Recent zbMATH articles in MSC 45https://www.zbmath.org/atom/cc/452021-04-16T16:22:00+00:00WerkzeugAttractiveness and exponential \(p\)-stability of neutral stochastic functional integro-differential equations driven by Wiener process and fBm with impulses effects.https://www.zbmath.org/1456.601732021-04-16T16:22:00+00:00"Hamit, Mahamat Hassan Mahamat"https://www.zbmath.org/authors/?q=ai:hamit.mahamat-hassan-mahamat"Allognissode, Fulbert Kuessi"https://www.zbmath.org/authors/?q=ai:allognissode.fulbert-kuessi"Mohamed, Mohamed salem"https://www.zbmath.org/authors/?q=ai:mohamed.mohamed-salem"Issaka, Louk-Man"https://www.zbmath.org/authors/?q=ai:issaka.louk-man"Diop, Mamadou Abdoul"https://www.zbmath.org/authors/?q=ai:diop.mamadou-abdoulSummary: In this work, we consider a class of neutral stochastic integro-differential equations driven by Wiener process and fractional Brownian motion with impulses effects. This paper deals with the global attractiveness and quasi-invariant sets for neutral stochastic integro-differential equations driven by Wiener process and fractional Brownian motion with impulses effects in Hilbert spaces. We use new integral inequalities combined with theories of resolvent operators to establish a set of sufficient conditions for the exponential \(p\)-stability of the mild solution of the considered equations. An example is presented to demonstrate the obtained theory.Nonlinear age-structured population models with nonlocal diffusion and nonlocal boundary conditions.https://www.zbmath.org/1456.350832021-04-16T16:22:00+00:00"Kang, Hao"https://www.zbmath.org/authors/?q=ai:kang.hao"Ruan, Shigui"https://www.zbmath.org/authors/?q=ai:ruan.shiguiSummary: In this paper, we develop some basic theory for age-structured population models with nonlocal diffusion and nonlocal boundary conditions. We first apply the theory of integrated semigroups and non-densely defined operators to a linear equation, study the spectrum, and analyze the asymptotic behavior via asynchronous exponential growth. Then we consider a semilinear equation with nonlocal diffusion and nonlocal boundary condition, use the method of characteristic lines to find the resolvent of the infinitesimal generator and the variation of constant formula, apply Krasnoselskii's fixed point theorem to obtain the existence of nontrivial steady states, and establish the stability of steady states. Finally we generalize these results to a nonlinear equation with nonlocal diffusion and nonlocal boundary condition.On existence theorems for generalized abstract measure integrodifferential equations.https://www.zbmath.org/1456.450022021-04-16T16:22:00+00:00"Dhage, Bapurao Chandrabhan"https://www.zbmath.org/authors/?q=ai:dhage.bapurao-chandrabhanSummary: In this paper, an existence and uniqueness results for a nonlinear abstract measure integrodifferential equation are proved via classical fixed point theorems of Schauder (see [\textit{A. Granas} and \textit{J. Dugundji}, Fixed point theory. New York, NY: Springer (2003; Zbl 1025.47002)]) and the author [Electron. J. Qual. Theory Differ. Equ. 2002, Paper No. 6, 9 p. (2002; Zbl 1029.47034)] under weaker Carathéodory condition. The existence for extremal solutions is also proved under certain Chandrabhan condition and using a hybrid fixed point theorem of the author [loc. cit.] in an ordered Banach space. Our existence results presented in this paper include the existence results of \textit{R. R. Sharma} [Proc. Am. Math. Soc. 32, 503--510 (1972; Zbl 0213.36201)], \textit{S. R. Joshi} [J. Math. Phys. Sci. 13, 497--506 (1979; Zbl 0435.34053)], \textit{G. R. Shendge} and \textit{S. R. Joshi} [Acta Math. Hung. 41, 53--59 (1983; Zbl 0536.34040)] and the author [J. Math. Phys. Sci. 20, 367--380 (1986; Zbl 0619.45005); with \textit{P. R. M. Reddy}, Jñānābha 49, No. 2, 82--93 (2019; Zbl 07273233)] on nonlinear abstract measure and abstract measure integrodifferential equations as special cases under weaker continuity condition.Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion.https://www.zbmath.org/1456.826362021-04-16T16:22:00+00:00"Leng, Yu"https://www.zbmath.org/authors/?q=ai:leng.yu"Tian, Xiaochuan"https://www.zbmath.org/authors/?q=ai:tian.xiaochuan"Trask, Nathaniel"https://www.zbmath.org/authors/?q=ai:trask.nathaniel"Foster, John T."https://www.zbmath.org/authors/?q=ai:foster.john-t-junHyers-Ulam stability of a nonautonomous semilinear equation with fractional diffusion.https://www.zbmath.org/1456.352242021-04-16T16:22:00+00:00"Villa-Morales, José"https://www.zbmath.org/authors/?q=ai:villa-morales.joseSummary: In this paper, we study the Hyers-Ulam stability of a nonautonomous semilinear reaction-diffusion equation. More precisely, we consider a nonautonomous parabolic equation with a diffusion given by the fractional Laplacian. We see that such a stability is a consequence of a Gronwall-type inequality.A note on transforming a plane strain first-kind Fredholm integral equation into an equivalent second-kind equation.https://www.zbmath.org/1456.450012021-04-16T16:22:00+00:00"Pomeranz, S."https://www.zbmath.org/authors/?q=ai:pomeranz.shirley-bSummary: Methods to convert Fredholm integral equations of the first kind into equivalent Fredholm integral equations of the second kind are used to study issues of existence and uniqueness of solutions. For some examples applied to plane strain problems, see [\textit{C. Constanda}, Proc. Am. Math. Soc. 123, No. 11, 3385--3396 (1995; Zbl 0847.35042); \textit{C. Constanda}, Direct and indirect boundary integral equation methods. Boca Raton, FL: Chapman \& Hall/CRC (2000; Zbl 0932.35001), Section 2.12]. In this paper, another technique to convert the Fredholm integral equation of the first kind that arises in a direct boundary integral formulation for the plane strain Dirichlet problem into an equivalent Fredholm integral equation of the second kind is developed. The technique presented in this paper generalizes work of \textit{Y. Yan} and \textit{I. H. Sloan} [J. Integral Equations Appl. 1, No. 4, 549--579 (1988; Zbl 0682.45001)] that was done for the scalar Laplace equation to the plane strain system of displacement equations.
For the entire collection see [Zbl 1330.65004].Legendre spectral Galerkin and multi-Galerkin methods for nonlinear Volterra integral equations of Hammerstein type.https://www.zbmath.org/1456.651862021-04-16T16:22:00+00:00"Mandal, Moumita"https://www.zbmath.org/authors/?q=ai:mandal.moumita"Nelakanti, Gnaneshwar"https://www.zbmath.org/authors/?q=ai:nelakanti.gnaneshwarSummary: In this paper, we discuss the superconvergence of the Galerkin solutions for second kind nonlinear integral equations of Volterra-Hammerstein type with a smooth kernel. Using Legendre polynomial bases, we obtain order of convergence \({\mathcal{O}}(n^{-r})\) for the Legendre Galerkin method in both \(L^2\)-norm and infinity norm, where \(n\) is the highest degree of the Legendre polynomial employed in the approximation and \(r\) is the smoothness of the kernel. The iterated Legendre Galerkin solutions converge with the order \({\mathcal{O}}(n^{-2r}),\) whose convergence order is the same as that of the multi-Galerkin solutions. We also prove that iterated Legendre multi-Galerkin method has order of convergence \({\mathcal{O}}(n^{-3r})\) in both \(L^2\)-norm and infinity norm. Numerical examples are given to demonstrate the efficacy of Galerkin and multi-Galerkin methods.A fuzzy transform method for numerical solution of fractional Volterra integral equations.https://www.zbmath.org/1456.651822021-04-16T16:22:00+00:00"Agheli, B."https://www.zbmath.org/authors/?q=ai:agheli.bahram"Firozja, M. Adabitabar"https://www.zbmath.org/authors/?q=ai:adabitabar-firozja.mSummary: Many researchers have used the numerical methods for the purpose of solving Volterra integral equations. In this research work, we have shown that it is possible to use fuzzy transform method \(( F\text{-transform} )\) to tackle with the fractional Volterra integral equation. The core idea of the technique of F-transforms is a fuzzy partition of a universe into fuzzy subsets. A function can be associated with a mapping from a set of fuzzy subsets to the set of its thus obtained average values. In numerical method, in order to approximate a function on a