Recent zbMATH articles in MSC 35Rhttps://www.zbmath.org/atom/cc/35R2021-02-27T13:50:00+00:00WerkzeugIncomplete iterative solution of subdiffusion.https://www.zbmath.org/1453.653262021-02-27T13:50:00+00:00"Jin, Bangti"https://www.zbmath.org/authors/?q=ai:jin.bangti"Zhou, Zhi"https://www.zbmath.org/authors/?q=ai:zhou.zhiSummary: In this work, we develop an efficient incomplete iterative scheme for the numerical solution of the subdiffusion model involving a Caputo derivative of order \(\alpha \in (0,1)\) in time. It is based on piecewise linear Galerkin finite element method in space and backward Euler convolution quadrature in time and solves one linear algebraic system inexactly by an iterative algorithm at each time step. We present theoretical results for both smooth and nonsmooth solutions, using novel weighted estimates of the time-stepping scheme. The analysis indicates that with the number of iterations at each time level chosen properly, the error estimates are nearly identical with that for the exact linear solver, and the theoretical findings provide guidelines on the choice. Illustrative numerical results are presented to complement the theoretical analysis.Estimating failure probabilities.https://www.zbmath.org/1453.653752021-02-27T13:50:00+00:00"ter Maten, E. Jan W."https://www.zbmath.org/authors/?q=ai:ter-maten.e-jan-w"Beelen, Theo G. J."https://www.zbmath.org/authors/?q=ai:beelen.theo-g-j"Di Bucchianico, Alessandro"https://www.zbmath.org/authors/?q=ai:di-bucchianico.alessandro"Pulch, Roland"https://www.zbmath.org/authors/?q=ai:pulch.roland"Römer, Ulrich"https://www.zbmath.org/authors/?q=ai:romer.ulrich"De Gersem, Herbert"https://www.zbmath.org/authors/?q=ai:de-gersem.herbert"Janssen, Rick"https://www.zbmath.org/authors/?q=ai:janssen.rick"Dohmen, Jos J."https://www.zbmath.org/authors/?q=ai:dohmen.jos-j"Tasić, Bratislav"https://www.zbmath.org/authors/?q=ai:tasic.bratislav"Gillon, Renaud"https://www.zbmath.org/authors/?q=ai:gillon.renaud"Wieers, Aarnout"https://www.zbmath.org/authors/?q=ai:wieers.aarnout"Deleu, Frederik"https://www.zbmath.org/authors/?q=ai:deleu.frederikSummary: System failure describes an undesired configuration of an engineering device, possibly leading to the destruction of material or a significant loss of performance and a consequent loss of yield. For systems subject to uncertainties, failure probabilities express the probability of this undesired configuration to take place. The accurate computation of failure probabilities, however, can be very difficult in practice. It may also become very costly, because of the many Monte Carlo samples that have to be taken, which may involve time consuming evaluations. In this chapter we present an overview of techniques to realistically estimate the amount of Monte Carlo runs that are needed to guarantee sharp bounds for relative errors of failure probabilities. They are presented for Monte Carlo sampling and for Importance Sampling. These error estimates apply to both non-parametric and parametric sampling. In the case of parametric sampling we propose a hybrid algorithm that combines simulations of full models and approximating response surface models. We illustrate this hybrid algorithm with a computation of bond wire fusing probabilities.
For the entire collection see [Zbl 1433.78001].A nonlinear moment model for radiative transfer equation in slab geometry.https://www.zbmath.org/1453.652482021-02-27T13:50:00+00:00"Fan, Yuwei"https://www.zbmath.org/authors/?q=ai:fan.yuwei"Li, Ruo"https://www.zbmath.org/authors/?q=ai:li.ruo"Zheng, Lingchao"https://www.zbmath.org/authors/?q=ai:zheng.lingchaoSummary: This paper is concerned with the approximation of the radiative transfer equation for a grey medium in the slab geometry by the moment method. We develop a novel moment model inspired by the classical \(P_N\) model and \(M_N\) model. The new model takes the ansatz of the \(M_1\) model as the weight function and follows the primary idea of the \(P_N\) model to approximate the specific intensity by expanding it around the weight function in terms of orthogonal polynomials. The weight function uses the information of the first two moments, which brings the new model the capability to approximate an anisotropic distribution. Mathematical properties of the moment model are investigated, and particularly the hyperbolicity and the characteristic structure of the Riemann problem of the model with three moments are studied in detail. Some numerical simulations demonstrate its numerical efficiency and show its superior in comparison to the \(P_N\) model.From fluctuating kinetics to fluctuating hydrodynamics: a \(\Gamma \)-convergence of large deviations functionals approach.https://www.zbmath.org/1453.820532021-02-27T13:50:00+00:00"Barré, J."https://www.zbmath.org/authors/?q=ai:barre.julien"Bernardin, C."https://www.zbmath.org/authors/?q=ai:bernardin.cedric"Chétrite, R."https://www.zbmath.org/authors/?q=ai:chetrite.raphael"Chopra, Y."https://www.zbmath.org/authors/?q=ai:chopra.y"Mariani, M."https://www.zbmath.org/authors/?q=ai:mariani.manuel-sebastian|mariani.maria-cristina|mariani.maria-christina|mariani.mauroSummary: We consider extended slow-fast systems of \(N\) interacting diffusions. The typical behavior of the empirical density is described by a nonlinear McKean-Vlasov equation depending on \(\varepsilon \), the scaling parameter separating the time scale of the slow variable from the time scale of the fast variable. Its atypical behavior is encapsulated in a large \(N\) Large Deviation Principle with a rate functional \(\mathscr{I}^\varepsilon \). We study the \(\Gamma \)-convergence of \(\mathscr{I}^\varepsilon\) as \(\varepsilon \rightarrow 0\) and show it converges to the rate functional appearing in the Macroscopic Fluctuations Theory for diffusive systems.Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks.https://www.zbmath.org/1453.650212021-02-27T13:50:00+00:00"Karumuri, Sharmila"https://www.zbmath.org/authors/?q=ai:karumuri.sharmila"Tripathy, Rohit"https://www.zbmath.org/authors/?q=ai:tripathy.rohit-k"Bilionis, Ilias"https://www.zbmath.org/authors/?q=ai:bilionis.ilias"Panchal, Jitesh"https://www.zbmath.org/authors/?q=ai:panchal.jitesh-hSummary: Stochastic partial differential equations (SPDEs) are ubiquitous in engineering and computational sciences. The stochasticity arises as a consequence of uncertainty in input parameters, constitutive relations, initial/boundary conditions, etc. Because of these functional uncertainties, the stochastic parameter space is often high-dimensional, requiring hundreds, or even thousands, of parameters to describe it. This poses an insurmountable challenge to response surface modeling since the number of forward model evaluations needed to construct an accurate surrogate grows exponentially with the dimension of the uncertain parameter space; a phenomenon referred to as the \textit{curse of dimensionality}. State-of-the-art methods for high-dimensional uncertainty propagation seek to alleviate the curse of dimensionality by performing dimensionality reduction in the uncertain parameter space. However, one still needs to perform forward model evaluations that potentially carry a very high computational burden. We propose a novel methodology for high-dimensional uncertainty propagation of elliptic SPDEs which lifts the requirement for a deterministic forward solver. Our approach is as follows. We parameterize the solution of the elliptic SPDE using a deep residual network (ResNet). In a departure from traditional squared residual (SR) based loss function for training the ResNet, we introduce a physics-informed loss function derived from variational principles. Specifically, our loss function is the expectation of the energy functional of the PDE over the stochastic variables. We demonstrate our solver-free approach through various examples where the elliptic SPDE is subjected to different types of high-dimensional input uncertainties. Also, we solve high-dimensional uncertainty propagation and inverse problems.Fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations.https://www.zbmath.org/1453.650622021-02-27T13:50:00+00:00"Bai, Zhong-Zhi"https://www.zbmath.org/authors/?q=ai:bai.zhongzhi"Lu, Kang-Ya"https://www.zbmath.org/authors/?q=ai:lu.kang-yaSummary: The discretizations of two- and three-dimensional spatial fractional diffusion equations with the shifted finite-difference formulas of the Grünwald-Letnikov type can result in discrete linear systems whose coefficient matrices are of the form \(D + T\), where \(D\) is a nonnegative diagonal matrix and \(T\) is a block-Toeplitz with Toeplitz-block matrix or a block-Toeplitz with each block being block-Toeplitz with Toeplitz-block matrix. For these discrete spatial fractional diffusion matrices, we construct diagonal and block-circulant with circulant-block splitting preconditioner for the two-dimensional case, and diagonal and block-circulant with each block being block-circulant with circulant-block splitting preconditioner for the three-dimensional case, to further accelerate the convergence rates of Krylov subspace iteration methods, and we analyze the eigenvalue distributions for the corresponding preconditioned matrices. Theoretical results show that except for a small number of outliners the eigenvalues of the preconditioned matrices are located within a complex disk centered at 1 with the radius being exactly less than 1, and numerical experiments demonstrate that these structured preconditioners can significantly improve the convergence behavior of the Krylov subspace iteration methods. Moreover, this approach is superior to the geometric multigrid method and the preconditioned conjugate gradient methods incorporated with the approximate inverse circulant-plus-diagonal preconditioners in both iteration counts and computing times.Sharp well-posedness and ill-posedness of the three-dimensional primitive equations of geophysics in Fourier-Besov spaces.https://www.zbmath.org/1453.351522021-02-27T13:50:00+00:00"Sun, Jinyi"https://www.zbmath.org/authors/?q=ai:sun.jinyi"Cui, Shangbin"https://www.zbmath.org/authors/?q=ai:cui.shangbinSummary: We study the well-posedness and ill-posedness for the Cauchy problem of the three-dimensional primitive equations describing the large-scale oceanic and atmospheric circulations. By using the Littlewood-Paley analysis technique, we prove that the Cauchy problem of the three-dimensional primitive equations with the Prandtl number \(P = 1\) is locally well-posed in the Fourier-Besov spaces \(\dot{F B}_{p, r}^{2 - \frac{3}{p}}(\mathbb{R}^3)\) for \(1 < p \leq \infty, 1 \leq r < \infty\) and \(\dot{F B}_{1, r}^{- 1}(\mathbb{R}^3)\) for \(1 \leq r \leq 2\), and is globally well-posed in these spaces when the initial data are small. We also verify that such problem is ill-posed in \(\dot{F B}_{1, r}^{- 1}(\mathbb{R}^3)\) for \(2 < r \leq \infty\), which implies that our work completes a dichotomy of well-posedness and ill-posedness for the three-dimensional primitive equations in the Fourier-Besov space framework.Optimal fractional differentiability for nonlinear parabolic measure data problems.https://www.zbmath.org/1453.351742021-02-27T13:50:00+00:00"Byun, Sun-Sig"https://www.zbmath.org/authors/?q=ai:byun.sun-sig"Cho, Namkyeong"https://www.zbmath.org/authors/?q=ai:cho.namkyeong"Song, Kyeong"https://www.zbmath.org/authors/?q=ai:song.kyeongSummary: We study a nonlinear parabolic equation with a finite Radon measure on the right-hand side. An optimal regularity assumption on the coefficients is identified to obtain a sharp fractional differentiability for such a parabolic measure data problem.Dynamics of stochastic reaction-diffusion equations.https://www.zbmath.org/1453.601212021-02-27T13:50:00+00:00"Kuehn, Christian"https://www.zbmath.org/authors/?q=ai:kuhn.christian"Neamţu, Alexandra"https://www.zbmath.org/authors/?q=ai:neamtu.alexandraThis survey article discusses some different approaches to solution theory and dynamical properties for stochastic partial differential equations (SPDEs). The current work essentially consists of three sections. Section 1 presents the introduction and a brief description of the article. Section 2 is concerned with the solution theory of SPDEs driven by general additive and multiplicative noise. The classical Itô theory, the random field approach and the recent developments in the context of rough paths and regularity structures are clearly illustrated. Section 3 is devoted to the long-time qualitative and quantitative dynamical behavior of SPDEs.
For the entire collection see [Zbl 1440.60053].
Reviewer: Marius Ghergu (Dublin)What is the fractional Laplacian? A comparative review with new results.https://www.zbmath.org/1453.351792021-02-27T13:50:00+00:00"Lischke, Anna"https://www.zbmath.org/authors/?q=ai:lischke.anna"Pang, Guofei"https://www.zbmath.org/authors/?q=ai:pang.guofei"Gulian, Mamikon"https://www.zbmath.org/authors/?q=ai:gulian.mamikon"Song, Fangying"https://www.zbmath.org/authors/?q=ai:song.fangying"Glusa, Christian"https://www.zbmath.org/authors/?q=ai:glusa.christian"Zheng, Xiaoning"https://www.zbmath.org/authors/?q=ai:zheng.xiaoning"Mao, Zhiping"https://www.zbmath.org/authors/?q=ai:mao.zhiping"Cai, Wei"https://www.zbmath.org/authors/?q=ai:cai.wei"Meerschaert, Mark M."https://www.zbmath.org/authors/?q=ai:meerschaert.mark-m"Ainsworth, Mark"https://www.zbmath.org/authors/?q=ai:ainsworth.mark"Karniadakis, George Em"https://www.zbmath.org/authors/?q=ai:karniadakis.george-emSummary: The fractional Laplacian in \(\mathbb{R}^d\), which we write as \((- \Delta)^{\alpha / 2}\) with \(\alpha \in(0, 2)\), has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. In contrast, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: ``What is the fractional Laplacian?'' Beginning from first principles, we compare several commonly used definitions of the fractional Laplacian theoretically, through their stochastic interpretations as well as their analytical properties. Then, we present quantitative comparisons using a sample of state-of-the-art methods. We discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems: radial basis function collocation (for the Riesz fractional Laplacian) and nonharmonic lifting (for the spectral fractional Laplacian). In our numerical studies, we aim to compare different definitions on bounded domains using a collection of benchmark problems. We consider the fractional Poisson equation with both zero and nonzero boundary conditions, where the fractional Laplacian is defined according to the Riesz definition, the spectral definition, the directional definition, and the horizon-based nonlocal definition. We verify the accuracy of the numerical methods used in the approximations for each operator, and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications.Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data.https://www.zbmath.org/1453.310142021-02-27T13:50:00+00:00"Bidaut-Véron, Marie-Françoise"https://www.zbmath.org/authors/?q=ai:bidaut-veron.marie-francoise"Nguyen, Quoc-Hung"https://www.zbmath.org/authors/?q=ai:nguyen.quoc-hung"Véron, Laurent"https://www.zbmath.org/authors/?q=ai:veron.laurentThis article is concerned with the quasilinear elliptic equation \(-\mathrm{div}(A(x,\nabla u))=|u|^{q_1-1} u|\nabla u|^{q_2}+\mu\) in \(\mathbb{R}^N\) and \(A(x,\nabla u)\sim |\nabla u|^{p-2}\nabla u\) in some sense. Here \(\mu\) is a Radon measure and \(q_1+q_2>p-1\). The authors obtain sufficient conditions in terms of the Wolff potential or the Riesz potentials of the measure \(\mu\) for the existence of solutions. Further, the potential estimates on the measure \(\mu\) are linked to Lipchitz estimates with respect to some Bessel or Riesz capacity.
Reviewer: Marius Ghergu (Dublin)Linearized compact difference methods combined with Richardson extrapolation for nonlinear delay Sobolev equations.https://www.zbmath.org/1453.652392021-02-27T13:50:00+00:00"Zhang, Chengjian"https://www.zbmath.org/authors/?q=ai:zhang.chengjian"Tan, Zengqiang"https://www.zbmath.org/authors/?q=ai:tan.zengqiangSummary: Delay Sobolev equations (DSEs) are a class of important models in fluid mechanics, thermodynamics and the other related fields. For solving this class of equations, in this paper, linearized compact difference methods (LCDMs) for one- and two-dimensional problems of DSEs are suggested. The solvability and convergence of the methods are analyzed and it is proved under some appropriate conditions that the methods are convergent of order two in time and order four in space. In order to improve the computational accuracy of LCDMs in time, we introduce the Richardson extrapolation technique, which leads to the improved LCDMs can reach the fourth-order accuracy in both time and space. Finally, with several numerical experiments, the theoretical accuracy and computational effectiveness of the proposed methods are further testified.Filtered stochastic Galerkin methods for hyperbolic equations.https://www.zbmath.org/1453.625792021-02-27T13:50:00+00:00"Kusch, Jonas"https://www.zbmath.org/authors/?q=ai:kusch.jonas"McClarren, Ryan G."https://www.zbmath.org/authors/?q=ai:mcclarren.ryan-g"Frank, Martin"https://www.zbmath.org/authors/?q=ai:frank.martinSummary: Uncertainty Quantification for nonlinear hyperbolic problems becomes a challenging task in the vicinity of shocks. Standard intrusive methods, such as Stochastic Galerkin (SG), lead to oscillatory solutions and can result in non-hyperbolic moment systems. The intrusive polynomial moment (IPM) method guarantees hyperbolicity but comes at higher numerical costs. In this paper, we filter the generalized polynomial chaos (gPC) coefficients of the SG approximation, which allows a numerically cheap reduction of oscillations. The derived filter is based on Lasso regression which sets small gPC coefficients of high order to zero. We adaptively and automatically choose the filter strength to obtain a zero-valued highest order moment. The filtered SG method is tested for Burgers' and the Euler equations. Results show a reduction of oscillations at shocks, which leads to an improved approximation of expectation values and the variance compared to SG and IPM.A low-rank approximated multiscale method for PDEs with random coefficients.https://www.zbmath.org/1453.654352021-02-27T13:50:00+00:00"Ou, Na"https://www.zbmath.org/authors/?q=ai:ou.na"Lin, Guang"https://www.zbmath.org/authors/?q=ai:lin.guang"Jiang, Lijian"https://www.zbmath.org/authors/?q=ai:jiang.lijianBinary relations, Bäcklund transformations, and wave packet propagation.https://www.zbmath.org/1453.810222021-02-27T13:50:00+00:00"Zharinov, V. V."https://www.zbmath.org/authors/?q=ai:zharinov.victor-vSummary: We propose a mathematical apparatus based on binary relations that expands the possibility of traditional analysis applied to problems in mathematical and theoretical physics. We illustrate the general constructions with examples with an algebraic description of Bäcklund transformations of nonlinear systems of partial differential equations and the dynamics of wave packet propagation.Approximate solution of fractional Black-Scholes European option pricing equation by using ETHPM.https://www.zbmath.org/1453.911062021-02-27T13:50:00+00:00"Bhadane, Pradip R."https://www.zbmath.org/authors/?q=ai:bhadane.pradip-r"Ghadle, Kirtiwant P."https://www.zbmath.org/authors/?q=ai:ghadle.kirtiwant-p"Hamoud, Ahmed A."https://www.zbmath.org/authors/?q=ai:hamoud.ahmed-abdullahSummary: We proposed a new reliable combination of new Homotopy Perturbation Method (HPM) and Elzaki transform called as Elzaki Transform Homotopy Perturbation Method (ETHPM) is designed to obtain a exact solution to the fractional Black-Scholes equation with boundary condition for a European option pricing problem. The fractional derivative is in Caputo sense and the nonlinear terms in Fractional Black-Scholes Equation can be handled by using HPM. The Black-Scholes formula is used as a model for valuing European or American call and put options on a non-dividend paying stock. The methods give an analytic solution of the fractional Black-Scholes equation in the form of a convergent series. Finally, some examples are included to demonstrate the validity and applicability of the proposed technique.Multiple solutions of Kazdan-Warner equation on graphs in the negative case.https://www.zbmath.org/1453.350912021-02-27T13:50:00+00:00"Liu, Shuang"https://www.zbmath.org/authors/?q=ai:liu.shuang"Yang, Yunyan"https://www.zbmath.org/authors/?q=ai:yang.yunyanOn a finite connecte graph \(G=(E,V)\), the authors consider the equation \(\Delta u+\kappa-K_\lambda e^{2u}=0\) where \(\kappa:V\to {\mathbb R}\) satisfies \(\int_V \kappa d\mu<\infty\), \(K_\lambda=K+\lambda\) and \(\lambda\in {\mathbb R}\), \(K:V\to {\mathbb R}\), \(\max_V K=0\). Using a variational approach, the authors obtain the existence of \(\lambda^*>0\) such that the above equation admits: (i) no solutions if \(\lambda>\lambda^*\); (ii) at least one solution if \(\lambda=\lambda^*\); (iii) at least two solutions for \(0<\lambda>\lambda^*\); (iv) a unique solution for \(\lambda\leq 0\).
