Non-local Poisson structures and applications to the theory of integrable systems

We develop a rigorous theory of non-local Poisson structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee applicability of the Lenard–Magri scheme of integrability to a pair of compatible non-local Poisson structures. We apply this scheme to several such pairs, proving thereby integrability of various evolution equations, as well as hyperbolic equations.     

A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity

In one space dimension, a non-local elastic model is based ona single integral law, giving the stress when the strain isknown at all spatial points. In this study, we first derivea higher-order Boussinesq equation using locally non-lineartheory of 1D non-local elasticity and then we are able to showthat under certain conditions the Cauchy problem is globallywell-posed.     

On the stability of the state 1 in the non-local Fisher–KPP equation in bounded domains

We consider the non-local Fisher–KPP equation on a bounded domain with Neumann boundary conditions. Thanks to a Lyapunov function, we prove that, under a general hypothesis on the kernel involved in the non-local term, the homogenous steady state 1 is globally asymptotically stable. This assumption happens to be linked to some conditions given in the literature, which ensure that travelling waves link 0 to 1.     

Irreducible maps of commutative rings

We give necessary conditions for a map to be irreducible (in the category of finitely generated, torsion free modules) over a non-local, commutative ring and sufficient conditions when the ring is Bass. In particular, we show that an irreducible map of ZG, where G is a square free abelian group, must be a monomorphism with a simple cokernel. We also show that local endomorphism rings are necessary and sufficient for the existence of almost split sequences over a commutative Bass ring and we explicitly describe the modules and the maps in those sequences. The results in this paper enable us to describe the Auslander-Reiten quiver of a non-local Bass ring in [8].     

Existence of mild solutions for semilinear evolution equations with non-local initial conditions

In this paper we study the existence of mild solutions for a class of semilinear evolution equations with non-local initial conditions and extend some related results in this direction.     

Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations.It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.     

Existence results for non-local operators of elliptic type

In this paper, we investigate the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. We make use of homological linking and Morse theory.     

On the Properties of ParabolicSolutions of Non-localEquations

In this short note, we investigate the properties of positive solutions for some non-local parabolic equations. The conditions on the global existence and blowup in finite time of solution are given.     

非局部抛物型方程解的性质

In this short note, we investigate the properties of positive solutions for some non-local parabolic equations. The conditions on the global existence and blow-up in finite time of solution are given.     

Abstract wave equations with generalized Wentzell boundary conditions

We introduce a general framework which allows to verify if abstract wave equations with generalized Wentzell boundary conditions are well-posed, i.e., are governed by a cosine family. As an example we study wave equations for second order differential operators on C[0,1] with non-local Wentzell-type boundary conditions. Moreover, in Appendix A we give a perturbation result for sine and cosine families.     

一类非线性的经济系统控制模型的稳定性分析

用边界比较法讨论了以CE S生产函数作为反馈的定常经济系统控制模型的稳定性,该模型为含有非局部条件的分布参数系统.并且证明了非定常模型的平衡解是全局渐进稳定的.     

带有非局部边界条件的反应——扩散方程组解的全局存在性

研究一类带有非局部边界条件的抛物型方程组解的全局存在性.主要通过构造迭代序列,利用比较原理得到定理的证明.     

一个反应——扩散方程组解的整体存在与有限时刻爆破

研究一类带有非局部边界条件的抛物型方程组解的整体存在与有限刻爆破.主要通过构造上、下解,利用比较原理得到定理的证明.     

Properties of Positive Solutions for Non-local Reaction–Diffusion Problems

In this paper we investigate the properties of positive solutions for three non-local reaction-diffusion problems. The conditions on the existence and non-existence of global positive solutions are given. Moreover, we prove that the blow-up set is the whole region when the non-linearity occurs in the equation.     

Determination of the source parameter in a heat equation with a non-local boundary condition

In this article we consider the inverse problem of identifying a time dependent unknown coefficient in a parabolic problem subject to initial and non-local boundary conditions along with an overspecified condition defined at a specific point in the spatial domain. Due to the non-local boundary condition, the system of linear equations resulting from the backward Euler approximation have a coefficient matrix that is a quasi-tridiagonal matrix. We consider an efficient method for solving the linear system and the predictor–corrector method for calculating the solution and updating the estimate of the unknown coefficient. Two model problems are solved to demonstrate the performance of the methods.     

On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions

Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In this paper, the problem of solving the one-dimensional wave equation subject to given initial and non-local boundary conditions is considered. These non-local conditions arise mainly when the data on the boundary cannot be measured directly. Several finite difference methods with low order have been proposed in other papers for the numerical solution of this one dimensional non-classic boundary value problem. Here, we derive a new family of efficient three-level algorithms with higher order to solve the wave equation and also use a Simpson formula with higher order to approximate the integral conditions. Additionally, the fourth-order formula is also adapted to nonlinear equations, in particular to the well-known nonlinear Klein–Gordon equations which many physical problems can be modeled with. Numerical results are presented and are compared with some existing methods showing the efficiency of the new algorithms.     

Boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations

In the present paper we study the unique solvability of two non-local boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations. The uniqueness of the solutions of the considered problems are proven by the “abc” method. Existence theorems for the solutions of these problems are proven by the method of integral equations. The obtained results can be used for studying local and non-local boundary-value problems for mixed-hyperbolic type equations with two and three lines of changing type.      

Pseudo-transparent boundary conditions for the diffusion equation. Part I

In this article, we introduce and study a family of artificial boundary conditions for the diffusion equation. These conditions are local in time and non-local in space. We describe the principle of the approximation, show that the corresponding approximate problems are mathematically well-posed and give convergence results, as well as error estimates, of the approximate solution to the exact one.     

On Elliptic Equations with General Non-Local Boundary Conditions in UMD Spaces

In this work, we give new results concerning existence, uniqueness and maximal regularity of the strict solution of a class of elliptic equations with non-local boundary conditions containing an unbounded linear operator. This study is performed in the framework of UMD Banach spaces.     

Maximal B-regular boundary value problems with parameters

This study focuses on non-local boundary value problems (BVP) for elliptic differential-operator equations (DOE) defined in Banach-valued Besov (B) spaces. Here equations and boundary conditions contain certain parameters. This study found some conditions that guarantee the maximal regularity and fredholmness in Banach-valued B-spaces uniformly with respect to these parameters. These results are applied to non-local boundary value problems for a regular elliptic partial differential equation with parameters on a cylindrical domain to obtain algebraic conditions that guarantee the same properties.