particular interval, only a restricted number of points have been employed. However, what makes the \(F\text{-transform}\) preferable to other methods is that it makes use of all points in this interval. Thus, a number of clear and specific examples have been enumerated for the purpose of illustrating the simplicity and the efficiency of the suggested method.Study of the algebra of smooth integro-differential operators with applications.https://www.zbmath.org/1456.160212021-04-16T16:22:00+00:00"Haghany, A."https://www.zbmath.org/authors/?q=ai:haghany.ahmad"Kassaian, Adel"https://www.zbmath.org/authors/?q=ai:kassaian.adelNumerical solution of Itô-Volterra integral equation by least squares method.https://www.zbmath.org/1456.650052021-04-16T16:22:00+00:00"Ahmadinia, M."https://www.zbmath.org/authors/?q=ai:ahmadinia.mahdi|ahmadinia.mehdi"Afshari, A. H."https://www.zbmath.org/authors/?q=ai:afshari.a-h"Heydari, M."https://www.zbmath.org/authors/?q=ai:heydari.mahdi|heydari.mehdi|heydari.majeed|heydari.mohammad-taghi|heydari.mohammad-hossien|heydari.mohammadhossein|heydari.masoud|heydari.maryam|heydari.mohammad-mehdi|heydari.maysam|heydari.mojganSummary: This paper presents a computational method based on least squares method and block pulse functions for solving Itô-Volterra integral equation. The Itô-Volterra integral equation is converted to a linear system of algebraic equations by the least squares method on the block pulse functions. The error analysis of the proposed method is investigated by providing theorems. Numerical examples show the accuracy and reliability of the presented method. The numerical results confirm that the presented method is more accurate than the block pulse functions operational matrix method.An infection age-space structured SIR epidemic model with Neumann boundary condition.https://www.zbmath.org/1456.352012021-04-16T16:22:00+00:00"Chekroun, Abdennasser"https://www.zbmath.org/authors/?q=ai:chekroun.abdennasser"Kuniya, Toshikazu"https://www.zbmath.org/authors/?q=ai:kuniya.toshikazuSummary: In this paper, we are concerned with an SIR epidemic model with infection age and spatial diffusion in the case of Neumann boundary condition. The original model is constructed as a nonlinear age structured system of reaction-diffusion equations. By using the method of characteristics, we reformulate the model into a system of a reaction-diffusion equation and a Volterra integral equation. For the reformulated system, we define the basic reproduction number \(\mathcal{R}_0\) by the spectral radius of the next generation operator, and show that if \(\mathcal{R}_0 < 1\), then the trivial disease-free steady state is globally attractive, whereas if \(\mathcal{R}_0 > 1\), then the disease in the system is persistent. Moreover, under an additional assumption that there exists a finite maximum age of infectiousness, we show the global attractivity of a constant endemic steady state for \(\mathcal{R}_0 > 1\).The scaling hypothesis for Smoluchowski's coagulation equation with bounded perturbations of the constant kernel.https://www.zbmath.org/1456.350652021-04-16T16:22:00+00:00"Cañizo, José A."https://www.zbmath.org/authors/?q=ai:canizo.jose-alfredo"Throm, Sebastian"https://www.zbmath.org/authors/?q=ai:throm.sebastianWhen the coagulation kernel \(K\) is homogeneous with a degree strictly smaller than one, it is expected that solutions to the coagulation equation
\[
\partial_\tau \phi(\tau,\xi) = \frac{1}{2} \int_0^\xi K(\xi-\eta,\eta) \phi(\tau,\xi-\eta) \phi(\tau,\eta)\ d\eta
- \int_0^\infty K(\xi,\eta) \phi(\tau,\xi) \phi(\tau,\eta)\ d\eta
\]
where \((\tau,\xi)\in (0,\infty)\times (0,\infty)\), with non-negative initial condition \(\phi_0\in L^1((0,\infty),\xi d\xi)\), behave in a self-similar way for large values of \(\tau\). This conjecture is up to now known to be true for the \textit{so-called} solvable kernels \(K(\xi,\eta)=2\) and \(K(\xi,\eta)=\xi+\eta\), see [\textit{G. Menon} and \textit{R. L. Pego}, Commun. Pure Appl. Math. 57, No. 9, 1197--1232 (2004; Zbl 1049.35048)]. Its validity is extended here to small perturbations of the constant kernel with homogeneity zero. In addition, a temporal decay rate is derived. More precisely, let \(W\in C((0,\infty)^2)\) be a symmetric function satisfying