Reviewer: Marius Ghergu (Dublin)Existence and stability of square-mean S-asymptotically periodic solutions to a fractional stochastic diffusion equation with fractional Brownian motion.https://www.zbmath.org/1453.601232021-02-27T13:50:00+00:00"Mu, Jia"https://www.zbmath.org/authors/?q=ai:mu.jia"Nan, Jiecuo"https://www.zbmath.org/authors/?q=ai:nan.jiecuo"Zhou, Yong"https://www.zbmath.org/authors/?q=ai:zhou.yongSummary: In this paper, a generalized Gronwall inequality is demonstrated, playing an important role in the study of fractional differential equations. In addition, with the fixed-point theorem and the properties of Mittag-Leffler functions, some results of the existence as well as asymptotic stability of square-mean \(S\)-asymptotically periodic solutions to a fractional stochastic diffusion equation with fractional Brownian motion are obtained. In the end, an example of numerical simulation is given to illustrate the effectiveness of our theory results.Spectral approach to the scattering map for the semi-classical defocusing Davey-Stewartson II equation.https://www.zbmath.org/1453.370622021-02-27T13:50:00+00:00"Klein, Christian"https://www.zbmath.org/authors/?q=ai:klein.christian"McLaughlin, Ken"https://www.zbmath.org/authors/?q=ai:mclaughlin.kenneth-d-t-r"Stoilov, Nikola"https://www.zbmath.org/authors/?q=ai:stoilov.nikola-mSummary: The inverse scattering approach for the defocusing Davey-Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier transforms complemented by a regularization of the singular parts by explicit analytic computation. The resulting algebraic equation is solved either by fixed point iterations or GMRES. Several examples for small values of the semi-classical parameter in the system are discussed.On the symmetry properties of a random passive scalar with and without boundaries, and their connection between hot and cold states.https://www.zbmath.org/1453.601162021-02-27T13:50:00+00:00"Camassa, Roberto"https://www.zbmath.org/authors/?q=ai:camassa.roberto"Kilic, Zeliha"https://www.zbmath.org/authors/?q=ai:kilic.zeliha"McLaughlin, Richard M."https://www.zbmath.org/authors/?q=ai:mclaughlin.richard-mSummary: We consider the evolution of a decaying passive scalar in the presence of a Gaussian white noise fluctuating linear shear flow known as the Majda Model. We focus on deterministic initial data and establish the short, intermediate, and long time symmetry properties of the evolving point wise probability measure (PDF) for the random passive scalar. We identify, for the cases of both point source and line source initial data, regions in the x-yplane outside of which the PDF skewness is sign definite for all time, while inside these regions we observe multiple sign changes corresponding to exchanges in symmetry between hot and cold leaning states using exact representation formula for the PDF at the origin, and away from the origin, using numerical evaluation of the exact available Mehler kernel formulae for the scalar's statistical moments. A new, rapidly convergent Monte-Carlo method is developed, dubbed Direct Monte-Carlo (DMC), using the available random Green's functions which allows for the fast construction of the PDF for single point statistics, as well as multi-point statistics including spatially integrated quantities natural for full Monte-Carlo simulations of the underlying stochastic differential equations (FMC). This new method demonstrates the full evolution of the PDF from short times, to its long time, limiting and collapsing universal distribution at arbitrary points in the plane. Further, this method provides a strong benchmark for FMC and we document numbers of field realization criteria for the FMC to faithfully compute this complete dynamics. Armed with this benchmark, we apply the FMC to a channel with a no-flux boundary condition enforced on parallel planes and observe a dramatically different long time state resulting from the existence of the wall. In particular, the channel case collapsing invariant measure has \textit{negative} skewness, with random states leaning heavily towards the hot state, in stark contrast to free space, where the limiting skewness is positive, with its states leaning heavily towards the cold state.A two-phase problem with Robin conditions on the free boundary.https://www.zbmath.org/1453.352002021-02-27T13:50:00+00:00"Guarino Lo Bianco, Serena"https://www.zbmath.org/authors/?q=ai:guarino-lo-bianco.serena"La Manna, Domenico Angelo"https://www.zbmath.org/authors/?q=ai:la-manna.domenico-angelo"Velichkov, Bozhidar"https://www.zbmath.org/authors/?q=ai:velichkov.bozhidarSummary: We study for the first time a two-phase free boundary problem in which the solution satisfies a Robin boundary condition. We consider the case in which the solution is continuous across the free boundary and we prove an existence and a regularity result for minimizers of the associated variational problem. Finally, in the appendix, we give an example of a class of Steiner symmetric minimizers.Stochastic neural field theory of wandering bumps on a sphere.https://www.zbmath.org/1453.351732021-02-27T13:50:00+00:00"Bressloff, Paul C."https://www.zbmath.org/authors/?q=ai:bressloff.paul-cSummary: We use a combination of group theoretic and perturbation methods to analyze the stochastic wandering of bump solutions in a neural field model on the sphere \(S^2\). We first construct an explicit bump solution in the absence of external inputs and noise, by taking the synaptic weight distribution to be the sum of first-order spherical harmonics. The corresponding neural field equation is equivariant under the action of the special orthogonal group \(S O(3)\), which implies that the bump is marginally stable with respect to rotations of the sphere. We then carry out an amplitude-phase decomposition of the solution in the presence of a weakly biased external input and weak noise, and use this to derive a pair of stochastic differential equations for the wandering of the bump, expressed in terms of angular coordinates on the sphere. The stochastic dynamics is a non-trivial generalization of the corresponding phase dynamics describing the wandering of a bump on a ring network with \(S O(2)\) symmetry, since \(S O(3)\) is non-abelian and \(S^2\) is a curved manifold.Causality and Bayesian network PDEs for multiscale representations of porous media.https://www.zbmath.org/1453.624222021-02-27T13:50:00+00:00"Um, Kimoon"https://www.zbmath.org/authors/?q=ai:um.kimoon"Hall, Eric J."https://www.zbmath.org/authors/?q=ai:hall.eric-joseph"Katsoulakis, Markos A."https://www.zbmath.org/authors/?q=ai:katsoulakis.markos-a"Tartakovsky, Daniel M."https://www.zbmath.org/authors/?q=ai:tartakovsky.daniel-mSummary: Microscopic (pore-scale) properties of porous media affect and often determine their macroscopic (continuum- or Darcy-scale) counterparts. Understanding the relationship between processes on these two scales is essential to both the derivation of macroscopic models of, e.g., transport phenomena in natural porous media, and the design of novel materials, e.g., for energy storage. Microscopic properties exhibit complex statistical correlations and geometric constraints that present challenges for the estimation of macroscopic quantities of interest (QoIs), e.g., in the context of global sensitivity analysis (GSA) of macroscopic QoIs with respect to microscopic material properties. We present a systematic way of building correlations into stochastic multiscale models through Bayesian Networks. The proposed framework allows us to construct the joint probability density function (PDF) of model parameters through causal relationships that are informed by domain knowledge and emulate engineering processes, e.g., the design of hierarchical nanoporous materials. These PDFs also serve as input for the forward propagation of parametric uncertainty thereby yielding a Bayesian Network PDE. To assess the impact of causal relationships and microscale correlations on macroscopic material properties, we propose a moment-independent GSA and corresponding effect rankings for Bayesian Network PDEs, based on the differential Mutual Information, that leverage the structure of Bayesian Networks and account for both correlated inputs and complex non-Gaussian (skewed, multimodal) QoIs. Our findings from numerical experiments, which feature a non-intrusive uncertainty quantification workflow, indicate two practical outcomes. First, the inclusion of correlations through structured priors based on causal relationships informed by domain knowledge impacts predictions of QoIs and has important implications for engineering design. Second, structured priors with non-trivial correlations yield different effect rankings than independent priors; these rankings are more consistent with the anticipated physics.Generalized Morrey regularity of \(2 b\)-parabolic systems.https://www.zbmath.org/1453.350372021-02-27T13:50:00+00:00"Palagachev, Dian K."https://www.zbmath.org/authors/?q=ai:palagachev.dian-k"Softova, Lubomira G."https://www.zbmath.org/authors/?q=ai:softova.lubomira-gSummary: We derive the Calderón-Zygmund property in generalized Morrey spaces for the strong solutions to \(2 b\)-order linear parabolic systems with discontinuous principal coefficients.Stationary currents in long-range interacting magnetic systems.https://www.zbmath.org/1453.820602021-02-27T13:50:00+00:00"Boccagna, Roberto"https://www.zbmath.org/authors/?q=ai:boccagna.robertoSummary: We construct a solution for the \(1d\) integro-differential stationary equation derived from a finite-volume version of the mesoscopic model proposed in [\textit{G. Giacomin} and \textit{J. L. Lebowitz}, J. Stat. Phys. 87, No. 1--2, 37--61 (1997; Zbl 0937.82037)]. This is the continuous limit of an Ising spin chain interacting at long range through Kac potentials, staying in contact at the two edges with reservoirs of fixed magnetizations. The stationary equation of the model is introduced here starting from the Lebowitz-Penrose free energy functional defined on the interval \([- \varepsilon^{-1}, \varepsilon^{-1}]\), \(\varepsilon > 0\). Below the critical temperature, and for \(\varepsilon\) small enough, we obtain a solution that is no longer monotone when opposite in sign, metastable boundary conditions are imposed. Moreover, the mesoscopic current flows along the magnetization gradient. This can be considered as an analytic proof of the existence of diffusion along the concentration gradient in one-component systems undergoing a phase transition, a phenomenon generally known as \textit{uphill diffusion}. In our proof uniqueness is lacking, and we have clues that the stationary solution obtained is not unique, as suggested by numerical simulations.Multiplicity and concentration of nontrivial solutions for fractional Schrödinger-Poisson system involving critical growth.https://www.zbmath.org/1453.351832021-02-27T13:50:00+00:00"Teng, Kaimin"https://www.zbmath.org/authors/?q=ai:teng.kaimin"Cheng, Yiqun"https://www.zbmath.org/authors/?q=ai:cheng.yiqunThis paper deals with following fractional Schrödinger-Poisson system:
\[
\begin{cases}
\varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u)+u^{2^*_s-1}&\quad \text{in}\ \mathbb{R}^3,\\
\varepsilon^{2t}(-\Delta)^t\phi=u^2&\quad \text{in}\ \mathbb{R}^3,\\
\end{cases}
\]
where \(s,t\in(0,1)\), \(s>\frac{3}{4}\), \(\varepsilon>0\) is a small parameter. Under some standard critical assumptions on \(f\) and local conditions on \(V\), they establish a family of positive solutions which concentrates around the set of local minima of \(V\) as \(\varepsilon\rightarrow0\). Moreover, they also obtain the multiplicity of solutions by employing the topology of the set where the potential \(V\) attains its minimum via Ljusternik-Schnirelmann theory.
Reviewer: Zhipeng Yang (Göttingen)Martingale solutions to stochastic nonlocal Cahn-Hilliard-Navier-Stokes equations with multiplicative noise of jump type.https://www.zbmath.org/1453.601172021-02-27T13:50:00+00:00"Deugoué, G."https://www.zbmath.org/authors/?q=ai:deugoue.gabriel"Ngana, A. Ndongmo"https://www.zbmath.org/authors/?q=ai:ngana.a-ndongmo"Medjo, T. Tachim"https://www.zbmath.org/authors/?q=ai:tachim-medjo.theodoreSummary: In this paper, we are interested in proving the existence of a weak martingale solution of the stochastic nonlocal Cahn-Hilliard-Navier-Stokes system driven by a pure jump noise in both 2D and 3D bounded domains. Our goal is achieved by using the classical Faedo-Galerkin approximation, a compactness method and a version of the Skorokhod embedding theorem for nonmetric spaces. In the 2D case, we prove the pathwise uniqueness of the solution and use the Yamada-Watanabe classical result to derive the existence of a strong solution.Uniqueness for identifying a space-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation from a single boundary point measurement.https://www.zbmath.org/1453.351982021-02-27T13:50:00+00:00"Wei, T."https://www.zbmath.org/authors/?q=ai:wei.tianwen|wei.tan|wei.tiangong|wei.tianhui|wei.tao|wei.timothy|wei.tingting|wei.tanyong|wei.tong|wei.tiehua|wei.tongli|wei.tianyou|wei.tangjian|wei.tian|wei.tongquan|wei.tengda|wei.ting|wei.tie"Yan, X. B."https://www.zbmath.org/authors/?q=ai:yan.xiaobin|yan.xiaobing|yan.xiongbin|yan.xiaobao|yan.xiangbinSummary: This paper is focused on a nonlinear inverse problem for identifying a space-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation by the measured data on a single boundary point for one-dimensional case. We give the definition of a weak solution and prove its existence for the corresponding direct problem by using the Fourier method. Based on the Gronwall inequality, analytic continuation and the Laplace transformation, we obtain the uniqueness for the inverse zeroth-order coefficient problem under some simple requirements to the Neumann boundary data.High energy semiclassical states for Kirchhoff problems with critical frequency.https://www.zbmath.org/1453.351752021-02-27T13:50:00+00:00"Zhang, Hui"https://www.zbmath.org/authors/?q=ai:zhang.hui.6"Zhang, Fubao"https://www.zbmath.org/authors/?q=ai:zhang.fubaoSummary: We study the following singularly perturbed Kirchhoff equation \[- ( \epsilon^2 a + \epsilon b \int_{\mathbb{R}^3} | \nabla u |^2 d x ) \Delta u + V ( x ) u = u^5 , \quad u \in D^{1 , 2} ( \mathbb{R}^3 ) ,\] where \(\epsilon > 0\) is a parameter, \( a , b > 0\), \(V \in L^{\frac{ 3}{ 2}} ( \mathbb{R}^3 )\) is nonnegative and of critical frequency. By means of variational methods, we investigate the relation between the number of high energy semiclassical states and the topology of the zero set of \(V\) for small \(\epsilon \).Analysis of a physically-relevant variable-order time-fractional reaction-diffusion model with Mittag-Leffler kernel.https://www.zbmath.org/1453.351852021-02-27T13:50:00+00:00"Zheng, Xiangcheng"https://www.zbmath.org/authors/?q=ai:zheng.xiangcheng"Wang, Hong"https://www.zbmath.org/authors/?q=ai:wang.hong.1"Fu, Hongfei"https://www.zbmath.org/authors/?q=ai:fu.hongfeiSummary: It is known that the well-posedness of time-fractional reaction-diffusion models with Mittag-Leffler kernel usually requires non-physical constraints on the initial data. In this paper, we propose a variable-order time-fractional reaction-diffusion equation with Mittag-Leffler kernel and prove that the aforementioned constraints could be eliminated by imposing the integer limit of the variable fractional order at the initial time, which mathematically demonstrates the physically-relevance of the variable-order modifications.A spectral approach for solving the nonclassical transport equation.https://www.zbmath.org/1453.821052021-02-27T13:50:00+00:00"Vasques, R."https://www.zbmath.org/authors/?q=ai:vasques.richard"Moraes, L. R. C."https://www.zbmath.org/authors/?q=ai:moraes.l-r-c"Barros, R. C."https://www.zbmath.org/authors/?q=ai:barros.rodrigo-c|barros.rui-c|de-barros.ricardo-c"Slaybaugh, R. N."https://www.zbmath.org/authors/?q=ai:slaybaugh.r-nSummary: This paper introduces a mathematical approach that allows one to numerically solve the nonclassical transport equation in a deterministic fashion using classical numerical procedures. The nonclassical transport equation describes particle transport for random statistically homogeneous systems in which the distribution function for free-paths between scattering centers is nonexponential. We use a spectral method to represent the nonclassical flux as a series of Laguerre polynomials in the free-path variable \(s\), resulting in a nonclassical equation that has the form of a classical transport equation. We present numerical results that validate the spectral approach, considering transport in slab geometry for both classical and nonclassical problems in the discrete ordinates formulation.Study of different approximations for solving heat transfer.https://www.zbmath.org/1453.800032021-02-27T13:50:00+00:00"Shestakov, A. A."https://www.zbmath.org/authors/?q=ai:shestakov.aleksandr-andreevichThe author considers equations for radiative heat transfer in media (transparent or not) at rest in a symmetric case. The aim is to compare various approximations and to derive conditions when these approximations are valid. The full equations are non-local (integro-differential).
