effective and accurate to analyse nonlinear fractional differential equations.Strength-duration relationship in an excitable medium.https://www.zbmath.org/1452.350582021-02-12T15:23:00+00:00"Bezekci, Burhan"https://www.zbmath.org/authors/?q=ai:bezekci.burhan"Biktashev, V. N."https://www.zbmath.org/authors/?q=ai:biktashev.vadim-nSummary: We consider the strength-duration relationship in one-dimensional spatially extended excitable media. In a previous study [\textit{I. Idris} and \textit{V. N. Biktashev}, ``Analytical approach to initiation of propagating fronts'', Phys Rev Lett 101, No. 24, Article ID 244101, 4 p. (2008; \url{doi:10.1103/PhysRevLett.101.244101})] set out to separate initial (or boundary) conditions leading to propagation wave solutions from those leading to decay solutions, an analytical criterion based on an approximation of the (center-)stable manifold of a certain critical solution was presented. The theoretical prediction in the case of strength-extent curve was later on extended to cover a wider class of excitable systems including multicomponent reaction-diffusion systems, systems with non-self-adjoint linearized operators and in particular, systems with moving critical solutions (critical fronts and critical pulses) [\textit{B. Bezekci} et al., ``Semianalytical approach to criteria for ignition of excitation waves'', Phys Rev E 92, No. 4, Article ID 042917, 28 p. (2015; \url{doi:10.1103/PhysRevE.92.042917})]. In the present work, we consider extension of the theory to the case of strength-duration curve.On the regularity of very weak solutions for linear elliptic equations in divergence form.https://www.zbmath.org/1452.350562021-02-12T15:23:00+00:00"La Manna, Domenico Angelo"https://www.zbmath.org/authors/?q=ai:la-manna.domenico-angelo"Leone, Chiara"https://www.zbmath.org/authors/?q=ai:leone.chiara"Schiattarella, Roberta"https://www.zbmath.org/authors/?q=ai:schiattarella.robertaThe authors consider a linear homogeneous elliptic equation in divergence form and show that its very weak solutions are actually weak. In order to get the result, they assume that the leading part coefficients satisfy a weak differentiability assumption, i.e., they belong to the space \(W^{1,n}(\Omega)\) and the continuity modulus satisfies a Dini-type continuity assumption.
Reviewer: Giuseppe Di Fazio (Catania)