\[
0 \le W(\xi,\eta) \leq 1\text{ and } W(\lambda\xi,\lambda\eta) = W(\xi,\eta), \qquad (\lambda,\xi,\eta)^3,
\]
and set \(K_\varepsilon = 2 + \varepsilon W\) for \(\varepsilon \ge 0\). It is shown that, for \(\varepsilon>0\) sufficiently small, there is a unique self-similar solution \((\tau,\xi) \mapsto (1+\tau)^{-2} G_\varepsilon(x(1+\tau)^{-1})\) such that \(\|G_\varepsilon\|_{L^1((0,\infty),\xi d\xi)}=1\) and \(G_\varepsilon\in L^1((0,\infty),\xi^k d\xi)\) for all \(k\ge 0\). It is further proved that this self-similar solution is stable in the following sense: given \(R>0\), \(k>2\), and a non-negative initial condition \(\phi_0\) satisfying
\[
\int_0^\infty \xi \phi_0(\xi)\ \mathrm{d}\xi = 1\,, \quad \int_0^\infty |\phi_0(\xi) -G_\varepsilon(\xi)| (1+\xi)^k\ \mathrm{d}\xi \le R\,,
\]
there are \(M>0\) and \(C>0\) depending only on \(R\) and \(k\) such that
\begin{align*}
& \int_0^\infty |(1+\tau)^2 \phi(\tau,x (1+\tau)) - G_\varepsilon(x)| (1+x)^k\ \mathrm{d}x \cr
& \qquad\qquad \le C (1+\tau)^{(1-2M\varepsilon)/2} \int_0^\infty |\phi_0(\xi) -G_\varepsilon(\xi)| (1+\xi)^k\ \mathrm{d}\xi
\end{align*}
for \(\tau\ge 0\). The proof relies on a refined study of the dynamics of the coagulation equation with constant kernel \(K_0\), building upon previous works on this particular case. In particular, a spectral gap for the linearised operator around the explicit self-similar profile \(G_0(x) = e^{-x}\) is obtained. Also, the stability of the self-similar profiles \((G_\varepsilon)\) with respect to \(\varepsilon\) is established.
Reviewer: Philippe Laurençot (Toulouse)Error estimates for second-order SAV finite element method to phase field crystal model.https://www.zbmath.org/1456.651262021-04-16T16:22:00+00:00"Wang, Liupeng"https://www.zbmath.org/authors/?q=ai:wang.liupeng"Huang, Yunqing"https://www.zbmath.org/authors/?q=ai:huang.yunqingSummary: In this paper, the second-order scalar auxiliary variable approach in time and linear finite element method in space are employed for solving the Cahn-Hilliard type equation of the phase field crystal model. The energy stability of the fully discrete scheme and the boundedness of numerical solution are studied. The rigorous error estimates of order \(O(\tau^2+h^2)\) in the sense of \(L^2\)-norm is derived. Finally, some numerical results are given to demonstrate the theoretical analysis.Approximate solution of Bagley-Torvik equations with variable coefficients and three-point boundary-value conditions.https://www.zbmath.org/1456.651832021-04-16T16:22:00+00:00"Huang, Q. A."https://www.zbmath.org/authors/?q=ai:huang.qiongao|huang.qiu-an"Zhong, X. C."https://www.zbmath.org/authors/?q=ai:zhong.xichang|zhong.xiaochun|zhong.xian-ci"Guo, B. L."https://www.zbmath.org/authors/?q=ai:guo.baolin|guo.boling|guo.baolongSummary: The fractional Bagley-Torvik equation with variable coefficients is investigated under three-point boundary-value conditions. By using the integration method, the considered problems are transformed into Fredholm integral equations of the second kind. It is found that when the fractional order is \(1< \alpha <2\), the obtained Fredholm integral equation is with a weakly singular kernel. When the fractional order is \(0< \alpha <1\), the given Fredholm integral equation is with a continuous kernel or a weakly singular kernel depending on the applied boundary-value conditions. The uniqueness of solution for the obtained Fredholm integral equation of the second kind with weakly singular kernel is addressed in continuous function spaces. A new numerical method is further proposed to solve Fredholm integral equations of the second kind with weakly singular kernels. The approximate solution is made and its convergence and error estimate are analyzed. Several numerical examples are computed to show the effectiveness of the solution procedures.Two applications of a generalization of an asymptotic fixed point theorem.https://www.zbmath.org/1456.060022021-04-16T16:22:00+00:00"Herzog, Gerd"https://www.zbmath.org/authors/?q=ai:herzog.gerd"Kunstmann, Peer Chr."https://www.zbmath.org/authors/?q=ai:kunstmann.peer-christianSummary: We present a variant of an asymptotic version of the Abian-Brown Fixed Point Theorem, and applications to recursively defined sequences and Hammerstein integral equations.Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels.https://www.zbmath.org/1456.651872021-04-16T16:22:00+00:00"Sajjadi, Sayed Arsalan"https://www.zbmath.org/authors/?q=ai:sajjadi.sayed-arsalan"Pishbin, Saeed"https://www.zbmath.org/authors/?q=ai:pishbin.saeedSummary: In this paper, we consider product integration method based on orthogonal polynomials to solve mixed system of Volterra integral equations of the first and second kind with weakly singular kernels. For investigation of the theoretical and numerical analysis of the mixed systems, the notions of the tractability index and \(\nu\)-smoothing property are extended for a weakly singular Volterra integral operator. Convergence analysis of the product integration method is derived. Finally, the proposed method is illustrated by two examples, which confirm the theoretical prediction of the error estimation.Dynamics for spherical spin glasses: disorder dependent initial conditions.https://www.zbmath.org/1456.828462021-04-16T16:22:00+00:00"Dembo, Amir"https://www.zbmath.org/authors/?q=ai:dembo.amir"Subag, Eliran"https://www.zbmath.org/authors/?q=ai:subag.eliranIn the paper the authors investigated the thermodynamic (\(N\to\infty\)), long-time (\(t\to\infty\)), behavior of a class of systems composed of \(N\) Langevin particles interacting with each other through a random potential.
Namely, the thermodynamic limit of the empirical correlation and response functions is derived for spherical mixed p-spin disordered mean-field models, starting uniformly within one of the spherical bands on which (at low temperature) the Gibbs measure concentrates for the pure p-spin models and mixed perturbations of them. Moreover, the large time asymptotics of the corresponding coupled non-linear integro-differential equations is related to the geometric structure of the Gibbs measures (at low temperature), and derive their FDT (Fluctuation Dissipation Theorem) solution (at high temperature).
Reviewer: Utkir A. Rozikov (Tashkent)Numerical solution of Volterra integro-differential equations with linear barycentric rational method.https://www.zbmath.org/1456.651852021-04-16T16:22:00+00:00"Li, Jin"https://www.zbmath.org/authors/?q=ai:li.jin.5|li.jin.4|li.jin.1|li.jin.3|li.jin.2|li.jin"Cheng, Yongling"https://www.zbmath.org/authors/?q=ai:cheng.yonglingSummary: In this article, we study linear barycentric rational method (LBM) to solve one-order Volterra integro-differential equations. With the help of linear barycentric rational interpolation, the matrix form of Volterra integro-differential equations is obtained. Then the convergence rate of LBM for solving one-order integro-differential equation is proved. Further more, Volterra integro-differential equation systems is also solved by the linear barycentric rational method. At last, several examples are presented to valid our theoretical analysis.On dual Bernstein polynomials and stochastic fractional integro-differential equations.https://www.zbmath.org/1456.601672021-04-16T16:22:00+00:00"Sayevand, Khosro"https://www.zbmath.org/authors/?q=ai:sayevand.khosro"Machado, J. Tenreiro"https://www.zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiro"Masti, Iman"https://www.zbmath.org/authors/?q=ai:masti.imanSummary: In recent years, random functional or stochastic equations have been reported in a large class of problems. In many cases, an exact analytical solution of such equations is not available and, therefore, is of great importance to obtain their numerical approximation. This study presents a numerical technique based on Bernstein operational matrices for a family of stochastic fractional integro-differential equations (SFIDE) by means of the trapezoidal rule. A relevant feature of this method is the conversion of the SFIDE into a linear system of algebraic equations that can be analyzed by numerical methods. An upper error bound, the convergence, and error analysis of the scheme are investigated. Three examples illustrate the accuracy and performance of the technique.An \(L^2\) to \(L^\infty\) framework for the Landau equation.https://www.zbmath.org/1456.351962021-04-16T16:22:00+00:00"Kim, Jinoh"https://www.zbmath.org/authors/?q=ai:kim.jinoh"Guo, Yan"https://www.zbmath.org/authors/?q=ai:guo.yan"Hwang, Hyung Ju"https://www.zbmath.org/authors/?q=ai:hwang.hyung-juThe authors consider the Landau equation with Coulomb potential: \(\partial