Reviewer: Ilya A. Chernov (Petrozavodsk)A parallelized computational model for multidimensional systems of coupled nonlinear fractional hyperbolic equations.https://www.zbmath.org/1453.652272021-02-27T13:50:00+00:00"Macías-Díaz, J. E."https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardoSummary: In this work, we consider a general multidimensional system of hyperbolic partial differential equations with fractional diffusion of the Riesz type, constant damping and coupled nonlinear reaction terms. The system generalizes many particular models from the physical sciences (including inhibitor-activator models in chemistry, diffusive nonlinear systems in population dynamics and relativistic wave equations), and considers the presence of an arbitrary number of both spatial dimensions and dependent variables. Motivated by the wide range of applications, we propose an explicit four-step finite-difference methodology to approximate the solutions of the continuous system. The properties of stability, boundedness and convergence of the scheme are proved rigorously using a discrete form of the fractional energy method. An efficient computational implementation of the scheme is also proposed in this work. It is important to recall that algorithms for space-fractional systems are computationally highly demanding. To alleviate this problem, a parallel implementation of our scheme is proposed using a vector reformulation of the numerical method. We provide some illustrative simulations on the formation of complex patterns in the two-dimensional scenario, and even in the computationally intense three-dimensional case. For the sake of convenience, an algorithmic presentation of our computational model is provided in this manuscript.Nonuniform Alikhanov linearized Galerkin finite element methods for nonlinear time-fractional parabolic equations.https://www.zbmath.org/1453.653502021-02-27T13:50:00+00:00"Zhou, Boya"https://www.zbmath.org/authors/?q=ai:zhou.boya"Chen, Xiaoli"https://www.zbmath.org/authors/?q=ai:chen.xiaoli"Li, Dongfang"https://www.zbmath.org/authors/?q=ai:li.dongfangSummary: The solutions of the nonlinear time fractional parabolic problems usually undergo dramatic changes at the beginning. In order to overcome the initial singularity, the temporal discretization is done by using the Alikhanov schemes on the nonuniform meshes. And the spatial discretization is achieved by using the finite element methods. The optimal error estimates of the fully discrete schemes hold without certain time-step restrictions dependent on the spatial mesh sizes. Such unconditionally optimal convergent results are proved by taking the global behavior of the analytical solutions into account. Numerical results are presented to confirm the theoretical findings.A bi-fidelity method for the multiscale Boltzmann equation with random parameters.https://www.zbmath.org/1453.653602021-02-27T13:50:00+00:00"Liu, Liu"https://www.zbmath.org/authors/?q=ai:liu.liu"Zhu, Xueyu"https://www.zbmath.org/authors/?q=ai:zhu.xueyuSummary: In this paper, we study the multiscale Boltzmann equation with multi-dimensional random parameters by a bi-fidelity stochastic collocation (SC) method developed in [\textit{A. Narayan} et al., SIAM J. Sci. Comput. 36, No. 2, 495--521 (2014; Zbl 1296.65013); \textit{X. Zhu} et al., SIAM/ASA J. Uncertain. Quantif. 2, 444--463 (2014; Zbl 1306.65010); ``A multi-fidelity collocation method for time-dependent parameterized problems'', AIAA SciTech Forum (2017)]. By choosing the compressible Euler system as the low-fidelity model, we adapt the bi-fidelity SC method to combine computational efficiency of the low-fidelity model with high accuracy of the high-fidelity (Boltzmann) model. With only a small number of high-fidelity asymptotic-preserving solver runs for the Boltzmann equation, the bi-fidelity approximation can capture well the macroscopic quantities of the solution to the Boltzmann equation in the random space. A priori estimate on the accuracy between the high- and bi-fidelity solutions together with a convergence analysis is established. Finally, we present extensive numerical experiments to verify the efficiency and accuracy of our proposed method.Travelling waves for reaction-diffusion equations forced by translation invariant noise.https://www.zbmath.org/1453.352022021-02-27T13:50:00+00:00"Hamster, C. H. S."https://www.zbmath.org/authors/?q=ai:hamster.c-h-s"Hupkes, H. J."https://www.zbmath.org/authors/?q=ai:hupkes.hermen-janSummary: Inspired by applications, we consider reaction-diffusion equations on \(\mathbb{R}\) that are stochastically forced by a small multiplicative noise term that is white in time, coloured in space and invariant under translations. We show how these equations can be understood as a stochastic partial differential equation (SPDE) forced by a cylindrical Q-Wiener process and subsequently explain how to study stochastic travelling waves in this setting. In particular, we generalize the phase tracking framework that was developed in [\textit{C. H. S. Hamster} and \textit{H. J. Hupkes}, SIAM J. Appl. Dyn. Syst. 18, No. 1, 205--278 (2019; Zbl 1414.35117)] for noise processes driven by a single Brownian motion. The main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed. We illustrate our approach by two fully worked-out examples, which highlight the predictive power of our expansions.Traveling and standing fronts on curved surfaces.https://www.zbmath.org/1453.351172021-02-27T13:50:00+00:00"Białecki, Sławomir"https://www.zbmath.org/authors/?q=ai:bialecki.slawomir"Nałęcz-Jawecki, Paweł"https://www.zbmath.org/authors/?q=ai:nalecz-jawecki.pawel"Kaźmierczak, Bogdan"https://www.zbmath.org/authors/?q=ai:kazmierczak.bogdan"Lipniacki, Tomasz"https://www.zbmath.org/authors/?q=ai:lipniacki.tomaszSummary: We analyze heteroclinic traveling waves propagating on two dimensional manifolds to show that the geometric modification of the front velocity is proportional to the geodesic curvature of the front line. As a result, on surfaces of concave domains, stable standing fronts can be formed on lines of constant geodesic curvature. These lines minimize the geometric functional describing the system's energy, consisting of terms proportional to the front line-length and to the inclosed surface area. Front stabilization at portions of surface with negative Gaussian curvature, provides a mechanism of pattern formation. In contrast to the mechanism associated with the Turing instability, the proposed mechanism requires only a single scalar bistable reaction-diffusion equation and connects the intrinsic surface geometry with the arising pattern. By considering a system of equations modeling boundary-volume interactions, we show that polarization of the boundary may induce a corresponding polarization in the volume.A fast algorithm for radiative transport in isotropic media.https://www.zbmath.org/1453.653622021-02-27T13:50:00+00:00"Ren, Kui"https://www.zbmath.org/authors/?q=ai:ren.kui"Zhang, Rongting"https://www.zbmath.org/authors/?q=ai:zhang.rongting"Zhong, Yimin"https://www.zbmath.org/authors/?q=ai:zhong.yiminSummary: Constructing efficient numerical solution methods for the equation of radiative transfer (ERT) remains as a challenging task in scientific computing despite of the tremendous development on the subject in recent years. We present in this work a simple fast computational algorithm for solving the ERT in isotropic media. The algorithm we developed has two steps. In the first step, we solve a volume integral equation for the angularly-averaged ERT solution using iterative schemes such as the GMRES method. The computation in this step is accelerated with a fast multipole method (FMM). In the second step, we solve a scattering-free transport equation to recover the angular dependence of the ERT solution. The algorithm does not require the underlying medium be homogeneous. We present numerical simulations under various scenarios to demonstrate the performance of the proposed numerical algorithm for both homogeneous and heterogeneous media.Superdiffusion in the presence of a reflecting boundary.https://www.zbmath.org/1453.351772021-02-27T13:50:00+00:00"Jesus, Carla"https://www.zbmath.org/authors/?q=ai:jesus.carla"Sousa, Ercília"https://www.zbmath.org/authors/?q=ai:sousa.erciliaSummary: We study the effect of having a reflecting boundary condition in a superdiffusive model. Firstly it is described how the problem formulation is affected by this type of physical boundary and then it is shown how to implement an implicit numerical method to compute the numerical solutions. The consistency and stability analysis of the numerical method are discussed. In the end numerical experiments are presented to show the performance of the scheme and to visualize the consequences of having a reflecting wall.Nonlocal problems for the fourth order impulsive partial differential equations.https://www.zbmath.org/1453.351862021-02-27T13:50:00+00:00"Assanova, Anar T."https://www.zbmath.org/authors/?q=ai:assanova.anar-turmaganbetkyzy"Abildayeva, Aziza D."https://www.zbmath.org/authors/?q=ai:abildayeva.aziza-d"Tleulessova, Agila B."https://www.zbmath.org/authors/?q=ai:tleulessova.agila-bSummary: Nonlocal problems for an impulsive system of fourth-order partial differential equations are investigated. By the method of introducing additional functions, the problems under study are reduced to an equivalent problem consisting of the impulsive system of second-order hyperbolic equations and integral relations. Algorithm for finding the approximate solutions to the equivalent problem is constructed and its convergence is proved. Sufficient conditions are obtained for the unique solvability of a nonlocal problem for the impulsive system of fourth-order partial differential equations. As an example, the conditions for the unique solvability of a periodic problem for the impulsive system of fourth-order partial differential equations are established.
For the entire collection see [Zbl 1445.34003].Inverse problem on conservation laws.https://www.zbmath.org/1453.351922021-02-27T13:50:00+00:00"Popovych, Roman O."https://www.zbmath.org/authors/?q=ai:popovych.roman-o"Bihlo, Alexander"https://www.zbmath.org/authors/?q=ai:bihlo.alexanderSummary: The explicit formulation of the general inverse problem on conservation laws is presented for the first time. Within this problem, one aims to derive the general form of systems of differential equations that admit a prescribed set of conservation laws. The particular cases of the inverse problem on first integrals of ordinary differential equations and on conservation laws for evolution equations are studied. We also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zeroth-order conservation-law characteristics. This particular case is further studied in the context of conservative first-order parameterization schemes for the two-dimensional incompressible Euler equations. We exhaustively classify conservative first-order parameterization schemes for the eddy-vorticity flux that lead to a class of closed, averaged Euler equations possessing conservation of generalized circulation, generalized momentum and energy.Exact Green's formula for the fractional Laplacian and perturbations.https://www.zbmath.org/1453.350512021-02-27T13:50:00+00:00"Grubb, Gerd"https://www.zbmath.org/authors/?q=ai:grubb.gerdThe Green formula is a basic tool in the mathematical analysis of large classes of differential or partial differential equations. This paper establishes the exact Green formula associated with the fractional Laplace operator. More generally, the author investigates powers of a general second-order strongly elliptic partial differential operator. Several examples illustrate both local and nonlocal particular cases.
Reviewer: Vicenţiu D. Rădulescu (Craiova)Numerical method for solving time-fractional multi-dimensional diffusion equations.https://www.zbmath.org/1453.653792021-02-27T13:50:00+00:00"Prakash, Amit"https://www.zbmath.org/authors/?q=ai:prakash.amit"Kumar, Manoj"https://www.zbmath.org/authors/?q=ai:yadav.manoj-kumar|kumar.manoj.1|kumar.manoj|kumar.manoj.2Summary: The key object of the current paper is to demonstrate a numerical technique to find the solution of fractional multi-dimensional diffusion equations that describe density dynamics in a material undergoing diffusion with the help of fractional variation iteration method (FVIM). Fractional variation iteration method is not confined to the minor parameter as usual perturbation method. This technique provides us analytical solution in the form of a convergent series with easily computable components. The advantage of this method over other method is that it does not require any linearisation, perturbation and restrictive assumptions.Second order difference schemes for time-fractional KdV-Burgers' equation with initial singularity.https://www.zbmath.org/1453.652102021-02-27T13:50:00+00:00"Cen, Dakang"https://www.zbmath.org/authors/?q=ai:cen.dakang"Wang, Zhibo"https://www.zbmath.org/authors/?q=ai:wang.zhibo"Mo, Yan"https://www.zbmath.org/authors/?q=ai:mo.yanSummary: In this paper, we study the numerical method for time-fractional KdV-Burgers' equation with initial singularity. The famous \(L 2- 1_\sigma\) formula on graded meshes is adopted to approximate the Caputo derivative. Meanwhile, a nonlinear finite difference method on uniform grids is deduced for spatial discretization. The proposed method is second order in time and first order in space. With the help of the fractional Grönwall inequality, the unconditional stability and convergence of the current scheme are analyzed based on some skills. To raise the accuracy in spatial direction, a second order method is then carefully deduced. At last, theoretical results are verified by numerical experiments.Solving spatial-fractional partial differential diffusion equations by spectral method.https://www.zbmath.org/1453.654602021-02-27T13:50:00+00:00"Nie, Ningming"https://www.zbmath.org/authors/?q=ai:nie.ningming"Huang, Jianfei"https://www.zbmath.org/authors/?q=ai:huang.jianfei"Wang, Wenjia"https://www.zbmath.org/authors/?q=ai:wang.wenjia"Tang, Yifa"https://www.zbmath.org/authors/?q=ai:tang.yifaSummary: This paper focuses on numerical solution of an initial-boundary value problem of spatial-fractional partial differential diffusion equation. The proposed numerical method is based on Legendre spectral method for Riemann-Liouville fractional derivative in space and a finite difference scheme in time. Numerical analysis of stability and convergence for our method is established rigourously. Finally, numerical results verify the validity of the theoretical analysis.Fast exponential time differencing/spectral-Galerkin method for the nonlinear fractional Ginzburg-Landau equation with fractional Laplacian in unbounded domain.https://www.zbmath.org/1453.653652021-02-27T13:50:00+00:00"Wang, Pengde"https://www.zbmath.org/authors/?q=ai:wang.pengdeSummary: This paper proposes a fast and efficient spectral-Galerkin method for the nonlinear complex Ginzburg-Landau equation involving the fractional Laplacian in \(\mathbb{R}^d\). By employing the Fourier-like bi-orthogonal mapped Chebyshev function as basis functions, the fractional Laplacian can be fully diagonalized. Then for the resulting diagonalized semi-discrete system, an exponential time differencing scheme is proposed for the temporal discretization. The obtained method can be fast implemented and has second order accuracy in time and algebraical accuracy in space. One- and two-dimensional numerical examples are tested to validate the accuracy and efficiency of the proposed method.Higher-order Wong-Zakai approximations of stochastic reaction-diffusion equations on \(\mathbb{R}^N\).https://www.zbmath.org/1453.350082021-02-27T13:50:00+00:00"Zhao, Wenqiang"https://www.zbmath.org/authors/?q=ai:zhao.wenqiang"Zhang, Yijin"https://www.zbmath.org/authors/?q=ai:zhang.yijin"Chen, Shangjie"https://www.zbmath.org/authors/?q=ai:chen.shangjieSummary: In this paper, we consider the higher-order Wong-Zakai approximations of the non-autonomous stochastic reaction-diffusion equation driven by additive/multiplicative white noises. The solutions between the approximation equation and stochastic reaction-diffusion equation are compared in higher-order spaces, in terms of the initial data. Based on these results and the known \(L^2\)-upper semi-continuity, we prove that the random attractor of the approximation random system converges to that of the non-autonomous stochastic reaction-diffusion equation with additive/multiplicative white noises in \(L^p(\mathbb{R}^N) \cap H^1(\mathbb{R}^N)\) when the size of the approximation shrinks to zero.Bifurcation analysis of pattern formation in a two-dimensional hybrid reaction-transport model.https://www.zbmath.org/1453.350242021-02-27T13:50:00+00:00"Carroll, Sam R."https://www.zbmath.org/authors/?q=ai:carroll.sam-r"Brooks, Heather Z."https://www.zbmath.org/authors/?q=ai:brooks.heather-z.1"Bressloff, Paul C."https://www.zbmath.org/authors/?q=ai:bressloff.paul-cSummary: Following up on our work on Turing pattern formation in a one-dimensional reaction-transport model, we explore the emergence of patterns in a two dimensional model described by a system of PDEs for passively transported diffusing particles (PT) and actively transported motor-driven particles (AT). We first propose a model where the actively transported particles are taken to be functions of spatial locations \(\mathbf{x} \in \mathbb{R}\) and velocity \(\mathbf{v} \). We then consider two special simplifying cases where particles are transported at (i) constant speeds \(v\) so that the concentration of AT particles is taken to be a function of spatial location and velocity angle \(\theta \in \mathbf{S}^1\) and (ii) discrete velocities (still with constant speed) in directions along a lattice tiling the plane. In the former case the system is equivariant with respect to the so called \textit{shift-twist} action of the Euclidean group \(\mathbf{E}(2)\) acting on functions on \(\mathbb{R}^2 \times \mathbf{S}^1\), while in the latter case it is equivariant with respect to the group \(\mathbf{D}_N <imes \mathbb{R}^2\) where \(\mathbf{D}_N (N = 4\) for square and \(N = 6\) for hexagonal) is the holohedry group of the lattice. In both cases, we use symmetric bifurcation theory to analyze the planforms emerging from a Turing bifurcation, should it occur. In the discrete velocity square lattice case, we are able to prove that a Turing bifurcation does indeed occur as the dimensionless parameter \(\gamma = \alpha D \slash v^2\) crosses some critical value. Here, \(D\) is the diffusion coefficient for the passively diffusing particles and \(\alpha\) is the switching rate between motor states.Some free boundary problem for two-phase inhomogeneous incompressible flows.https://www.zbmath.org/1453.351402021-02-27T13:50:00+00:00"Saito, Hirokazu"https://www.zbmath.org/authors/?q=ai:saito.hirokazu"Shibata, Yoshihiro"https://www.zbmath.org/authors/?q=ai:shibata.yoshihiro"Zhang, Xin"https://www.zbmath.org/authors/?q=ai:zhang.xin.4|zhang.xin.1|zhang.xin.3The motion of two immiscible viscous incompressible liquids with moving interfaces are studied in the paper. Two fluids occupy the time depended domain \(\Omega_t\subset\mathbb{R}^N\) \((N\geq2)\). The first fluid occupies the domain \(\Omega_{1t}\) bounded by two disjoint surfaces \(\Gamma_{1t}\) and \(\Gamma_{t}\), where \(\Gamma_{1t}\) is a free surface and \(\Gamma_{t}\) is the interface between fluids. The second fluid occupies the domain \(\Omega_{2t}\) bounded by the interface \(\Gamma_{t}\) and fixed boundary \(\Gamma_2\). Three typical physical situations are considered.
\begin{itemize}
\item[1)] The droplet \(\Omega_{2t}\) is surrounded by the bonded layer of \(\Omega_{1t}\). In this case \(\Gamma_2=\emptyset\).
\item[2)] The fluid rod \(\Omega_{2t}\) is surrounded by the layer of \(\Omega_{1t}\) in container with solid boundary \(\Gamma_2\). In this case \(\Gamma_{1t}=\emptyset\).
\item[3)] The infinite layer \(\Omega_{1t}\) is located above the infinite layer of \(\Omega_{2t}\). Here \(\Gamma_2\) is a bottom.
\end{itemize}
The motion is governed by the equations.
\[
\begin{array}{l}
\frac{\partial}{\partial t}\left(\rho v\right)+\mbox{Div}\,(\rho v\otimes v)-\mbox{Div}\,\mathbb{T}(v,p)=
\rho f\quad \mbox{in}\ \Omega_t,\vphantom{\begin{array}{c}I\\I\end{array}}\\
\frac{\partial \rho}{\partial t}+\mbox{div}\,(\rho v)=0,\quad \mbox{div}\, v=0
\quad \mbox{in}\ \Omega_t,\vphantom{\begin{array}{c}I\\I\end{array}}\\
\left[ \mathbb{T}(v,p)n_t \right]=[v]=0,\quad V_t=v\cdot n_t\quad \mbox{on}\ \Gamma_{t},
\vphantom{\begin{array}{c}I\\I\end{array}}\\
\mathbb{T}(v_1,p_1)n_{1t}=0,\quad V_{1t}=v_1\cdot n_{1t}\quad \mbox{on}\ \Gamma_{1t},
\vphantom{\begin{array}{c}I\\I\end{array}}\\
v_2=0\quad \mbox{on}\ \Gamma_{2},
\vphantom{\begin{array}{c}I\\I\end{array}}\\
(\rho,v)\Big|_{t=0}=(\rho_0,v_0),
\end{array}
\]
where \(\rho\) is the density, \(v\) is the velocity field, \(p\) is the pressure,
\[
\mathbb{T}(v,p)=\mu(\rho(x,t))\mathbb{D}(v)-p\mathbb{I}
\]
is the stress tensor, \(\mathbb{D}(v)\) is the double deformation tensor, \(f\) is the given vector of external force, \(n_t\) is the outward unit normal to the moving interface \(\Gamma_t\), \(n_{1t}\) is the outward unit normal to the free surface \(\Gamma_{1t}\), \(V_t\) and \(V_{1t}\) stand for the normal velocities of \(\Gamma_t\) and \(\Gamma_{1t}\), respectively. Square brackets denote the jump of the vector across some surface. \((\rho,v,p,\Omega_t)\) are unknowns.
The authors establish the local solvability of the problem within the maximal \(L_p\)--\(L_q\) regularity. For piecewise constant density and viscosity coefficient, they construct the long time solution from the small initial states in the case 1) of the bounded droplet.
Reviewer: Il'ya Sh. Mogilevskii (Tver')Radon measures as solutions of the Cauchy problem for evolution equations.https://www.zbmath.org/1453.351372021-02-27T13:50:00+00:00"Colombeau, Mathilde"https://www.zbmath.org/authors/?q=ai:colombeau.mathildeThe author studies space-periodic solutions to the compressible multi-D Navier-Stokes system. Existence of a weak asymptotic solution is established with the help of a special differential-difference approximation scheme. It is shown that the density and momentum components are represented by Radon measures in the limit as the approximation parameter vanishes while the velocity remains bounded. The author demonstrates that the presented methods can be also applied for more general types of evolution equations.
Reviewer: Evgeniy Panov (Novgorod)Mathematical analysis and numerical methods for multiscale kinetic equations with uncertainties.https://www.zbmath.org/1453.820712021-02-27T13:50:00+00:00"Jin, Shi"https://www.zbmath.org/authors/?q=ai:jin.shi.1|jin.shiMathematical analysis of finite volume preserving scheme for nonlinear Smoluchowski equation.https://www.zbmath.org/1453.652602021-02-27T13:50:00+00:00"Singh, Mehakpreet"https://www.zbmath.org/authors/?q=ai:singh.mehakpreet"Matsoukas, Themis"https://www.zbmath.org/authors/?q=ai:matsoukas.themis"Walker, Gavin"https://www.zbmath.org/authors/?q=ai:walker.gavinSummary: This study presents the convergence analysis of the recently developed finite volume preserving scheme [\textit{L. Forestier-Coste} and \textit{S. Mancini}, SIAM J. Sci. Comput. 34, No. 6, B840--B860 (2012; Zbl 1259.82054)] for approximating a coalescence or Smoluchowski equation. The idea of the finite volume scheme is to preserve the total volume in the system by modifying the coalescence kernel using the notion of overlapping bins (cells). The consistency of the finite volume scheme is examined thoroughly in order to prove second-order convergence on uniform, non-uniform smooth and locally uniform grids independently of the aggregation kernel. The theoretical observations of order of convergence is verified using the experimental order of convergence for analytically tractable kernels.Near-field imaging of inhomogeneities in a stratified ocean waveguide.https://www.zbmath.org/1453.653942021-02-27T13:50:00+00:00"Liu, Keji"https://www.zbmath.org/authors/?q=ai:liu.kejiSummary: In this work, we have derived the general representation of the scattered field in the three-layered ocean waveguide. Moreover, a generalized direct sampling method which does not rely on the equivalent condition of density, and a novel MUSIC method as an alternative approach, have been proposed for the reconstructions of penetrable obstacles in the stratified ocean waveguide. In practice, the methods are capable of recovering the scatterers of different shapes and locations, robust against noise, computationally fairly cheap and easy to carry out. We can consider them as simple and fast algorithms to supply satisfactory initial positions for the application of any existing more refined and precise but computationally more demanding techniques to achieve accurate reconstructions of physical features of objects.Asymptotic analysis of a coupled system of nonlocal equations with oscillatory coefficients.https://www.zbmath.org/1453.350182021-02-27T13:50:00+00:00"Scott, James M."https://www.zbmath.org/authors/?q=ai:scott.james-m"Mengesha, Tadele"https://www.zbmath.org/authors/?q=ai:mengesha.tadeleIsogeometric analysis for surface PDEs with extended loop subdivision.https://www.zbmath.org/1453.654132021-02-27T13:50:00+00:00"Pan, Qing"https://www.zbmath.org/authors/?q=ai:pan.qing"Rabczuk, Timon"https://www.zbmath.org/authors/?q=ai:rabczuk.timon"Xu, Gang"https://www.zbmath.org/authors/?q=ai:xu.gang"Chen, Chong"https://www.zbmath.org/authors/?q=ai:chen.chongSummary: We investigate the isogeometric analysis for surface PDEs based on the extended Loop subdivision approach. The basis functions consisting of quartic box-splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly, and construct the solution space for dependent variables as well, which is consistent with the concept of isogeometric analysis. The subdivision process is equivalent to the \(h\)-refinement of NURBS-based isogeometric analysis. The performance of the proposed method is evaluated by solving various surface PDEs, such as surface Laplace-Beltrami harmonic/biharmonic/triharmonic equations, which are defined on the limit surfaces of extended Loop subdivision for different initial control meshes. Numerical experiments show that the proposed method has desirable performance in terms of the accuracy, convergence and computational cost for solving the above surface PDEs defined on both open and closed surfaces. The proposed approach is proved to be second-order accuracy in the sense of \(L^2\)-norm with theoretical and/or numerical results, which is also outperformed over the standard linear finite element by several numerical comparisons.Adaptive spectral solution method for the Landau and Lenard-Balescu equations.https://www.zbmath.org/1453.653632021-02-27T13:50:00+00:00"Scullard, Christian R."https://www.zbmath.org/authors/?q=ai:scullard.christian-r"Hickok, Abigail"https://www.zbmath.org/authors/?q=ai:hickok.abigail"Sotiris, Justyna O."https://www.zbmath.org/authors/?q=ai:sotiris.justyna-o"Tzolova, Bilyana M."https://www.zbmath.org/authors/?q=ai:tzolova.bilyana-m"Van Heyningen, R. Loek"https://www.zbmath.org/authors/?q=ai:van-heyningen.r-loek"Graziani, Frank R."https://www.zbmath.org/authors/?q=ai:graziani.frank-rSummary: We present an adaptive spectral method for solving the Landau/Fokker-Planck equation for electron-ion systems. The heart of the algorithm is an expansion in Laguerre polynomials, which has several advantages, including automatic conservation of both energy and particles without the need for any special discretization or time-stepping schemes. One drawback of such an expansion is the \(O(N^3)\) memory requirement, where \(N\) is the number of polynomials used. This can impose an inconvenient limit in cases of practical interest, such as when two particle species have widely separated temperatures. The algorithm we describe here addresses this problem by periodically re-projecting the solution onto a judicious choice of new basis functions that are still Laguerre polynomials but have arguments adapted to the current physical conditions. This results in a reduction in the number of polynomials needed, at the expense of increased solution time. Because the equations are solved with little difficulty, this added time is not of much concern compared to the savings in memory. To demonstrate the algorithm, we solve several relaxation problems that could not be computed with the spectral method without re-projection. Another major advantage of this method is that it can be used for collision operators more complicated than that of the Landau equation, and we demonstrate this here by using it to solve the non-degenerate quantum Lenard-Balescu (QLB) equation for a hydrogen plasma. We conclude with some comparisons of temperature relaxation problems solved with the latter equation and the Landau equation with a Coulomb logarithm inspired by the properties of the QLB operator. We find that with this choice of Coulomb logarithm, there is little difference between using the two equations for these particular systems.Solving a Bernoulli type free boundary problem with random diffusion.https://www.zbmath.org/1453.351992021-02-27T13:50:00+00:00"Brügger, Rahel"https://www.zbmath.org/authors/?q=ai:brugger.rahel"Croce, Roberto"https://www.zbmath.org/authors/?q=ai:croce.roberto"Harbrecht, Helmut"https://www.zbmath.org/authors/?q=ai:harbrecht.helmutThe paper deals with a Bernoulli's free boundary problem under modelling uncertainties. The forward problem is governed by an overdetermined diffusive equation. The basic idea consists in rewriting it as a shape optimization problem. In particular, a shape functional measuring the expected energy is minimized with respect to the shape of the domain. The associated shape gradient is derived by using the standard velocity method. The resulting sensitivity is given by quantities concentrated on the moving boundary depending on the solution to the state equation, which fits the so-called Hadamard's Structure Theorem. In addition, random diffusion is considered, which may represent modelling uncertainties, for instance. The resulting high-dimensional integral is approximated by the quasi-Monte Carlo method. A level-set method driven by the obtained shape derivative is proposed. Finally, a nice set of numerical experiments is presented, showing that uncertainties in the diffusive coefficient may lead to significantly different minimizers, as expected.
Reviewer: Antonio André Novotny (Petrópolis)An efficient algorithm for a class of stochastic forward and inverse Maxwell models in \(\mathbb{R}^3\).https://www.zbmath.org/1453.650172021-02-27T13:50:00+00:00"Ganesh, M."https://www.zbmath.org/authors/?q=ai:ganesh.mahadevan"Hawkins, S. C."https://www.zbmath.org/authors/?q=ai:hawkins.stuart-collin"Volkov, D."https://www.zbmath.org/authors/?q=ai:volkov.darkoSummary: We describe an efficient algorithm for reconstruction of the electromagnetic parameters of an unbounded dielectric medium from noisy cross section data induced by a point source in \(\mathbb{R}^3\). The efficiency of our Bayesian inverse algorithm for the parameters is based on developing an offline high order forward stochastic model and also an associated deterministic dielectric media Maxwell solver. Underlying the inverse/offline approach is our high order fully discrete Galerkin algorithm for solving an equivalent surface integral equation reformulation that is stable for all frequencies. The efficient algorithm includes approximating the likelihood distribution in the Bayesian model by a decomposed fast generalized polynomial chaos (gPC) model as a surrogate for the forward model. Offline construction of the gPC model facilitates fast online evaluation of the posterior distribution of the dielectric medium parameters. Parallel computational experiments demonstrate the efficiency of our deterministic, forward stochastic, and inverse dielectric computer models.Long-time asymptotics for homoenergetic solutions of the Boltzmann equation: collision-dominated case.https://www.zbmath.org/1453.351332021-02-27T13:50:00+00:00"James, Richard D."https://www.zbmath.org/authors/?q=ai:james.richard-d"Nota, Alessia"https://www.zbmath.org/authors/?q=ai:nota.alessia"Velázquez, Juan J. L."https://www.zbmath.org/authors/?q=ai:velazquez.juan-j-lThe paper concerns studies on the long-time asymptotics of the homoenergetic solutions for the Boltzmann equation when the collision kernel describes the interactions between non-Maxwellian molecules. For the equation \(\partial_t g-(L(t)\omega)\cdot\partial_\omega g=\mathbb{C}g(\omega)\), when the collision term \(\mathbb{C}g(\omega)\) is much larger than the hyperbolic term \((L(t)\omega)\cdot\partial_\omega g\) (collision-dominated) as \(t\rightarrow\infty\), they formally obtainthat the corresponding distribution of particle velocities for the associated homoenergetic flows can be approximated by a family approximately Maxwellians with a time-dependent temperature by the standard Hilbert expansion. Here the homogeneity of the collision kernel and the particular form of the hyperbolic term are used.
It is also interesting to prove rigorously the existence of solutions of the Boltzmann equation with the asymptotic properties obtained in this paper as in [\textit{R. D. James} et al., Arch. Ration. Mech. Anal. 231, No. 2, 787--843 (2019; Zbl 1430.82012)], where they gave the rigorous proof of existence of self-similar solutions when the collision kernel describes the interactions between Maxwell molecules, and the terms \((L(t)w)\cdot\partial_\omega g\) and \(\mathbb{C}g(\omega)\) have the comparable size as \(t\rightarrow\infty\).
Reviewer: Zhigang Wu (Shanghai)\((2+1)\)-dimensional interface dynamics: mixing time, hydrodynamic limit and anisotropic KPZ growth.https://www.zbmath.org/1453.820462021-02-27T13:50:00+00:00"Toninelli, Fabio"https://www.zbmath.org/authors/?q=ai:toninelli.fabio-lucioGaussian wave packet transform based numerical scheme for the semi-classical Schrödinger equation with random inputs.https://www.zbmath.org/1453.650202021-02-27T13:50:00+00:00"Jin, Shi"https://www.zbmath.org/authors/?q=ai:jin.shi"Liu, Liu"https://www.zbmath.org/authors/?q=ai:liu.liu"Russo, Giovanni"https://www.zbmath.org/authors/?q=ai:russo.giovanni.1|russo.giovanni"Zhou, Zhennan"https://www.zbmath.org/authors/?q=ai:zhou.zhennanSummary: In this work, we study the semi-classical limit of the Schrödinger equation with random inputs, and show that the semi-classical Schrödinger equation produces \(O(\varepsilon)\) oscillations in the random variable space. With the Gaussian wave packet transform, the original Schrödinger equation is mapped to an ordinary differential equation (ODE) system for the wave packet parameters coupled with a partial differential equation (PDE) for the quantity \(w\) in rescaled variables. Further, we show that the \(w\) equation does not produce \(\epsilon\) dependent oscillations, and thus it is more amenable for numerical simulations. We propose multi-level sampling strategy in implementing the Gaussian wave packet transform, where in the most costly part, i.e. simulating the \(w\) equation, it is sufficient to use \(\epsilon\) independent samples. We also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm, and hopefully shed light on possible future directions.Stochastic multiscale flux basis for Stokes-Darcy flows.https://www.zbmath.org/1453.761772021-02-27T13:50:00+00:00"Ambartsumyan, Ilona"https://www.zbmath.org/authors/?q=ai:ambartsumyan.ilona"Khattatov, Eldar"https://www.zbmath.org/authors/?q=ai:khattatov.eldar"Wang, ChangQing"https://www.zbmath.org/authors/?q=ai:wang.changqing"Yotov, Ivan"https://www.zbmath.org/authors/?q=ai:yotov.ivanSummary: Three algorithms are developed for uncertainty quantification in modeling coupled Stokes and Darcy flows. The porous media may consist of multiple regions with different properties. The permeability is modeled as a non-stationary stochastic variable, with its log represented as a sum of local Karhunen-Loève (KL) expansions. The problem is approximated by stochastic collocation on either tensor-product or sparse grids, coupled with a multiscale mortar mixed finite element method for the spatial discretization. A non-overlapping domain decomposition algorithm reduces the global problem to a coarse scale mortar interface problem, which is solved by an iterative solver, for each stochastic realization. In the traditional domain decomposition implementation, each subdomain solves a local Dirichlet or Neumann problem in every interface iteration. To reduce this cost, two additional algorithms based on deterministic or stochastic multiscale flux basis are introduced. The basis consists of the local flux (or velocity trace) responses from each mortar degree of freedom. It is computed for each subdomain independently before the interface iteration begins. The use of the multiscale flux basis avoids the need for subdomain solves on each iteration. The deterministic basis is computed at each stochastic collocation and used only at this realization. The stochastic basis is formed by further looping over all local realizations of a subdomain's KL region before the stochastic collocation begins. It is reused over multiple realizations. Numerical tests are presented to illustrate the performance of the three algorithms, with the stochastic multiscale flux basis showing significant savings in computational cost compared to the other two algorithms.Uncollided flux techniques for arbitrary finite element meshes.https://www.zbmath.org/1453.654072021-02-27T13:50:00+00:00"Hanuš, Milan"https://www.zbmath.org/authors/?q=ai:hanus.milan"Harbour, Logan H."https://www.zbmath.org/authors/?q=ai:harbour.logan-h"Ragusa, Jean C."https://www.zbmath.org/authors/?q=ai:ragusa.jean-c"Adams, Michael P."https://www.zbmath.org/authors/?q=ai:adams.michael-p"Adams, Marvin L."https://www.zbmath.org/authors/?q=ai:adams.marvin-lSummary: The uncollided angular flux can be difficult to compute accurately in discrete-ordinate radiation transport codes, especially in weakly-scattering configurations with localized sources. It has long been recognized that an analytical or semi-analytical treatment of the uncollided flux, coupled with a discrete-ordinate solution for the collided flux, can yield dramatic improvements in solution accuracy and computational efficiency. In this paper, we present such an algorithm for the semi-analytical calculation of the uncollided flux. This algorithm is unique in several aspects: (1) it applies to arbitrary polyhedral cells (and can be thus coupled with collided flux solvers that support arbitrary polyhedral meshes without the need for explicit tetrahedral re-meshing), (2) it provides accurate uncollided solutions near sources, (3) it is devised with parallel implementation in mind, and (4) it minimizes the total number of traced rays and maintains a reasonable ray density on each local subdomain. This paper provides a complete derivation of the algorithm and demonstrates its important features on a set of simple examples and a standard transport benchmark. Assessment of its parallel performance will be the subject of a subsequent paper.On simultaneous recovery of sources/obstacles and surrounding mediums by boundary measurements.https://www.zbmath.org/1453.351622021-02-27T13:50:00+00:00"Fang, Xiaoping"https://www.zbmath.org/authors/?q=ai:fang.xiaoping"Deng, Youjun"https://www.zbmath.org/authors/?q=ai:deng.youjun"Tsui, Wing-Yan"https://www.zbmath.org/authors/?q=ai:tsui.wing-yan"Zhang, Zaiyun"https://www.zbmath.org/authors/?q=ai:zhang.zaiyunThe authors study the inverse problem of recovering a source embedded within an inhomogeneous medium by knowledge of the field generated by the source outside the medium. They provide a survey of recent results on these type of inverse problems, as well as some of the techniques involved, such as the method of layer potentials and the Lippmann-Schwinger equation. Applications to thermoacoustic and photoacoustic tomography are also discussed. Emphasis is placed on the fact that low frequency measurements are key in obtaining uniqueness.
Reviewer: Eric Stachura (Marietta)Fast and accurate algorithms for the computation of spherically symmetric nonlocal diffusion operators on lattices.https://www.zbmath.org/1453.654432021-02-27T13:50:00+00:00"Li, Yu"https://www.zbmath.org/authors/?q=ai:li.yu"Slevinsky, Richard Mikaël"https://www.zbmath.org/authors/?q=ai:slevinsky.richard-mikaelSummary: We present a unified treatment of the Fourier spectra of spherically symmetric nonlocal diffusion operators. We develop numerical and analytical results for the class of kernels with weak algebraic singularity as the distance between source and target tends to 0. Rapid algorithms are derived for their Fourier spectra with the computation of each eigenvalue independent of all others. The algorithms are trivially parallelizable, capable of leveraging more powerful compute environments, and the accuracy of the eigenvalues is individually controllable. The algorithms include a Maclaurin series and a full divergent asymptotic series valid for any \(d\) spatial dimensions. Using Drummond's sequence transformation, we prove linear complexity recurrence relations for degree-graded sequences of numerators and denominators in the rational approximations to the divergent asymptotic series. These relations are important to ensure that the algorithms are efficient, and also increase the numerical stability compared with the conventional algorithm with quadratic complexity.Imposing jump conditions on nonconforming interfaces for the correction function method: a least squares approach.https://www.zbmath.org/1453.653842021-02-27T13:50:00+00:00"Marques, Alexandre Noll"https://www.zbmath.org/authors/?q=ai:marques.alexandre-noll"Nave, Jean-Christophe"https://www.zbmath.org/authors/?q=ai:nave.jean-christophe"Rosales, Rodolfo Ruben"https://www.zbmath.org/authors/?q=ai:rosales.rodolfo-rubenSummary: We introduce a technique that simplifies the problem of imposing jump conditions on interfaces that are not aligned with a computational grid in the context of the \textit{Correction Function Method} (CFM). The CFM offers a general framework to solve Poisson's equation in the presence of discontinuities to high order of accuracy, while using a compact discretization stencil. A key concept behind the CFM is enforcing the jump conditions in a least squares sense. This concept requires computing integrals over sections of the interface, which is a challenge in 3-D when only an implicit representation of the interface is available (e.g., the zero contour of a level set function). The technique introduced here is based on a new formulation of the least squares procedure that relies only on integrals over domains that are amenable to simple quadrature after local coordinate transformations. We incorporate this technique into a fourth order accurate implementation of the CFM, and show examples of solutions to Poisson's equation with imposed jump conditions computed in 2-D and 3-D.Algorithm for some anomalously diffusive hyperbolic systems in molecular dynamics: theoretical analysis and pattern formation.https://www.zbmath.org/1453.652742021-02-27T13:50:00+00:00"Macías-Díaz, J. E."https://www.zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardo"Hendy, A. S."https://www.zbmath.org/authors/?q=ai:hendy.ahmed-sSummary: Departing from a two-dimensional hyperbolic system that describes the interaction between some activator and inhibitor substances in chemical reactions, we investigate a general form of that model using a finite-difference approach. The model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms. The presence of two-dimensional diffusive terms consisting of fractional operators of the Riesz type is considered here, using spatial differentiation orders in the set \((0, 1) \cup(1, 2]\). We impose initial conditions on a closed and bounded rectangle, and a finite-difference methodology based on the use of fractional centered differences is proposed. Among the most important results of this work, we prove the existence and the uniqueness of the solutions of the numerical method, and establish analytically the second-order consistency of our scheme. Moreover, the discrete energy method is employed to prove the stability and the quadratic convergence of the technique. Some numerical simulations obtained through our method show the presence of Turing patterns and wave instabilities, in agreement with some reports found in the literature on superdiffusive hyperbolic activator-inhibitor systems. We show numerically that the presence of Turing patterns is independent of the size of the spatial domain. As a new application, we show that Turing patterns are also present in subdiffusive scenarios.Minimization based formulations of inverse problems and their regularization.https://www.zbmath.org/1453.651292021-02-27T13:50:00+00:00"Kaltenbacher, Barbara"https://www.zbmath.org/authors/?q=ai:kaltenbacher.barbaraA tool for symmetry breaking and multiplicity in some nonlocal problems.https://www.zbmath.org/1453.351822021-02-27T13:50:00+00:00"Musina, Roberta"https://www.zbmath.org/authors/?q=ai:musina.roberta"Nazarov, Alexander I."https://www.zbmath.org/authors/?q=ai:nazarov.alexander-iSummary: We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function \(u\) on \(\mathbb{R}^n\) and of its perturbation \(u \varphi_\mu\), where \(\varphi_\mu\) is a suitably chosen eigenfunction of the Laplace-Beltrami operator on the sphere \(\mathbb{S}^{n-1}\), thus providing a technical but rather powerful tool to detect symmetry breaking and multiplicity phenomena in variational equations driven by the fractional Laplace operator. A concrete application to a problem related to the fractional Caffarelli-Kohn-Nirenberg inequality is given.Graph Merriman-Bence-Osher as a semidiscrete implicit Euler scheme for graph Allen-Cahn flow.https://www.zbmath.org/1453.652092021-02-27T13:50:00+00:00"Budd, Jeremy"https://www.zbmath.org/authors/?q=ai:budd.jeremy"Van Gennip, Yves"https://www.zbmath.org/authors/?q=ai:van-gennip.yvesThis article discusses a link between formulations of the Merriman-Bence-Osher scheme for diffusion generated motion and the Allen-Cahn gradient flow of the Ginzburg-Landau functional on finite graphs. Further results linking these flows with a graph formulation of mean curvature flow are discussed.