_{t}F+v\cdot \nabla _{x}F=Q(F,F)=\nabla v\cdot \int_{\mathbb{R}^{3}}\phi
(v-v^{\prime })[F(v^{\prime })\nabla _{v}F(v)-F(v)\nabla _{v}F(v^{\prime
})]dv\), posed in \((0,\infty )\times \mathbb{T}^{3}\), where \(\mathbb{T}^{3}\)
is the 3D torus, \(F(t,x,v)\geq 0\) is the spatially periodic distribution
function for particles, and \(\phi \) is the non-negative matrix defined as \(
\phi ^{ij}(v)=\{\delta _{i,j}-\frac{v_{i}v_{j}}{\left\vert v\right\vert ^{2}}
\}\left\vert v\right\vert ^{-1}\). They introduce the normalized Maxwellian \(
\mu (v)=e^{-\left\vert v\right\vert ^{2}}\)\ and writing \(F(t,x,v)=\mu
(v)+f(t,x,v)\) they observe that \(f\) satisfies \(f_{t}+v\cdot \partial
_{x}f+Lf=\Gamma (f,f)\), where \(L=-A-K\) is the linear operator with \(Af=\mu
^{-1/2}\partial _{i}\{\mu ^{1/2}\sigma ^{ij}[\partial _{j}f+v_{j}f]\}\), \(
Kf=-\mu ^{-1/2}\partial _{i}\{\mu \phi ^{ij}\ast \mu ^{1/2}[\partial
_{j}f+v_{j}f]\}\), and \(\Gamma (g,f)=\partial _{i}[\{\phi ^{ij}\ast \lbrack
\mu ^{1/2}g]\}\partial _{j}f]+\{\phi ^{ij}\ast \lbrack v_{i}\mu
^{1/2}g]\}\partial _{j}f-\partial _{i}[\{\phi ^{ij}\ast \lbrack \mu
^{1/2}\partial _{j}g]\}f]+\{\phi ^{ij}\ast \lbrack v_{i}\mu ^{1/2}\partial
_{j}g]\}f\). The initial condition \(f(0,x,v)=f_{0}(x,v)\) is added, where \(
f_{0}\) satisfies the conservation laws \(\int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}f_{0}(x,v)\sqrt{\mu }=\int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}v_{i}f_{0}(x,v)\sqrt{\mu }=\int \int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}\left\vert v\right\vert ^{2}f_{0}(x,v)\sqrt{\mu }=0\). The authors
define the notion of weak solution to this problem as a function \(
f(t,x,v)\in L^{\infty }((0,\infty )\times \mathbb{T}^{3}\times \mathbb{R}
^{3},w^{\vartheta }(v)dtdxdv)\), which satisfies \(\int_{0}^{T}\left\Vert
f(s)\right\Vert _{\sigma ,\vartheta }^{2}ds<+\infty \) and a variational
formulation issued from the above equation. Here \(\left\Vert f(s)\right\Vert
_{\sigma ,\vartheta }^{2}=\int \int_{\mathbb{T}^{3}\times \mathbb{R}
^{3}}w^{2\vartheta }[\sigma ^{ij}\partial _{i}f\partial _{j}f+\sigma
^{ij}v_{i}v_{j}f^{2}]dvdx\). The main result of the paper proves the
existence of a unique weak solution to this problem, if the initial data \(
f_{0}\) satisfies \(\left\Vert f_{0}\right\Vert _{\infty ,\vartheta }^{2}\leq
\varepsilon _{0}\) and \(\left\Vert -v\cdot \nabla _{v}f_{0}+\overline{A}
_{f_{0}}f_{0}\right\Vert _{\infty ,\vartheta }+\left\Vert
D_{v}f_{0}\right\Vert _{\infty ,\vartheta }<\infty \) for some \(\varepsilon
_{0}\in (0,1]\) and some positive \(\vartheta \). This weak solution satisfies
different estimates. For the proof, the authors first consider the
linearized Landau equation \(\partial _{t}f+v\cdot \partial _{x}f+Lf=\Gamma
(g,f)\), for some bounded function \(g\). They establish a uniform \(L^{2}\)
-estimate on a\ classical solution to the original problem and to this
linearized problem if \(\left\Vert g\right\Vert _{\infty }\) is small enough,
from which they then deduce a uniform \(L^{\infty }\)-estimate and a \(%
C^{0,\alpha }\)-estimate, through \(L^{2}-L^{\infty }\) estimates for the
solution of auxiliary linear problems. This allows deriving an Hölder
estimate and a \(S^{p}\)-estimate for the solution to the linearized problem,
where \(\left\Vert f\right\Vert _{S^{p}(\Omega )}=\left\Vert f\right\Vert _{L^{p}(\Omega )}+\left\Vert
D_{v}f\right\Vert _{L^{p}(\Omega )}+\left\Vert D_{vv}f\right\Vert
_{L^{p}(\Omega )}+\left\Vert (-\partial _{t}-v\cdot \nabla _{x})f\right\Vert
_{L^{p}(\Omega )}\), with \(\Omega =(0,\infty )\times \mathbb{T}^{3}\times
\mathbb{R}^{3}\).