Reviewer: Marius Ghergu (Dublin)Legendre wavelets direct method for the numerical solution of time-fractional order telegraph equations.https://www.zbmath.org/1453.653692021-02-27T13:50:00+00:00"Xu, Xiaoyong"https://www.zbmath.org/authors/?q=ai:xu.xiaoyong"Xu, Da"https://www.zbmath.org/authors/?q=ai:xu.daSummary: In this paper, a Legendre wavelet collocation method for solving a class of time-fractional order telegraph equation defined by Caputo sense is discussed. Fractional integral formula of a single Legendre wavelet in the Riemann-Liouville sense is derived by means of shifted Legendre polynomials. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence analysis and error analysis of the proposed method are investigated. Several examples are presented to show the applicability and accuracy of the proposed method.On stochastic Galerkin approximation of the nonlinear Boltzmann equation with uncertainty in the fluid regime.https://www.zbmath.org/1453.650192021-02-27T13:50:00+00:00"Hu, Jingwei"https://www.zbmath.org/authors/?q=ai:hu.jingwei"Jin, Shi"https://www.zbmath.org/authors/?q=ai:jin.shi"Shu, Ruiwen"https://www.zbmath.org/authors/?q=ai:shu.ruiwenSummary: The Boltzmann equation may contain uncertainties in initial/boundary data or collision kernel. To study the impact of these uncertainties, a stochastic Galerkin (sG) method was proposed in [the first two authors, ibid. 315, 150--168 (2016; Zbl 1349.82088)] and studied in the kinetic regime. When the system is close to the fluid regime (the Knudsen number is small), the method would become prohibitively expensive due to the stiff collision term. In this work, we develop efficient sG methods for the Boltzmann equation that work for a wide range of Knudsen numbers, and investigate, in particular, their behavior in the fluid regime.A QSC method for fractional subdiffusion equations with fractional boundary conditions and its application in parameters identification.https://www.zbmath.org/1453.653592021-02-27T13:50:00+00:00"Liu, Jun"https://www.zbmath.org/authors/?q=ai:liu.jun.1|liu.jun.3|liu.jun.2|liu.jun.5|liu.jun.4|liu.jun"Fu, Hongfei"https://www.zbmath.org/authors/?q=ai:fu.hongfei"Zhang, Jiansong"https://www.zbmath.org/authors/?q=ai:zhang.jiansongSummary: A quadratic spline collocation (QSC) method combined with \(L 1\) time discretization, named QSC-\( L 1\), is proposed to solve fractional subdiffusion equations with artificial boundary conditions. A novel norm-based stability and convergence analysis is carefully discussed, which shows that the QSC-\( L 1\) method is unconditionally stable in a discrete space-time norm, and has a convergence order \(\mathcal{O} (\tau^{2-\alpha} + h^2)\), where \(\tau\) and \(h\) are the temporal and spatial step sizes, respectively. Then, based on fast evaluation of the Caputo fractional derivative (see, [\textit{S. Jiang} et al., ``Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations'', Commun. Comput. Phys. 21, No. 3, 650--678 (2017; \url{doi:10.4208/cicp.oa-2016-0136})]), a fast version of QSC-\(L1\) which is called QSC-F\(L1\) is proposed to improve the computational efficiency. Two numerical examples are provided to support the theoretical results. Furthermore, an inverse problem is considered, in which some parameters of the fractional subdiffusion equations need to be identified. A Levenberg-Marquardt (L-M) method equipped with the QSC-F\(L1\) method is developed for solving the inverse problem. Numerical tests show the effectiveness of the method even for the case that the observation data is contaminated by some levels of random noise.High order positivity-preserving discontinuous Galerkin schemes for radiative transfer equations on triangular meshes.https://www.zbmath.org/1453.653482021-02-27T13:50:00+00:00"Zhang, Min"https://www.zbmath.org/authors/?q=ai:zhang.min.2|zhang.min.7|zhang.min.4|zhang.min|zhang.min.5|zhang.min.6|zhang.min.3|zhang.min.1"Cheng, Juan"https://www.zbmath.org/authors/?q=ai:cheng.juan"Qiu, Jianxian"https://www.zbmath.org/authors/?q=ai:qiu.jianxianSummary: It is an important and challenging issue for the numerical solution of radiative transfer equations to maintain both high order accuracy and positivity. For the two-dimensional radiative transfer equations, \textit{D. Ling} et al. give a counterexample [J. Sci. Comput. 77, No. 3, 1801--1831 (2018; Zbl 1407.65196)] showing that unmodulated discontinuous Galerkin (DG) solver based either on the \(P^k\) or \(Q^k\) polynomial spaces could generate negative cell averages even if the inflow boundary value and the source term are both positive (and, for time dependent problems, also a nonnegative initial condition). Therefore the positivity-preserving frameworks in [\textit{X. Zhang} and \textit{C.-W. Shu}, J. Comput. Phys. 229, No. 23, 8918--8934 (2010; Zbl 1282.76128)] and [\textit{X. Zhang} et al., J. Sci. Comput. 50, No. 1, 29--62 (2012; Zbl 1247.65131)] which are based on the value of cell averages being positive cannot be directly used to obtain a high order conservative positivity-preserving DG scheme for the radiative transfer equations neither on rectangular meshes nor on triangular meshes. In [\textit{D. Yuan} et al., SIAM J. Sci. Comput. 38, No. 5, A2987--A3019 (2016; Zbl 1351.65105)], when the cell average of DG schemes is negative, a rotational positivity-preserving limiter is constructed which could keep high order accuracy and positivity in the one-dimensional radiative transfer equations with \(P^k\) polynomials and could be straightforwardly extended to two-dimensional radiative transfer equations on rectangular meshes with \(Q^k\) polynomials (tensor product polynomials). This paper presents an extension of the idea of the above mentioned one-dimensional rotational positivity-preserving limiter algorithm to two-dimensional high order positivity-preserving DG schemes for solving steady and unsteady radiative transfer equations on triangular meshes with \(P^k\) polynomials. The extension of this method is conceptually plausible but highly nontrivial. We first focus on finding a special quadrature rule on a triangle which should satisfy some conditions. The most important one is that the quadrature points can be arranged on several line segments, on which we can use the one-dimensional rotational positivity-preserving limiter. Since the number of the quadrature points is larger than the number of basis functions of \(P^k\) polynomial space, we determine a \(k\)-th polynomial by a \(L_2\)-norm Least Square subject to its cell average being equal to the weighted average of the values on the quadrature points after using the rotational positivity-preserving limiter. Since the weights used here are the quadrature weights which are positive, then the cell average of the modified polynomial is nonnegative. And the final modified polynomial can be obtained by using the two-dimensional scaling positivity-preserving limiter on the triangular element. We theoretically prove that our rotational positivity-preserving limiter on triangular meshes could keep both high order accuracy and positivity. It is relatively simple to implement, and also does not affect convergence to weak solutions. The numerical results validate the high order accuracy and the positivity-preserving properties of our schemes. The advantage of the triangular meshes on handling complex domain is also presented in our numerical examples.On the \((N+1)\)-dimensional local fractional reduced differential transform method and its applications.https://www.zbmath.org/1453.351802021-02-27T13:50:00+00:00"Liu, Jian-gen"https://www.zbmath.org/authors/?q=ai:liu.jiangen"Yang, Xiao-jun"https://www.zbmath.org/authors/?q=ai:yang.xiao-jun"Feng, Yi-ying"https://www.zbmath.org/authors/?q=ai:feng.yiying"Cui, Ping"https://www.zbmath.org/authors/?q=ai:cui.pingSummary: In this paper, we generalize the \((N+1)\)-dimensional local fractional reduced differential transform method (LFRDTM) within the local fractional derivative sense. First, we show some new properties, lemmas, theorems and corollaries for the \((N+1)\)-dimensional LFRDTM. Second, these new properties, lemmas and theorems can be proved immediately after. Thirdly, we used two examples to state that this approach is efficient and simple to find numerical solutions to higher dimensional local fractional partial differential equations. Finally, we can be seen that this work can be looked as an extension of the prior work.A 2D kernel determination problem in a visco-elastic porous medium with a weakly horizontally inhomogeneity.https://www.zbmath.org/1453.351882021-02-27T13:50:00+00:00"Durdiev, Durdimurod Kalandarovich"https://www.zbmath.org/authors/?q=ai:durdiev.durdimurod-kalandarovich"Rahmonov, Askar Ahmadovich"https://www.zbmath.org/authors/?q=ai:rahmonov.askar-ahmadovichSummary: We consider a system of hyperbolic integro-differential equations of SH waves in a visco-elastic porous medium. In this work, it is assumed that the visco-elastic porous medium has weakly horizontally inhomogeneity. The direct problem is the initial-boundary problem: the initial data is equal to zero, and the Neumann-type boundary condition is specified at the half-plane boundary and is an impulse function. As additional information, the oscillation mode of the half-plane line is given. It is assumed that the unknown kernel has the form \(K(x,t)=K_0(t)+ \epsilon xK_1(t)+\dots\), where \(\epsilon\) is a small parameter. In this work, we construct a method for finding \(K_0,K_1\) up to a correction of the order of \(O(\epsilon^2)\).On the theory of integral manifolds for some delayed partial differential equations with nondense domain.https://www.zbmath.org/1453.351762021-02-27T13:50:00+00:00"Jendoubi, C."https://www.zbmath.org/authors/?q=ai:jendoubi.chirazSummary: Integral manifolds are very useful in studying the dynamics of nonlinear evolution equations. We consider a nondensely defined partial differential equation
\[\frac{du}{dt}=\left(A+B(t)\right)u(t)+f\left(t,{u}_t\right),\quad t\in \mathbb{R},\tag{1}\]
where \(( A,D (A))\) satisfies the Hille-Yosida condition, \((B(t))_{t \in R}\) is a family of operators in \(L(D(A) ,X )\) satisfying certain measurability and boundedness conditions, and the nonlinear forcing term \(f\) satisfies the inequality \(\Vert f(t, \varphi ) - f(t, \psi ) \Vert \leq \phi (t) \Vert \varphi - \psi \Vert c\), where \(\phi\) belongs to admissible spaces and \(\varphi,\psi \in C := C([- r, 0], X)\). We first present an exponential convergence result between the stable manifold and every mild solution of (1). Then we prove the existence of center-unstable manifolds for these solutions. Our main methods are invoked by the extrapolation theory and the Lyapunov-Perron method based on the properties of admissible functions.Fractional traveling wave solutions of the \((2+1)\)-dimensional fractional complex Ginzburg-Landau equation via two methods.https://www.zbmath.org/1453.350412021-02-27T13:50:00+00:00"Lu, Peng-Hong"https://www.zbmath.org/authors/?q=ai:lu.peng-hong"Wang, Ben-Hai"https://www.zbmath.org/authors/?q=ai:wang.ben-hai"Dai, Chao-Qing"https://www.zbmath.org/authors/?q=ai:dai.chaoqingSummary: A \((2+1)\)-dimensional fractional complex Ginzburg-Landau equation is solved via fractional Riccati method and fractional bifunction method, and exact traveling wave solutions including soliton solution and combined soliton solutions are constructed based on Mittag-Leffler function. A series of fractional orders is used to demonstrate the graphical representation and physical interpretation of the resulting solutions. The role of the fractional order is revealed.Identification of the source function for a seawater intrusion problem in unconfined aquifer.https://www.zbmath.org/1453.652922021-02-27T13:50:00+00:00"Slimani, S."https://www.zbmath.org/authors/?q=ai:slimani.said|slimani.souad|slimani.safia"Medarhri, I."https://www.zbmath.org/authors/?q=ai:medarhri.ibtissam"Najib, K."https://www.zbmath.org/authors/?q=ai:najib.khalid"Zine, A."https://www.zbmath.org/authors/?q=ai:zine.abdelmalekSummary: In this work, we study the inverse source problem of a seawater intrusion problem in an unconfined aquifer with sharp-diffuse interfaces. The model associated with the direct problem is nonlinear. We aim to reconstruct the source term following the technique used in [\textit{M. Kulbay} et al., Inverse Probl. Sci. Eng. 25, No. 2, 279--308 (2017; Zbl 1362.80005)]. As this technique is based on variable separation, a fixed-point strategy is adopted to linearize the problem. Numerical convergence is proven using some examples.Dissipativity and stability for semilinear anomalous diffusion equations involving delays.https://www.zbmath.org/1453.350292021-02-27T13:50:00+00:00"Ke, Tran Dinh"https://www.zbmath.org/authors/?q=ai:ke.tran-dinh"Thuy, Lam Tran Phuong"https://www.zbmath.org/authors/?q=ai:thuy.lam-tran-phuongSummary: We analyze the dissipativity and stability of solutions to a class of semilinear anomalous diffusion equations involving delays. The existence of absorbing set, the stability, and weak stability will be shown under suitable assumptions on the nonlinearity. Our analysis is based on new Halanay-type inequality, local estimates, and fixed-point arguments.Double stabilities of pullback random attractors for stochastic delayed \(p\)-Laplacian equations.https://www.zbmath.org/1453.350332021-02-27T13:50:00+00:00"Zhang, Qiangheng"https://www.zbmath.org/authors/?q=ai:zhang.qiangheng"Li, Yangrong"https://www.zbmath.org/authors/?q=ai:li.yangrongSummary: We provide a method to study the double stabilities of a pullback random attractor (PRA) generated from a stochastic partial differential equation (PDE) with delays, such a PRA is actually a family of compact random sets \(A_\varrho (t,\cdot)\), where \(t\) is the current time and \(\varrho\) is the memory time. We study its longtime stability, which means the attractor semiconverges to a compact set as the current time tends to minus infinity, and also its zero-memory stability, which means the delayed attractor semiconverges to the nondelayed attractor as the memory time tends to zero. The stochastic nonautonomous \(p\)-Laplacian equation with variable delays on an unbounded domain will be applied to illustrate the method and some suitable assumptions about the nonlinearity and time-dependent delayed forces can ensure existence, backward compactness, and double stabilities of a PRA.Identification of physical parameters in pressing of rapeseeds from the oil flux measurements.https://www.zbmath.org/1453.351872021-02-27T13:50:00+00:00"Bacha, Rekia Meriem Ahmed"https://www.zbmath.org/authors/?q=ai:bacha.rekia-meriem-ahmed"El Badia, Abdellatif"https://www.zbmath.org/authors/?q=ai:el-badia.abdellatif"El Hajj, Ahmad"https://www.zbmath.org/authors/?q=ai:el-hajj.ahmad"Mottelet, Stephane"https://www.zbmath.org/authors/?q=ai:mottelet.stephaneSummary: In this paper, we are interested in the study of an inverse problem that occurs during the pressing of rapeseeds, where some physical parameters influence on rapeseed oil extraction yield. Our objective is to identify the consolidation coefficient of the press cake, the inverse characteristic time of consolidation in press cake, and inside the rapeseed in order to increase the rapeseed oil extraction yield. Three questions will be addressed: the identifiability, the identification, and the stability of the inverse problem. Finally, we provide numerical results to confirm the theoretical results.Semi-discrete finite difference schemes for the nonlinear Cauchy problems of the normal form.https://www.zbmath.org/1453.651172021-02-27T13:50:00+00:00"Higashimori, Nobuyuki"https://www.zbmath.org/authors/?q=ai:higashimori.nobuyuki"Fujiwara, Hiroshi"https://www.zbmath.org/authors/?q=ai:fujiwara.hiroshiSummary: We consider the Cauchy problems of nonlinear partial differential equations of the normal form in the class of analytic functions. We apply semi-discrete finite difference approximation which discretizes the problems only with respect to the time variable, and we give a proof for its convergence. The result implies that there are cases of convergence of finite difference schemes applied to ill-posed Cauchy problems.A note on a posteriori error bounds for numerical solutions of elliptic equations with a piecewise constant reaction coefficient having large jumps.https://www.zbmath.org/1453.654112021-02-27T13:50:00+00:00"Korneev, V. G."https://www.zbmath.org/authors/?q=ai:korneev.vadim-glebovichSummary: We have derived guaranteed, robust, and fully computable a posteriori error bounds for approximate solutions of the equation \(\Delta \Delta u + \Bbbk^2 u = f\), where the coefficient \(\Bbbk \geqslant 0\) is a constant in each subdomain (finite element) and chaotically varies between subdomains in a sufficiently wide range. For finite element solutions, these bounds are robust with respect to \(\Bbbk \in [0,ch^{-2}]\), \(c = \text{const}\), and possess some other good features. The coefficients in front of two typical norms on their right-hand sides are only insignificantly worse than those obtained earlier for \(\Bbbk \equiv\text{const}\). The bounds can be calculated without resorting to the equilibration procedures, and they are sharp for at least low-order methods. The derivation technique used in this paper is similar to the one used in our preceding papers (2017--2019) for obtaining a posteriori error bounds that are not improvable in the order of accuracy.A transform based local RBF method for 2D linear PDE with Caputo-Fabrizio derivative.https://www.zbmath.org/1453.351782021-02-27T13:50:00+00:00"Kamran"https://www.zbmath.org/authors/?q=ai:kamran.muhammad-ahmad|kamran.farrukh|kamran.niloofar-n|kamran.fakhar|kamran.tayyab|kamran.niky|kamran.muhammad-sarwar|kamran.r|kamran.mohsin|kamran.kazem"Ali, Amjad"https://www.zbmath.org/authors/?q=ai:ali.amjad.1|ali.amjad"Gómez-Aguilar, José Francisco"https://www.zbmath.org/authors/?q=ai:gomez-aguilar.jose-franciscoSummary: The present work aims to approximate the solution of linear time fractional PDE with Caputo Fabrizio derivative. For the said purpose Laplace transform with local radial basis functions is used. The Laplace transform is applied to obtain the corresponding time independent equation in Laplace space and then the local RBFs are employed for spatial discretization. The solution is then represented as a contour integral in the complex space, which is approximated by trapezoidal rule with high accuracy. The application of Laplace transform avoids the time stepping procedure which commonly encounters the time instability issues. The convergence of the method is discussed also we have derived the bounds for the stability constant of the differentiation matrix of our proposed numerical scheme. The efficiency of the method is demonstrated with the help of numerical examples. For our numerical experiments we have selected three different domains, in the first test case the square domain is selected, for the second test the circular domain is considered, while for third case the L-shape domain is selected.Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal.https://www.zbmath.org/1453.350252021-02-27T13:50:00+00:00"Eigentler, Lukas"https://www.zbmath.org/authors/?q=ai:eigentler.lukas"Sherratt, Jonathan A."https://www.zbmath.org/authors/?q=ai:sherratt.jonathan-aAuthors' abstract: Vegetation patterns are a characteristic feature of semi-arid regions. On hillsides these patterns occur as stripes running parallel to the contours. The Klausmeier model, a coupled reaction-advection-diffusion system, is a deliberately simple model describing the phenomenon. In this paper, we replace the diffusion term describing plant dispersal by a more realistic nonlocal convolution integral to account for the possibility of long-range dispersal of seeds. Our analysis focuses on the rainfall level at which there is a transition between uniform vegetation and pattern formation. We obtain results, valid
to leading order in the large parameter comparing the rate of water flow downhill to the rate of plant dispersal, for a negative exponential dispersal kernel. Our results indicate that both a wider dispersal of seeds and an increase in dispersal rate inhibit the formation of patterns. Assuming an evolutionary trade-off between these two quantities, mathematically motivated by the limiting behaviour of the convolution term, allows us to make comparisons to existing results for the original reaction-advection-diffusion system. These comparisons show that the nonlocal model always predicts a larger parameter region supporting pattern formation. We then numerically extend the results to other dispersal kernels, showing that the tendency to form patterns depends on the type of decay of the kernel.
Reviewer: Vladimir V. Kisil (Leeds)A correction function method for Poisson problems with interface jump conditions.https://www.zbmath.org/1453.350542021-02-27T13:50:00+00:00"Marques, Alexandre Noll"https://www.zbmath.org/authors/?q=ai:marques.alexandre-noll"Nave, Jean-Christophe"https://www.zbmath.org/authors/?q=ai:nave.jean-christophe"Rosales, Rodolfo Ruben"https://www.zbmath.org/authors/?q=ai:rosales.rodolfo-rubenSummary: In this paper we present a method to treat interface jump conditions for constant coefficients Poisson problems that allows the use of standard ''black box'' solvers, without compromising accuracy. The basic idea of the new approach is similar to the Ghost Fluid Method (GFM). The GFM relies on corrections applied on nodes located across the interface for discretization stencils that straddle the interface. If the corrections are solution-independent, they can be moved to the right-hand-side (RHS) of the equations, producing a problem with the same linear system as if there were no jumps, only with a different RHS. However, achieving high accuracy is very hard (if not impossible) with the ''standard'' approaches used to compute the GFM correction terms.
In this paper we generalize the GFM correction terms to a correction function, defined on a band around the interface. This function is then shown to be characterized as the solution to a PDE, with appropriate boundary conditions. This PDE can, in principle, be solved to any desired order of accuracy. As an example, we apply this new method to devise a 4th order accurate scheme for the constant coefficients Poisson equation with discontinuities in 2D. This scheme is based on (i) the standard 9-point stencil discretization of the Poisson equation, (ii) a representation of the correction function in terms of bicubics, and (iii) a solution of the correction function PDE by a least squares minimization. Several applications of the method are presented to illustrate its robustness dealing with a variety of interface geometries, its capability to capture sharp discontinuities, and its high convergence rate.Müntz pseudo-spectral method: theory and numerical experiments.https://www.zbmath.org/1453.653582021-02-27T13:50:00+00:00"Khosravian-Arab, Hassan"https://www.zbmath.org/authors/?q=ai:khosravian-arab.hassan"Eslahchi, M. R."https://www.zbmath.org/authors/?q=ai:eslahchi.mohammad-rezaSummary: This paper presents two new non-classical Lagrange basis functions which are based on the new Jacobi-Müntz functions presented by the authors recently. These basis functions are, in fact, generalized forms of the newly generated Jacobi-based functions. With respect to these non-classical Lagrange basis functions, two non-classical interpolants are introduced and their error bounds are proved in detail. The pseudo-spectral differentiation (and integration) matrices have been extracted in two different manners. Some numerical experiments are provided to show the efficiency and capability of these newly generated non-classical Lagrange basis functions.Quantum Hamilton equations from stochastic optimal control theory.https://www.zbmath.org/1453.490142021-02-27T13:50:00+00:00"Köppe, Jeanette"https://www.zbmath.org/authors/?q=ai:koppe.jeanette"Patzold, Markus"https://www.zbmath.org/authors/?q=ai:patzold.markus"Beyer, Michael"https://www.zbmath.org/authors/?q=ai:beyer.michael"Grecksch, Wilfried"https://www.zbmath.org/authors/?q=ai:grecksch.wilfried"Paul, Wolfgang"https://www.zbmath.org/authors/?q=ai:paul.wolfgang-j|paul.wolfgangStochastic Schrödinger equations.https://www.zbmath.org/1453.490092021-02-27T13:50:00+00:00"Grecksch, Wilfried"https://www.zbmath.org/authors/?q=ai:grecksch.wilfried"Lisei, Hannelore"https://www.zbmath.org/authors/?q=ai:lisei.hanneloreNumerical approximation of the Frobenius-Perron operator using the finite volume method.https://www.zbmath.org/1453.652592021-02-27T13:50:00+00:00"Norton, Richard A."https://www.zbmath.org/authors/?q=ai:norton.richard-a"Fox, Colin"https://www.zbmath.org/authors/?q=ai:fox.colin-d"Morrison, Malcolm E."https://www.zbmath.org/authors/?q=ai:morrison.malcolm-eHigh-order quasi-compact difference schemes for fractional diffusion equations.https://www.zbmath.org/1453.652382021-02-27T13:50:00+00:00"Yu, Yanyan"https://www.zbmath.org/authors/?q=ai:yu.yanyan"Deng, Weihua"https://www.zbmath.org/authors/?q=ai:deng.weihua"Wu, Yujiang"https://www.zbmath.org/authors/?q=ai:wu.yujiangSummary: The continuous time random walk (CTRW) underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution, its corresponding Fokker-Planck equation has a space fractional derivative, which characterizes Lévy flights. Sometimes the infinite variance of Lévy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more `physical' and the tempered space fractional diffusion equation appears. This paper provides the basic strategy of deriving the high-order quasi-compact discretizations for the space fractional derivative and the tempered space fractional derivative. The fourth-order quasi-compact discretization for the space fractional derivative is applied to solve a space fractional diffusion equation, and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart, the fourth-order quasi-compact scheme, and the convergence orders are verified numerically.A posteriori error estimates for fully discrete finite element method for generalized diffusion equation with delay.https://www.zbmath.org/1453.653462021-02-27T13:50:00+00:00"Wang, Wansheng"https://www.zbmath.org/authors/?q=ai:wang.wansheng"Yi, Lijun"https://www.zbmath.org/authors/?q=ai:yi.lijun.1|yi.lijun"Xiao, Aiguo"https://www.zbmath.org/authors/?q=ai:xiao.aiguoSummary: In this paper, we derive several a posteriori error estimators for generalized diffusion equation with delay in a convex polygonal domain. The Crank-Nicolson method for time discretization is used and a continuous, piecewise linear finite element space is employed for the space discretization. The a posteriori error estimators corresponding to space discretization are derived by using the interpolation estimates. Two different continuous, piecewise quadratic reconstructions are used to obtain the error due to the time discretization. To estimate the error in the approximation of the delay term, linear approximations of the delay term are used in a crucial way. As a consequence, a posteriori upper and lower error bounds for fully discrete approximation are derived for the first time. In particular, long-time a posteriori error estimates are obtained for stable systems. Numerical experiments are presented which confirm our theoretical results.Multiple solutions for a fractional elliptic problem with critical growth.https://www.zbmath.org/1453.351812021-02-27T13:50:00+00:00"Miyagaki, O. H."https://www.zbmath.org/authors/?q=ai:miyagaki.olimpio-hiroshi"Motreanu, D."https://www.zbmath.org/authors/?q=ai:motreanu.dumitru"Pereira, F. R."https://www.zbmath.org/authors/?q=ai:pereira.fabio-rSummary: The paper focuses on a nonlocal Dirichlet problem with asymmetric nonlinearities. The equation is driven by the fractional Laplacian \(( - \Delta )^s\) for \(s \in(0, 1)\) and exhibits a sublinear term containing a parameter \(\lambda \), a linear term interfering with the spectrum of \(( - \Delta )^s\) and a superlinear term with fractional critical growth. The corresponding local problem governed by the standard Laplacian operator was investigated by F. O. de Paiva and A. E. Presoto. It can be recovered by letting \(s \uparrow 1\). The statement given here in the nonlocal setting is also related to extensively studied topics for local elliptic operators as the Brezis-Nirenberg problem and asymmetric nonlinearities. We go beyond the case of the standard Laplacian taking advantage of recent contributions on nonlocal fractional equations. Our main result establishes the existence of at least three nontrivial solutions, with one nonnegative and one nonpositive, provided the parameter \(\lambda > 0\) is sufficiently small. In order to overcome the difficulties in the nonlocal setting we develop new arguments that are substantially different from those used in previous works.Modeling anomalous diffusion. From statistics to mathematics.https://www.zbmath.org/1453.350012021-02-27T13:50:00+00:00"Deng, Weihua"https://www.zbmath.org/authors/?q=ai:deng.weihua"Hou, Ru"https://www.zbmath.org/authors/?q=ai:hou.ru"Wang, Wanli"https://www.zbmath.org/authors/?q=ai:wang.wanli"Xu, Pengbo"https://www.zbmath.org/authors/?q=ai:xu.pengboDiffusion has been widely studied phenomena in natural science after it was first systematically studied by Thomas Grahm. Later a physiologist Adolf Fick proposed his law of diffusion in 1855. It has wide applications ranging from physiology to many industrial processes. It is a basic transport process involved in the evolution of many non equilibrium systems toward equilibrium. In a normal diffusion process, the mean square displacement is proportional to time. They follow Fick's law of diffusion. However there exist several processes where the square of the mean square displacement is proportional to some power of time. These processes are termed as anomalous diffusion. There are several examples in the nature, e.g. protein diffusion within cells, diffusion through porous media, etc. Anomalous diffusion has been a topic of current research from modeling and simulation point of view.
The book by Weihua Deng, Ru Hou, Wanli Wang and Pengbo Xu on Modeling Anomalous Diffusion introduces the anomalous diffusion phenomena from a physical and atomistic way, by considering the random walk of the diffusing particles. Both the microscopic models (stochastic processes) and macroscopic models (partial differential equations) have been built up. The relationships between the two kinds of models are clarified, and based on these models, some statistical observables are analyzed. A specific anomalous and nonergodic diffusion process can be modeled by several different microscopic models, e.g. the continuous time random walk (CTRW) model and subordinated Langevin equation.
In its eight chapters, the book introduces stochastic models, Fokker-Planck equations, Feynman-Kac equations, aging Fokker-Planck and Feynman-Kac equation, fractional reaction diffusion equations, etc., to develop skills of the students in developing anomalous diffusion models and carrying out simulations of the processes of the natural world. Once the microscopic models of the anomalous and nonergodic diffusions are established, it is natural to analyze the models for uncovering the potential mechanism, explaining the observed natural phenomena and extending their applications.
The book is intended for the advanced graduate level students who may find it useful in carrying out modeling and simulation of anomalous diffusion. The book is recommended for the universities and research establishments.
Reviewer: K. N. Shukla (Gurgaon)X-ray transform and boundary rigidity for asymptotically hyperbolic manifolds.https://www.zbmath.org/1453.351902021-02-27T13:50:00+00:00"Graham, C. Robin"https://www.zbmath.org/authors/?q=ai:graham.c-robin"Guillarmou, Colin"https://www.zbmath.org/authors/?q=ai:guillarmou.colin"Stefanov, Plamen"https://www.zbmath.org/authors/?q=ai:stefanov.plamen-d"Uhlmann, Gunther"https://www.zbmath.org/authors/?q=ai:uhlmann.gunther-aSummary: We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.Monitoring Lévy-process crossovers.https://www.zbmath.org/1453.820732021-02-27T13:50:00+00:00"dos Santos, Maike A. F."https://www.zbmath.org/authors/?q=ai:dos-santos.maike-a-f"Nobre, Fernando D."https://www.zbmath.org/authors/?q=ai:nobre.fernando-d"Curado, Evaldo M. F."https://www.zbmath.org/authors/?q=ai:curado.evaldo-m-fSummary: The crossover among two or more types of diffusive processes represents a vibrant theme in nonequilibrium statistical physics. In this work we propose two models to generate crossovers among different Lévy processes: in the first model we change gradually the order of the derivative in the Laplacian term of the diffusion equation, whereas in the second one we consider a combination of fractional-derivative diffusive terms characterized by coefficients that change in time. The proposals are illustrated by considering semi-analytical (i.e., analytical together with numerical) procedures to follow the time-dependent solutions. We find changes between two different regimes and it is shown that, far from the crossover regime, both models yield qualitatively similar results, although these changes may occur in different forms for the two models. The models introduced herein are expected to be useful for describing crossovers among distinct diffusive regimes that occur frequently in complex systems.Non-contact analysis of magnetic fields of biological objects: algorithms for data recording and processing.https://www.zbmath.org/1453.921752021-02-27T13:50:00+00:00"Primin, M. A."https://www.zbmath.org/authors/?q=ai:primin.m-a"Nedayvoda, I. V."https://www.zbmath.org/authors/?q=ai:nedayvoda.i-vSummary: Based on low-temperature SQUID sensors, an ultra-sensitive magnetometric system has been created for the analysis of nanoparticles in biological objects. The main features of the SQUID magnetometric system and information technology during registration and analysis of magnetic signals from organs of laboratory animals are considered. Experimental data on the operation of the magnetometric system and algorithms of data recording and processing in the study of physical models (small animals) with nanoparticles are presented.Spectrum of Dirichlet BDIDE operator.https://www.zbmath.org/1453.653982021-02-27T13:50:00+00:00"Mohamed, N. A."https://www.zbmath.org/authors/?q=ai:mohamed.nurul-akmal"Ibrahim, N. F."https://www.zbmath.org/authors/?q=ai:ibrahim.nur-fadhilah"Mohamed, N. F."https://www.zbmath.org/authors/?q=ai:faried.nashat"Mohamed, N. H."https://www.zbmath.org/authors/?q=ai:mohamed.nurul-hudaSummary: In this paper, we present the distribution of some maximal eigenvalues that are obtained numerically from the discrete Dirichlet Boundary Domain Integro-Dierential Equation (BDIDE) operator. We also discuss the convergence of the discrete Dirichlet BDIDE that corresponds with the obtained absolute value of the largest eigenvalues of the discrete BDIDE operator. There are three test domains that are considered in this paper, i.e., a square, a circle, and a parallelogram. In our numerical test, the eigenvalues disperse as the power of the variable coecient increases. Not only that, we also note that the dispersion of the eigenvalues corresponds with the characteristic size of the test domains. It enables us to predict the convergence of an iterative method. This is an advantage as it enables the use of an iterative method in solving Dirichlet BDIDE as an alternative to the direct methods.Conditional well-posedness for an inverse source problem in the diffusion equation using the variational adjoint method.https://www.zbmath.org/1453.351962021-02-27T13:50:00+00:00"Sun, Chunlong"https://www.zbmath.org/authors/?q=ai:sun.chunlong"Liu, Qian"https://www.zbmath.org/authors/?q=ai:liu.qian"Li, Gongsheng"https://www.zbmath.org/authors/?q=ai:li.gongshengSummary: This article deals with an inverse problem of determining a linear source term in the multidimensional diffusion equation using the variational adjoint method. A variational identity connecting the known data with the unknown is established based on an adjoint problem, and a conditional uniqueness for the inverse source problem is proved by the approximate controllability to the adjoint problem under the condition that the unknowns can keep orders locally. Furthermore, a bilinear form is set forth also based on the variational identity and then a norm for the unknowns is well-defined by which a conditional Lipschitz stability is established.Modified immersed boundary method for flows over randomly rough surfaces.https://www.zbmath.org/1453.760352021-02-27T13:50:00+00:00"Kwon, Chunsong"https://www.zbmath.org/authors/?q=ai:kwon.chunsong"Tartakovsky, Daniel M."https://www.zbmath.org/authors/?q=ai:tartakovsky.daniel-mSummary: Many phenomena, ranging from biology to electronics, involve flow over rough or irregular surfaces. We treat such surfaces as random fields and use an immersed boundary method (IBM) with discrete (random) forcing to solve resulting stochastic flow problems. Our approach relies on the Uhlmann formulation of the fluid-solid interaction force; computational savings stem from the modification of the time advancement scheme that obviates the need to solve the Poisson equation for pressure at each sub-step. We start by testing the proposed algorithm on two classical benchmark problems. The first deals with the Wannier problem of Stokesian flow around a cylinder in the vicinity of a moving plate. The second problem considers steady-state and transient flows over a stationary circular cylinder. Our simulation results show that our algorithm achieves second-order temporal accuracy. It is faster than the original IBM, while yielding consistent estimates of such quantities of interest as the drag and lift coefficients, the length of a recirculation zone in a cylinder's wake, and the Strouhal number. Then we use the proposed IBM algorithm to model flow over cylinders whose surface is either (deterministically) corrugated or (randomly) rough.multiUQ: an intrusive uncertainty quantification tool for gas-liquid multiphase flows.https://www.zbmath.org/1453.761812021-02-27T13:50:00+00:00"Turnquist, Brian"https://www.zbmath.org/authors/?q=ai:turnquist.brian-p"Owkes, Mark"https://www.zbmath.org/authors/?q=ai:owkes.markSummary: Uncertainty quantification (UQ) of fluid flows offers the ability to understand the impact of variation in fluid properties, boundary conditions, and initial conditions on simulation results. In this work, an open-source program called multiUQ is developed which performs UQ using an intrusive approach applied to gas-liquid multiphase flows. Intrusive methods require modifying the governing equations by incorporating stochastic (uncertain) variables. This adds complexity but reduces computational cost compared to non-intrusive methods (e.g. Monte Carlo). To date, much of the work on intrusive UQ has focused on single phase flows. We extend this work by adding capabilities for gas-liquid flows which include a stochastic conservative level set method to capture the location of the phase interface, computing a stochastic curvature, and development of a stochastic surface tension force. Several test cases are presented which illustrate the strength of the framework. Both deterministic and stochastic channel flow cases converge to analytic results and demonstrate the accuracy of the level set transport. Zalesak's disk and the deformation test cases further highlight the abilities of the transport method as well as the robustness of the reinitialization equation, which maintains the level set profile. Deterministic and stochastic oscillating droplet test cases paired with analytic results, solve a true multiphase flow problem, and highlight the abilities of the UQ framework. Finally, results from a stochastic atomizing jet show droplet breakup and merging for cases of uncertainty about the surface tension coefficient and incoming velocity.The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation.https://www.zbmath.org/1453.601222021-02-27T13:50:00+00:00"Lions, Pierre-Louis"https://www.zbmath.org/authors/?q=ai:lions.pierre-louis"Souganidis, Panagiotis E."https://www.zbmath.org/authors/?q=ai:souganidis.panagiotis-eSummary: We study the asymptotics of Allen-Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is governed by an interface moving with curvature dependent normal velocity which is additively perturbed by time white noise. The result is global in time and does not require any regularity assumptions on the evolving front. The main tools are (i) the notion of stochastic (pathwise) solution for nonlinear degenerate parabolic equations with multiplicative rough (stochastic) time dependence, which has been developed by the authors, and (ii) the theory of generalized front propagation put forward by the second author and collaborators to establish the onset of moving fronts in the asymptotics of reaction-diffusion equations.Identification of physical processes via combined data-driven and data-assimilation methods.https://www.zbmath.org/1453.627972021-02-27T13:50:00+00:00"Chang, Haibin"https://www.zbmath.org/authors/?q=ai:chang.haibin"Zhang, Dongxiao"https://www.zbmath.org/authors/?q=ai:zhang.dongxiaoSummary: With the advent of modern data collection and storage technologies, data-driven approaches have been developed for discovering the governing partial differential equations (PDE) of physical problems. However, in the extant works the model parameters in the equations are either assumed to be known or have a linear dependency. Therefore, most of the realistic physical processes cannot be identified with the current data-driven PDE discovery approaches. In this study, an innovative framework is developed that combines data-driven and data-assimilation methods for simultaneously identifying physical processes and inferring model parameters. Spatiotemporal measurement data are first divided into a training data set and a testing data set. Using the training data set, a data-driven method is developed to learn the governing equation of the considered physical problem by identifying the occurred (or dominated) processes and selecting the proper empirical model. Through introducing a prediction error of the learned governing equation for the testing data set, a data-assimilation method is devised to estimate the uncertain model parameters of the selected empirical model. For the contaminant solute transport problem investigated, the results demonstrate that the proposed method can adequately identify the considered physical processes via concurrently discovering the corresponding governing equations and inferring uncertain parameters of nonlinear models, even in the presence of measurement errors. This work helps to broaden the applicable area of the research of data driven discovery of governing equations of physical problems.A discontinuous Galerkin method for the Aw-Rascle traffic flow model on networks.https://www.zbmath.org/1453.653102021-02-27T13:50:00+00:00"Buli, Joshua"https://www.zbmath.org/authors/?q=ai:buli.joshua"Xing, Yulong"https://www.zbmath.org/authors/?q=ai:xing.yulongSummary: Macroscopic models for flows strive to depict the physical world by considering quantities of interest at the aggregate level versus focusing on each discrete particle in the system. Many practical problems of interest such as the blood flow in the circulatory system, irrigation channels, supply chains, and vehicular traffic on freeway systems can all be modeled using hyperbolic conservation laws that track macroscopic quantities through a network. In this paper we consider the latter, specifically the second-order Aw-Rascle (AR) traffic flow model on a network, and propose a discontinuous Galerkin (DG) method for solving the AR system of hyperbolic partial differential equations with appropriate coupling conditions at the junctions. On each road, the standard DG method with Lax-Friedrichs flux is employed, and at the junction, we solve an optimization problem to evaluate the numerical flux of the DG method. As the choice of well-posed coupling conditions for the AR model is not unique, we test different coupling conditions at the junctions. Numerical examples are provided to demonstrate the high-order accuracy, and comparison of results between the first-order Lighthill-Whitham-Richards model and the second-order AR model. The ability of the model to capture the capacity drop phenomenon is also explored.Classification of solutions of an equation related to a conformal log Sobolev inequality.https://www.zbmath.org/1453.350362021-02-27T13:50:00+00:00"Frank, Rupert L."https://www.zbmath.org/authors/?q=ai:frank.rupert-l"König, Tobias"https://www.zbmath.org/authors/?q=ai:konig.tobias"Tang, Hanli"https://www.zbmath.org/authors/?q=ai:tang.hanliSummary: We classify all finite energy solutions of an equation which arises as the Euler-Lagrange equation of a conformally invariant logarithmic Sobolev inequality on the sphere due to Beckner. Our proof uses an extension of the method of moving spheres from \(\mathbb{R}^n\) to \(\mathbb{S}^n\) and a classification result of Li and Zhu. Along the way we prove a small volume maximum principle and a strong maximum principle for the underlying operator which is closely related to the logarithmic Laplacian.Homotopy series solutions to time-space fractional coupled systems.https://www.zbmath.org/1453.653812021-02-27T13:50:00+00:00"Zhang, Jin"https://www.zbmath.org/authors/?q=ai:zhang.jin|zhang.jin.2|zhang.jin.3|zhang.jin.1"Cai, Ming"https://www.zbmath.org/authors/?q=ai:cai.ming"Chen, Bochao"https://www.zbmath.org/authors/?q=ai:chen.bochao"Wei, Hui"https://www.zbmath.org/authors/?q=ai:wei.huiSummary: We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.Difference scheme for partial differential equations of fractional order with a nonlinear differentiation operator.https://www.zbmath.org/1453.652322021-02-27T13:50:00+00:00"Solodushkin, Svyatoslav"https://www.zbmath.org/authors/?q=ai:solodushkin.svyatoslav-i"Gorbova, Tatiana"https://www.zbmath.org/authors/?q=ai:gorbova.tatiana"Pimenov, Vladimir"https://www.zbmath.org/authors/?q=ai:pimenov.vladimir-gSummary: A fractional differential equation in partial derivatives with non-linearity in differentiation operator is considered. We developed a numerical method which has the second order of convergence in time and first order in space and could be considered as a fractional analog of Crank-Nicolson method. Nonlinear high dimensional systems which arise on each time layer are solved iteratively. The method is proven to be consistent and unconditionally stable. Results of numerical examples coincides with theoretical ones.