Reviewer: Alain Brillard (Riedisheim)On the approximate solution of a class of nonlinear multidimensional weakly singular integral equations.https://www.zbmath.org/1456.651812021-04-16T16:22:00+00:00"Agachev, Yu. R."https://www.zbmath.org/authors/?q=ai:agachev.yurii-romanovich"Gubaidullina, R. K."https://www.zbmath.org/authors/?q=ai:gubaidullina.r-kSummary: In this work, for a class of nonlinear weakly singular integral equations defined in the space of quadratically summable functions in a circle, the rationale for the general projection method is given. Using the obtained general results, the convergence of the well-known Galerkin method is proved.Schoenberg coefficients and curvature at the origin of continuous isotropic positive definite kernels on spheres.https://www.zbmath.org/1456.420072021-04-16T16:22:00+00:00"Arafat, Ahmed"https://www.zbmath.org/authors/?q=ai:arafat.ahmed"Gregori, Pablo"https://www.zbmath.org/authors/?q=ai:gregori.pablo"Porcu, Emilio"https://www.zbmath.org/authors/?q=ai:porcu.emilioSummary: We consider the class \(\Psi_d\) of continuous functions that define isotropic covariance functions in the \(d\)-dimensional sphere \(\mathbb{S}^d\). We provide a new recurrence formula for the solution of Problem 1 in \textit{T. Gneiting} [``Strictly and non-strictly positive definite functions on spheres: online supplement'' (2013), \url{https://projecteuclid.org/download/suppdf_1/euclid.bj/1377612854}], solved by \textit{J. Fiedler} [``From Fourier to Gegenbauer: dimension walks on spheres'' (2013), Preprint, \url{arXiv:1303.6856}]. In addition, we have improved the current bounds for the curvature at the origin of locally supported covariances (Problem 3 in T. Gneiting [loc. cit.]), which is of applied interest at least for \(d=2\).Forward-backward SDEs with jumps and classical solutions to nonlocal quasilinear parabolic PDEs.https://www.zbmath.org/1456.601622021-04-16T16:22:00+00:00"Shamarova, Evelina"https://www.zbmath.org/authors/?q=ai:shamarova.evelina"Sá Pereira, Rui"https://www.zbmath.org/authors/?q=ai:sa-pereira.ruiSummary: We obtain an existence and uniqueness theorem for fully coupled forward-backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of \textit{O. A. Ladyzhenskaya} et al. [Linear and quasi-linear equations of parabolic type. Translated from the Russian by S. Smith. American Mathematical Society (AMS), Providence, RI (1968; Zbl 0174.15403)] to non-local PDEs of this class. Namely, we obtain the existence and uniqueness of a classical solution to both the Cauchy problem and the initial-boundary value problem for non-local quasilinear parabolic second-order PDEs.Splitting methods for Fokker-Planck equations related to jump-diffusion processes.https://www.zbmath.org/1456.650662021-04-16T16:22:00+00:00"Gaviraghi, Beatrice"https://www.zbmath.org/authors/?q=ai:gaviraghi.beatrice"Annunziato, Mario"https://www.zbmath.org/authors/?q=ai:annunziato.mario"Borzì, Alfio"https://www.zbmath.org/authors/?q=ai:borzi.alfioSummary: A splitting implicit-explicit (SIMEX) scheme for solving a partial integro-differential Fokker-Planck equation related to a jump-diffusion process is investigated. This scheme combines the method of Chang-Cooper for spatial discretization with the Strang-Marchuk splitting and first- and second-order time discretization methods. It is proven that the SIMEX scheme is second-order accurate, positive preserving, and conservative. Results of numerical experiments that validate the theoretical results are presented. (This chapter is a summary of the paper [\textit{B. Gaviraghi} et al., Appl. Math. Comput. 294, 1--17 (2017; Zbl 1411.65110)]; all theoretical statements in this summary are proved in that reference.)