For the entire collection see [Zbl 1445.34003].Blow-up results for space-time fractional stochastic partial differential equations.https://www.zbmath.org/1453.601132021-02-27T13:50:00+00:00"Asogwa, Sunday A."https://www.zbmath.org/authors/?q=ai:asogwa.sunday-a"Mijena, Jebessa B."https://www.zbmath.org/authors/?q=ai:mijena.jebessa-b"Nane, Erkan"https://www.zbmath.org/authors/?q=ai:nane.erkanSummary: Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type, \[ \partial^{\beta}_t u_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}[b(u)+ \sigma(u)\stackrel{\cdot}{F}(t,x)]\] in \((d + 1)\) dimensions, where \(\nu > 0\), \(\beta \in (0, 1)\), \(\alpha \in (0, 2]\). The operator \(\partial^{\beta }_t\) is the Caputo fractional derivative while \(- (- \Delta)^{ \alpha /2}\) is the generator of an isotropic \(\alpha \)-stable Lévy process and \(I^{1- \beta }\) is the Riesz fractional integral operator. The forcing noise denoted by \(\stackrel{\cdot }{F}(t,x)\) is a Gaussian noise. These equations might be used as a model for materials with random thermal memory. We derive non-existence (blow-up) of global random field solutions under some additional conditions, most notably on \(b, \sigma\) and the initial condition. Our results complement those of
\textit{P.-L. Chow} [Commun. Stoch. Anal. 3, No. 2, 211--222 (2009; Zbl 1331.35405); J. Differ. Equations 250, No. 5, 2567--2580 (2011; Zbl 1213.35410)], and \textit{M. Foondun} et al. [Fract. Calc. Appl. Anal. 19, No. 6, 1527--1553 (2016; Zbl 1355.60084)], \textit{M. Foondun} and \textit{R. D. Parshad} [Proc. Am. Math. Soc. 143, No. 9, 4085--4094 (2015; Zbl 1322.60110)] among others.Exact solutions for the Wick-type stochastic Schamel-Korteweg-de Vries equation.https://www.zbmath.org/1453.601252021-02-27T13:50:00+00:00"Wang, Xueqin"https://www.zbmath.org/authors/?q=ai:wang.xueqin"Shang, Yadong"https://www.zbmath.org/authors/?q=ai:shang.yadong"Di, Huahui"https://www.zbmath.org/authors/?q=ai:di.huahuiSummary: We consider the Wick-type stochastic Schamel-Korteweg-de Vries equation with variable coefficients in this paper. With the aid of symbolic computation and Hermite transformation, by employing the \((G'/G,1/G)\)-expansion method, we derive the new exact travelling wave solutions, which include hyperbolic and trigonometric solutions for the considered equations.On pointwise exponentially weighted estimates for the Boltzmann equation.https://www.zbmath.org/1453.351322021-02-27T13:50:00+00:00"Gamba, Irene M."https://www.zbmath.org/authors/?q=ai:gamba.irene-m"Pavlović, Nataša"https://www.zbmath.org/authors/?q=ai:pavlovic.natasa"Tasković, Maja"https://www.zbmath.org/authors/?q=ai:taskovic.majaThe paper concerns studies on the propagation in time of weighted \(L^\infty\) bounds for solutions to the homogeneous Boltzmann equation without cut-off when these solutions satisfy the propagation in time of weighted \(L^1\) bounds. In order to point out that the propagation in weighted-\(L^\infty\) norm relies on the propagation in weighted-\(L^1\) norms, the authors consider certain general weights, which include exponential and Mittag-Leffler weights. Since the non-cutoff collision operator cannot be split into the gain and loss terms, they first introduce a new splitting of the collision integral with weighted functions. Then they give the definition of so-called \(w\)-suitable\ solution), which is a weak solution satisfying a particular condition on \(L^\infty\)-weighted norms. After that, they prove a priori results for such solutions based on the regularity theory of integro-differential equations in the recent work of \textit{L. Silvestre} [Commun. Math. Phys. 348, No. 1, 69--100 (2016; Zbl 1352.35091)]. Consequently, the pointwise exponential weighted estimates are obtained.
Reviewer: Zhigang Wu (Shanghai)Combined effects for fractional Schrödinger-Kirchhoff systems with critical nonlinearities.https://www.zbmath.org/1453.351842021-02-27T13:50:00+00:00"Xiang, Mingqi"https://www.zbmath.org/authors/?q=ai:xiang.mingqi"Rădulescu, Vicenţiu D."https://www.zbmath.org/authors/?q=ai:radulescu.vicentiu-d"Zhang, Binlin"https://www.zbmath.org/authors/?q=ai:zhang.binlinSummary: In this paper, we investigate the existence of solutions for critical Schrödinger-Kirchhoff type systems driven by nonlocal integro-differential operators. As a particular case, we consider the following system:
\[ \begin{gathered} M\left([(u,v)]_{s,p}^p + \Vert(u,v)\Vert_{p,V}^p\right) ((-\Delta)_p^s u + V(x)\vert u\vert^{p-2}u) = \lambda H_u(x,u,v) + \frac{\alpha}{p_s^*} \vert v\vert^\beta \vert u\vert^{\alpha -2} u \quad\text{in } \mathbb{R}^N, \\
M\left([(u,v)]_{s,p}^p + \Vert(u,v)\Vert_{p,V}^p\right) ((-\Delta)_p^s v + V(x)\vert u\vert^{p-2}u) = \lambda H_v(x,u,v) + \frac{\beta}{p_s^*} \vert u\vert^\alpha \vert v\vert^{\beta-2} v \quad\text{in } \mathbb R^N, \end{gathered} \]
where \(\Delta^s_p\) is the fractional \(p\)-Laplace operator with \(0 < s < 1 < p < N/s\), \(\alpha, \beta > 1\) with \(\alpha + \beta= p^*_s\), \(M: \mathbb{R}^+_0\rightarrow \mathbb{R}^+_0\) is a continuous function, \(V: \mathbb{R}^N \rightarrow \mathbb{R}^+\) is a continuous function, \(\lambda > 0\) is a real parameter. By applying the mountain pass theorem and Ekeland's variational principle, we obtain the existence and asymptotic behaviour of solutions for the above systems under some suitable assumptions. A distinguished feature of this paper is that the above systems are degenerate, that is, the Kirchhoff function could vanish at zero. To the best of our knowledge, this is the first time to exploit the existence of solutions for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities in \(\mathbb{R}^N\).Existence of weak solutions of the aggregation equation with the \(p ( \cdot )\)-Laplacian.https://www.zbmath.org/1453.351092021-02-27T13:50:00+00:00"Vildanova, V. F."https://www.zbmath.org/authors/?q=ai:vildanova.venera-fidarisovna"Mukminov, F. Kh."https://www.zbmath.org/authors/?q=ai:mukminov.farit-khSummary: We consider an elliptic-parabolic aggregation equation of the form
\[b(u)_t=\operatorname{div}\left({\left|\nabla u\right|}^{p(x)-2}\nabla u-b(u)G(u)\right)+\gamma \left(x,b(u)\right),\]
where \(b\) is a nondecreasing function and \(G(u)\) is an integral operator. The condition on the boundary of a bounded domain \(\Omega\) ensures that the mass of the population \(\int u(x, t) dx = \mathrm{const}\) is preserved for \(\gamma = 0\). We prove the existence of a weak solution of the problem with a nonnegative bounded initial function in the cylinder \(\Omega \times (0, T)\). A formula for the guaranteed time \(T\) of the existence of the solution is obtained.Analysis of a delayed and diffusive oncolytic M1 virotherapy model with immune response.https://www.zbmath.org/1453.921302021-02-27T13:50:00+00:00"Elaiw, A. M."https://www.zbmath.org/authors/?q=ai:elaiw.ahmed-m"Al Agha, A. D."https://www.zbmath.org/authors/?q=ai:al-agha.a-dSummary: Oncolytic virotherapy (OVT) is a promising therapeutic approach that uses replication-competent viruses to target and kill tumor cells. Alphavirus M1 is a selective oncolytic virus which showed high efficacy against tumor cells. \textit{Z. Wang} et al. [Math. Biosci. 276, 19--27 (2016; Zbl 1341.92035)] studied an ordinary differential equation (ODE) model to verify the potent efficacy of M1 virus. Our purpose is to extend their model to include the effect of time delays and anti-tumor immune response. Also, we assume that all elements of the extended model undergo diffusion in a bounded domain. We study the existence, non-negativity and boundedness of solutions in order to verify the well-posedness of the model. We calculate all possible equilibrium points and determine the threshold conditions required for their existence and stability. These points reflect three different fates for OVT: partial success, complete success, or complete failure. We prove the global asymptotic stability of all equilibrium points by constructing suitable Lyapunov functionals, and verify the corresponding instability conditions. We conduct some numerical simulations to confirm the analytical results and show the crucial role of time delays and immune response in the success of OVT.Semiclassical sampling and discretization of certain linear inverse problems.https://www.zbmath.org/1453.351952021-02-27T13:50:00+00:00"Stefanov, Plamen"https://www.zbmath.org/authors/?q=ai:stefanov.plamen-dApproximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations.https://www.zbmath.org/1453.653452021-02-27T13:50:00+00:00"Torres-Sánchez, Alejandro"https://www.zbmath.org/authors/?q=ai:torres-sanchez.alejandro"Santos-Oliván, Daniel"https://www.zbmath.org/authors/?q=ai:santos-olivan.daniel"Arroyo, Marino"https://www.zbmath.org/authors/?q=ai:arroyo.marinoSummary: We introduce a new method, the Local Monge Parametrizations (LMP) method, to approximate tensor fields on general surfaces given by a collection of local parametrizations, e.g. as in finite element or NURBS surface representations. Our goal is to use this method to solve numerically tensor-valued partial differential equations (PDEs) on surfaces. Previous methods use scalar potentials to numerically describe vector fields on surfaces, at the expense of requiring higher-order derivatives of the approximated fields and limited to simply connected surfaces, or represent tangential tensor fields as tensor fields in 3D subjected to constraints, thus increasing the essential number of degrees of freedom. In contrast, the LMP method uses an optimal number of degrees of freedom to represent a tensor, is general with regards to the topology of the surface, and does not increase the order of the PDEs governing the tensor fields. The main idea is to construct maps between the element parametrizations and a local Monge parametrization around each node. We test the LMP method by approximating in a least-squares sense different vector and tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply the LMP method to two physical models on surfaces, involving a tension-driven flow (vector-valued PDE) and nematic ordering (tensor-valued PDE), on different topologies. The LMP method thus solves the long-standing problem of the interpolation of tensors on general surfaces with an optimal number of degrees of freedom.Kernel-based collocation methods for heat transport on evolving surfaces.https://www.zbmath.org/1453.653562021-02-27T13:50:00+00:00"Chen, Meng"https://www.zbmath.org/authors/?q=ai:chen.meng.1|chen.meng"Ling, Leevan"https://www.zbmath.org/authors/?q=ai:ling.leevanSummary: We propose algorithms for solving convective-diffusion partial differential equations (PDEs), which model surfactant concentration and heat transport on evolving surfaces, based on extrinsic kernel-based meshless collocation methods. The algorithms can be classified into two categories: one collocates PDEs extrinsically and analytically, and the other approximates surface differential operators by meshless pseudospectral approaches. The former is specifically designed to handle PDEs on evolving surfaces defined by parametric equations, and the latter works on surface evolutions based on point clouds. After some convergence studies and comparisons, we demonstrate that the proposed method can solve challenging PDEs posed on surfaces with high curvatures with discontinuous initial conditions with correct physics.An averaging approach to the Smoluchowski-Kramers approximation in the presence of a varying magnetic field.https://www.zbmath.org/1453.820632021-02-27T13:50:00+00:00"Cerrai, Sandra"https://www.zbmath.org/authors/?q=ai:cerrai.sandra"Wehr, Jan"https://www.zbmath.org/authors/?q=ai:wehr.jan"Zhu, Yichun"https://www.zbmath.org/authors/?q=ai:zhu.yichunSummary: We study the small mass limit of the equation describing planar motion of a charged particle of a small mass \(\mu\) in a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity \(\epsilon > 0\). We show that for all small but fixed frictions the small mass limit of \(q_{\mu, \epsilon}\) gives the solution \(q_\epsilon\) to a stochastic first order equation, containing a noise-induced drift term. Then, by using a generalization of the classical averaging theorem for Hamiltonian systems by \textit{M. I. Freidlin} and \textit{A. D. Wentzell} [Random perturbations of dynamical systems. Translated from the Russian by J Szücs. 3rd ed. Berlin: Springer (2012; Zbl 1267.60004)], we take the limit of the slow component of the motion \(q_\epsilon\) and we prove that it converges weakly to a Markov process on the graph obtained by identifying all points in the same connected components of the level sets of the magnetic field intensity function.A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations.https://www.zbmath.org/1453.653702021-02-27T13:50:00+00:00"Zhang, Hui"https://www.zbmath.org/authors/?q=ai:zhang.hui.5|zhang.hui.7|zhang.hui.2|zhang.hui.1|zhang.hui.11|zhang.hui|zhang.hui.6|zhang.hui.9|zhang.hui.3|zhang.hui.8|zhang.hui.10|zhang.hui.4"Jiang, Xiaoyun"https://www.zbmath.org/authors/?q=ai:jiang.xiaoyun"Zeng, Fanhai"https://www.zbmath.org/authors/?q=ai:zeng.fanhai"Karniadakis, George Em"https://www.zbmath.org/authors/?q=ai:karniadakis.george-emSummary: The reaction-diffusion model can generate a wide variety of spatial patterns, which has been widely applied in chemistry, biology, and physics, even used to explain self-regulated pattern formation in the developing animal embryo. In this work, a second-order stabilized semi-implicit time-stepping Fourier spectral method for the reaction-diffusion systems of equations with space described by the fractional Laplacian is developed. We adopt the temporal-spatial error splitting argument to illustrate that the proposed method is stable without imposing the CFL condition, and an optimal \(L^2\)-error estimate in space is proved. We also analyze the linear stability of the stabilized semi-implicit method and obtain a practical criterion to choose the time step size to guarantee the stability of the semi-implicit method. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen-Cahn, Gray-Scott and FitzHugh-Nagumo models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator, which are quite different from the patterns of the corresponding integer-order model.Finite difference/finite element method for tempered time fractional advection-dispersion equation with fast evaluation of Caputo derivative.https://www.zbmath.org/1453.653112021-02-27T13:50:00+00:00"Cao, Jiliang"https://www.zbmath.org/authors/?q=ai:cao.jiliang"Xiao, Aiguo"https://www.zbmath.org/authors/?q=ai:xiao.aiguo"Bu, Weiping"https://www.zbmath.org/authors/?q=ai:bu.weipingSummary: In this paper, a class of fractional advection-dispersion equations with Caputo tempered fractional derivative are considered numerically. An efficient algorithm for the evaluation of Caputo tempered fractional derivative is proposed to sharply reduce the computational work and storage, and this is of great significance for large-scale problems. Based on the nonsmooth regularity assumptions, a semi-discrete form is obtained by finite difference method in time, and its stability and convergence are investigated. Then by finite element method, we derive the corresponding fully discrete scheme and discuss its convergence. At last, some numerical examples, based on different domains, are presented to demonstrate effectiveness of numerical schemes and confirm the theoretical analysis.A numerically stable spherical harmonics solution for the neutron transport equation in a spherical shell.https://www.zbmath.org/1453.654522021-02-27T13:50:00+00:00"Garcia, R. D. M."https://www.zbmath.org/authors/?q=ai:garcia.roberto-d-mSummary: A numerically stable version of the spherical harmonics \((P_N)\) method for solving the one-speed neutron transport equation with \(L\) th order anisotropic scattering in a spherical shell is developed. Implementing a stable \(P_N\) solution for this problem is a challenging task for which no satisfactory answer has been given in the literature. The approach used in this work follows and generalizes a previous work on a problem whose domain is defined by the exterior of a sphere. First, a transformation is used to reduce the original transport equation in spherical geometry to a plane-geometry-like transport equation, where the angular redistribution term in spherical geometry is treated as a source. Then, a \(P_N\) solution in plane geometry given by a combination of the solution of the associated homogeneous equation and a particular solution is developed. This is followed by a post-processing step which is very effective in improving the \(P_N\) solution. An additional amount of work with respect to that required for solving problems in plane geometry occurs in the form of a system of \(N+1\) Volterra integral equations of the second kind that has to be solved for the coefficients of the particular solution. The proposed approach has, in any case, the merit of avoiding the ill-conditioning caused by the presence of modified spherical Bessel functions in the standard \(P_N\) solution, as demonstrated by numerical results tabulated for some test cases.Fast upwind and Eulerian-Lagrangian control volume schemes for time-dependent directional space-fractional advection-dispersion equations.https://www.zbmath.org/1453.652472021-02-27T13:50:00+00:00"Du, Ning"https://www.zbmath.org/authors/?q=ai:du.ning"Guo, Xu"https://www.zbmath.org/authors/?q=ai:guo.xu"Wang, Hong"https://www.zbmath.org/authors/?q=ai:wang.hong.1Summary: We develop control volume methods for two-dimensional time-dependent advection-dominated directional space-fractional advection-dispersion equations with the directional space-fractional derivative weighted in all the directions by a probability measure in the unit circle, which are used to model the anisotropic superdiffusive transport of solutes in groundwater moving through subsurface heterogeneous porous media.