For the entire collection see [Zbl 1390.91011].Generalization analysis of Fredholm kernel regularized classifiers.https://www.zbmath.org/1456.681482021-04-16T16:22:00+00:00"Gong, Tieliang"https://www.zbmath.org/authors/?q=ai:gong.tieliang"Xu, Zongben"https://www.zbmath.org/authors/?q=ai:xu.zongben"Chen, Hong"https://www.zbmath.org/authors/?q=ai:chen.hong.1|chen.hongSummary: Recently, a new framework, Fredholm learning, was proposed for semisupervised learning problems based on solving a regularized Fredholm integral equation. It allows a natural way to incorporate unlabeled data into learning algorithms to improve their prediction performance. Despite rapid progress on implementable algorithms with theoretical guarantees, the generalization ability of Fredholm kernel learning has not been studied. In this letter, we focus on investigating the generalization performance of a family of classification algorithms, referred to as Fredholm kernel regularized classifiers. We prove that the corresponding learning rate can achieve \(\mathcal{O}(l^{-1})\) (\(l \) is the number of labeled samples) in a limiting case. In addition, a representer theorem is provided for the proposed regularized scheme, which underlies its applications.The fine error estimation of collocation methods on uniform meshes for weakly singular Volterra integral equations.https://www.zbmath.org/1456.651842021-04-16T16:22:00+00:00"Liang, Hui"https://www.zbmath.org/authors/?q=ai:liang.hui"Brunner, Hermann"https://www.zbmath.org/authors/?q=ai:brunner.hermannSummary: It is well known that for the second-kind Volterra integral equations (VIEs) with weakly singular kernel, if we use piecewise polynomial collocation methods of degree \(m\) to solve it numerically, due to the weak singularity of the solution at the initial time \(t=0\), only \(1-\alpha\) global convergence order can be obtained on uniform meshes, comparing with \(m\) global convergence order for VIEs with smooth kernel. However, in this paper, we will see that at mesh points, the convergence order can be improved, and it is better and better as \(n\) increasing. In particular, 1 order can be recovered for \(m=1\) at the endpoint. Some superconvergence results are obtained for iterated collocation methods, and a representative numerical example is presented to illustrate the obtained theoretical results.Mixed-type Galerkin variational principle and numerical simulation for a generalized nonlocal elastic model.https://www.zbmath.org/1456.651612021-04-16T16:22:00+00:00"Jia, Lueling"https://www.zbmath.org/authors/?q=ai:jia.lueling"Chen, Huanzhen"https://www.zbmath.org/authors/?q=ai:chen.huanzhen"Wang, Hong"https://www.zbmath.org/authors/?q=ai:wang.hong.1Summary: A mixed-type Galerkin variational principle is proposed for a generalized nonlocal elastic model. The solvability and regularity of its solution is naturally derived through the Lax-Milgram lemma, from which a solvability criterion is inferred for a Fredholm integral equation of the first kind. A mixed-type finite element procedure is therefore developed and the existence and uniqueness of the discrete solution is proved. This compensates the lack of solvability proof for the collocation-finite difference scheme proposed in [\textit{N. Du} et al., J. Comput. Phys. 297, 72--83 (2015; Zbl 1349.76455)]. Numerical error bounds for the unknown and the intermediate variable are proved. By carefully exploring the structure of the coefficient matrices of the numerical method, we develop a fast conjugate gradient algorithm , which reduces the computations to \(\mathcal O(N\log N)\) per iteration and the memory to \(\mathcal O(N)\). The use of the preconditioner significantly reduces the number of iterations. Numerical results show the utility of the method.