We develop a fast upwind control volume method for the governing equation to eliminate the spurious numerical oscillations that often occur in space-centered numerical discretizations of advection term, which are relatively straightforward to implement. We also develop a Eulerian-Lagrangian control-volume method for the governing equation, which symmetrizes the governing equation by combining the time-derivative term and the advection term into a material derivative term along characteristic curves. Both methods are locally mass-conservative, which are essential in these applications.
Due to the nonlocal nature of the directional space-fractional differential operators, corresponding numerical discretizations usually generate full stiffness matrices. Conventional direct solvers tend to require \(O(N^2)\) memory requirement and have \(O(N^3)\) computational complexity per time step, where \(N\) is the number of spatial unknowns, which is computationally significantly more expensive than the numerical approximations of integer-order advection-diffusion equations. Based on the analysis of the structure of stiffness matrix, we propose a fast Krylov subspace iterative solver to accelerate the numerical approximations of both the upwind and Eulerian-Lagrangian control volume methods, which reduce computational complexity from \(O(N^3)\) by a direct solver to \(O(N\log N)\) per Krylov subspace iteration per time step and a memory requirement from \(O(N^2)\) to \(O(N)\). Numerical results are presented to show the utility of the methods.Numerical computations of split Bregman method for fourth order total variation flow.https://www.zbmath.org/1453.651422021-02-27T13:50:00+00:00"Giga, Yoshikazu"https://www.zbmath.org/authors/?q=ai:giga.yoshikazu"Ueda, Yuki"https://www.zbmath.org/authors/?q=ai:ueda.yukiSummary: The split Bregman framework for Osher-Solé-Vese (OSV) model and fourth order total variation flow are studied. We discretize the problem by piecewise constant function and compute \(\nabla (-\Delta_{\operatorname{av}})^{-1}\) approximately and exactly. Furthermore, we provide a new shrinkage operator for Spohn's fourth order model. Numerical experiments are demonstrated for fourth order problems under periodic boundary condition.Networks of coadjoint orbits: from geometric to statistical mechanics.https://www.zbmath.org/1453.820042021-02-27T13:50:00+00:00"Arnaudon, Alexis"https://www.zbmath.org/authors/?q=ai:arnaudon.alexis"Takao, So"https://www.zbmath.org/authors/?q=ai:takao.soSummary: A class of network models with symmetry group \( G \) that evolve as a Lie-Poisson system is derived from the framework of geometric mechanics, which generalises the classical Heisenberg model studied in statistical mechanics. We considered two ways of coupling the spins: one via the momentum and the other via the position and studied in details the equilibrium solutions and their corresponding nonlinear stability properties using the energy-Casimir method. We then took the example \( G = \mathrm{SO}(3)\) and saw that the momentum-coupled system reduces to the classical Heisenberg model with massive spins and the position-coupled case reduces to a new system that has a broken symmetry group \(\mathrm{SO}(3)/\mathrm{SO}(2)\) similar to the heavy top. In the latter system, we numerically observed an interesting synchronisation-like phenomenon for a certain class of initial conditions. Adding a type of noise and dissipation that preserves the coadjoint orbit of the network model, we found that the invariant measure is given by the Gibbs measure, from which the notion of temperature is defined. We then observed a surprising 'triple-humped' phase transition in the heavy top-like lattice model, where the spins switched from one equilibrium position to another before losing magnetisation as we increased the temperature. This work is only a first step towards connecting geometric mechanics with statistical mechanics and several interesting problems are open for further investigation.Efficient preconditioner updates for semilinear space-time fractional reaction-diffusion equations.https://www.zbmath.org/1453.652842021-02-27T13:50:00+00:00"Bertaccini, Daniele"https://www.zbmath.org/authors/?q=ai:bertaccini.daniele"Durastante, Fabio"https://www.zbmath.org/authors/?q=ai:durastante.fabioSummary: The numerical solution of fractional partial differential equations poses significant computational challenges in regard to efficiency as a result of the nonlocality of the fractional differential operators. In this work we consider the numerical solution of nonlinear space-time fractional reaction-diffusion equations integrated in time by fractional linear multistep formulas. The Newton step needed to advance in (fractional) time requires the solution of sequences of large and dense linear systems because of the fractional operators in space. A preconditioning updating strategy devised recently is adapted and the spectrum of the underlying operators is briefly analyzed. Because of the quasilinearity of the problem, each Jacobian matrix of the Newton equations can be written as the sum of a multilevel Toeplitz plus a diagonal matrix and produced exactly in the code. Numerical tests with a population dynamics problem show that the proposed approach is fast and reliable with respect to standard direct, unpreconditioned, multilevel circulant/Toeplitz and ILU preconditioned iterative solvers.
For the entire collection see [Zbl 1416.65008].Global dynamics of a TB model with classes age structure and environmental transmission.https://www.zbmath.org/1453.922912021-02-27T13:50:00+00:00"Dang, Yan-Xia"https://www.zbmath.org/authors/?q=ai:dang.yanxia"Wang, Juan"https://www.zbmath.org/authors/?q=ai:wang.juan"Li, Xue-Zhi"https://www.zbmath.org/authors/?q=ai:li.xuezhi"Ghosh, Mini"https://www.zbmath.org/authors/?q=ai:ghosh.miniSummary: In this article, an age structured SVEIR epidemic model for TB is formulated and analyzed by considering three types of ages e.g., latent age, infection age and vaccination age. The presented model also incorporates the environmental transmission of TB. The dynamics of the disease is governed by a system of differential-integral equations. We assume that the vaccines for TB are partially effective. Some vaccinated individuals get permanent immunity to this disease, but some vaccinated individuals lose its protective power over a time and become susceptible again. The dynamical property of the model is established by using LaSalle's invariance principle and constructing suitable Lyapunov functions. It has been shown that the dynamics of the model is governed by basic reproductive number \(\mathcal{R}(\xi)\). The disease-free equilibrium is globally stable if the basic reproductive number \(\mathcal{R}(\xi)<1\). The endemic equilibrium is locally and globally stable if \(\mathcal{R}(\xi)>1\). As the basic reproduction number plays an important role in determining the stability of the system, reducing this number below one through vaccination can lead to decrease in the transmission of this disease. Additionally, contaminated environment also contributes to the increase in \(\mathcal{R}(\xi)\), so we also need to keep the environment clean to decrease the basic reproduction number \(\mathcal{R}(\xi)\) below one. These types of control measures are easy to implement in our society and certainly this will improve the well-being of the society.
For the entire collection see [Zbl 1446.65004].An analysis of a mathematical fractional model of hybrid viscous nanofluids and its application in heat and mass transfer.https://www.zbmath.org/1453.351442021-02-27T13:50:00+00:00"Ali, Rizwan"https://www.zbmath.org/authors/?q=ai:ali.rizwan"Asjad, Muhammad Imran"https://www.zbmath.org/authors/?q=ai:asjad.muhammad-imran"Akgül, Ali"https://www.zbmath.org/authors/?q=ai:akgul.aliOne of the well-know problems in hydrodynamics is modelling free convection of viscous fluid in rectangular channel with non-equal temperatures given on its walls. In the present paper, the authors generalize this problem and examine water- and engine-oil based nanofluids containing copper or aluminium particles.
As these media have more complicated properties than classical fluids, three governing laws are generalized using Caputo fractional-order derivatives in time: Newton's law for viscous stresses, Fourier's law of heat conduction and Fick's law for diffusion. To solve the resulting equations, the Laplace transform is used.
The authors examine dependencies of temperature, concentration and velocity on coordinate, time and orders of Caputo derivatives and illustrate them by numerous plots.
Reviewer: Aleksey Syromyasov (Saransk)Symbolic manipulation of flows of nonlinear evolution equations, with application in the analysis of split-step time integrators.https://www.zbmath.org/1453.651162021-02-27T13:50:00+00:00"Auzinger, Winfried"https://www.zbmath.org/authors/?q=ai:auzinger.winfried"Hofstätter, Harald"https://www.zbmath.org/authors/?q=ai:hofstatter.harald"Koch, Othmar"https://www.zbmath.org/authors/?q=ai:koch.othmarSummary: We describe a package realized in the Julia programming language which performs symbolic manipulations applied to nonlinear evolution equations, their flows, and commutators of such objects. This tool was employed to perform contrived computations arising in the analysis of the local error of operator splitting methods. It enabled the proof of the convergence of the basic method and of the asymptotical correctness of a defect-based error estimator. The performance of our package is illustrated on several examples.
For the entire collection see [Zbl 1346.68010].Numerical method for solving the inverse problem of nonisothermal filtration.https://www.zbmath.org/1453.761872021-02-27T13:50:00+00:00"Badertdinova, E. R."https://www.zbmath.org/authors/?q=ai:badertdinova.e-r"Khairullin, M. Kh."https://www.zbmath.org/authors/?q=ai:khairullin.m-kh"Shamsiev, M. N."https://www.zbmath.org/authors/?q=ai:shamsiev.m-n"Khairullin, R. M."https://www.zbmath.org/authors/?q=ai:khairullin.r-mSummary: In this work a mathematical model of thermohydrodynamic processes occurring in the oil reservoir and the horizontal well bore is developed. On the basis of the proposed model and regularization methods, a computational algorithm for solving the inverse coefficient problem is proposed. Data of temperature changes, registered simultaneously by several depth gauges installed in the different locations of the horizontal part of the well bore, is taken as the initial information.Metastability of stochastic partial differential equations and Fredholm determinants.https://www.zbmath.org/1453.352012021-02-27T13:50:00+00:00"Berglund, Nils"https://www.zbmath.org/authors/?q=ai:berglund.nilsSummary: Metastability occurs when a thermodynamic system, such assupercooled water (which is liquid even below freezing point), lands on the ``wrong'' side of a phase transition, and remains in a state which differs from its equilibrium state for a considerable time. There are numerous mathematical models describing this phenomenon, including lattice models with stochastic dynamics. In this text, we will be interested in metastability in parabolic stochastic partial differential equations (SPDEs). Some of these equations are ill posed, and only thanks to very recent progress in the theory of so-called singular SPDEs does one know how to construct solutions via a renormalisation procedure. The study of metastability in these systems reveals unexpected links with the theory of spectral determinants, including Fredholm and Carleman-Fredholm determinants.Deep neural network expression of posterior expectations in Bayesian PDE inversion.https://www.zbmath.org/1453.351912021-02-27T13:50:00+00:00"Herrmann, Lukas"https://www.zbmath.org/authors/?q=ai:herrmann.lukas"Schwab, Christoph"https://www.zbmath.org/authors/?q=ai:schwab.christoph"Zech, Jakob"https://www.zbmath.org/authors/?q=ai:zech.jakobBlow-up results for a quasilinear von Karman equation of memory type with acoustic boundary conditions.https://www.zbmath.org/1453.350342021-02-27T13:50:00+00:00"Lee, Mi Jin"https://www.zbmath.org/authors/?q=ai:lee.mi-jin"Kang, Jum-Ran"https://www.zbmath.org/authors/?q=ai:kang.jum-ranSummary: We study the blow-up result of a quasilinear von Karman equation of memory type with acoustic boundary conditions. For the asymptotic behavior of solution to the von Karman equation, many authors have been considered. We prove the finite time blow-up result of solution under suitable condition on the initial data.Hölder-logarithmic stability in Fourier synthesis.https://www.zbmath.org/1453.420082021-02-27T13:50:00+00:00"Isaev, Mikhail"https://www.zbmath.org/authors/?q=ai:isaev.mikhail-ismailovitch"Novikov, Roman G."https://www.zbmath.org/authors/?q=ai:novikov.roman-gHyper-differential sensitivity analysis for inverse problems constrained by partial differential equations.https://www.zbmath.org/1453.351972021-02-27T13:50:00+00:00"Sunseri, Isaac"https://www.zbmath.org/authors/?q=ai:sunseri.isaac"Hart, Joseph"https://www.zbmath.org/authors/?q=ai:hart.joseph-l"van Bloemen Waanders, Bart"https://www.zbmath.org/authors/?q=ai:van-bloemen-waanders.bart-g"Alexanderian, Alen"https://www.zbmath.org/authors/?q=ai:alexanderian.alenA second order Calderón's method with a correction term and \textit{a priori} information.https://www.zbmath.org/1453.351942021-02-27T13:50:00+00:00"Shin, Kwancheol"https://www.zbmath.org/authors/?q=ai:shin.kwancheol"Mueller, Jennifer L."https://www.zbmath.org/authors/?q=ai:mueller.jennifer-lShocks make the Riemann problem for the full Euler system in multiple space dimensions ill-posed.https://www.zbmath.org/1453.351432021-02-27T13:50:00+00:00"Klingenberg, Christian"https://www.zbmath.org/authors/?q=ai:klingenberg.christian|klingenberg.christian-peter"Kreml, Ondřej"https://www.zbmath.org/authors/?q=ai:kreml.ondrej"Mácha, Václav"https://www.zbmath.org/authors/?q=ai:macha.vaclav"Markfelder, Simon"https://www.zbmath.org/authors/?q=ai:markfelder.simonLocal theory for spatio-temporal canards and delayed bifurcations.https://www.zbmath.org/1453.350202021-02-27T13:50:00+00:00"Avitabile, Daniele"https://www.zbmath.org/authors/?q=ai:avitabile.daniele"Desroches, Mathieu"https://www.zbmath.org/authors/?q=ai:desroches.mathieu"Veltz, Romain"https://www.zbmath.org/authors/?q=ai:veltz.romain"Wechselberger, Martin"https://www.zbmath.org/authors/?q=ai:wechselberger.martinA compact difference scheme for fourth-order fractional sub-diffusion equations with Neumann boundary conditions.https://www.zbmath.org/1453.652372021-02-27T13:50:00+00:00"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 compact finite difference scheme with global convergence order \(O(\tau^2+h^4)\) is derived for fourth-order fractional sub-diffusion equations subject to Neumann boundary conditions. The difficulty caused by the fourth-order derivative and Neumann boundary conditions is carefully handled. The stability and convergence of the proposed scheme are studied by the energy method. Theoretical results are supported by numerical experiments.A shape optimization approach for electrical impedance tomography with point measurements.https://www.zbmath.org/1453.780052021-02-27T13:50:00+00:00"Albuquerque, Yuri Flores"https://www.zbmath.org/authors/?q=ai:albuquerque.yuri-flores"Laurain, Antoine"https://www.zbmath.org/authors/?q=ai:laurain.antoine"Sturm, Kevin"https://www.zbmath.org/authors/?q=ai:sturm.kevinThe authors consider an electrical impedance tomography inverse problem which is analysed in the form of a shape optimization problem, where the discontinuity interface is the unknown. A conductivity equation is considered with Neumann and Dirichlet boundary conditions. Generalizations to a Banach space framework are included, and the authors compute the distributed shape derivative and prove its validity for conductivity inclusions which are only open. Moreover, in the case Lipschitz polygonal or \(C^1\) inclusions, boundary expressions of the shape derivative are also obtained. A numerical algorithm is proposed, based on the distributed shape derivative, and the efficiency of the approach is described.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes.https://www.zbmath.org/1453.351892021-02-27T13:50:00+00:00"Esteve, Carlos"https://www.zbmath.org/authors/?q=ai:esteve.carlos"Zuazua, Enrique"https://www.zbmath.org/authors/?q=ai:zuazua.enriqueA time multidomain spectral method for valuing affine stochastic volatility and jump diffusion models.https://www.zbmath.org/1453.653612021-02-27T13:50:00+00:00"Moutsinga, Claude Rodrigue Bambe"https://www.zbmath.org/authors/?q=ai:moutsinga.claude-rodrigue-bambe"Pindza, Edson"https://www.zbmath.org/authors/?q=ai:pindza.edson"Maré, Eben"https://www.zbmath.org/authors/?q=ai:mare.ebenA time-spectral domain decomposition method is developed that accommodates differential equations arising from financial models of affine type. The affine structure of the financial models is used to avoid solving the multi-dimensional partial integro-differential equation (PIDE) but rather to solve a system of Riccati equations. The method is based on the Tau-matrix approach using a differentiation matrix method on a time interval divided into disjoint domains. The Riccati equations are solved in the frequency domain using an operational matrix based on Chebyshev polynomials. In this way, the original problem is transformed into an iterative system of algebraic equations that is easier to solve. Three numerical examples are implemented and solutions are compared to numerical solutions from Chebfun [\textit{R. B. Platte} and \textit{L. N. Trefethen}, Math. Ind. 15, 69--87 (2010; Zbl 1220.65100)]. The numerical results show that the method maintains its spectral convergence even for large time-space intervals. The method can be applied to other affine models with jumps.
Reviewer: Bülent Karasözen (Ankara)Fixed angle inverse scattering for almost symmetric or controlled perturbations.https://www.zbmath.org/1453.351932021-02-27T13:50:00+00:00"Rakesh"https://www.zbmath.org/authors/?q=ai:rakesh.shanti-lal|rakesh.nitin|rakesh.leela|rakesh.|rakesh.kumar|rakesh.s-g"Salo, Mikko"https://www.zbmath.org/authors/?q=ai:salo.mikkoThe D-bar method for electrical impedance tomography -- demystified.https://www.zbmath.org/1453.780072021-02-27T13:50:00+00:00"Mueller, J. L."https://www.zbmath.org/authors/?q=ai:mueller.jennifer-l"Siltanen, Samuli"https://www.zbmath.org/authors/?q=ai:siltanen.samuliThe authors present a survey on certain computational inversion methods for the inverse problem of electrical impedance tomography. Namely, they describe the so-called D-bar methods (in 2D and 3D), based on noniterative methods involving complex geometrical optics solutions, D-bar equations, and nonlinear Fourier transforms.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)