Existence of the solution to a nonlocal-in-time evolutional problem

This work is devoted to the study of a nonlocal-in-time evolutional problem for the first order differential equation in Banach space. Our primary approach, although stems from the convenient technique based on the reduction of a nonlocal problem to its classical initial value analogue, uses more advanced analysis. That is a validation of the correctness in definition of the general solution representation via the Dunford-Cauchy formula. Such approach allows us to reduce the given existence problem to the problem of locating zeros of a certain entire function. It results in the necessary and sufficient conditions for the existence of a generalized (mild) solution to the given nonlocal problem. Aside of that we also present new sufficient conditions which in the majority of cases generalize existing results.Comment: This article is an extended translation of the part of Dmytro Sytnyk's PhD Thesi

On the stability of a weighted finite difference scheme for wave equation with nonlocal boundary conditions

We consider the stability of a weighted finite difference scheme for a linear hyperbolic equation with nonlocal integral boundary condition. By studying the spectrum of the transition matrix of the three-layered difference scheme we obtain a sufficient stability condition in a special matrix norm. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014)

On the stability of explicit finite difference schemes for a pseudoparabolic equation with nonlocal conditions

A new explicit conditionally consistent finite difference scheme for one-dimensional third-order linear pseudoparabolic equation with nonlocal conditions is constructed. The stability of the finite difference scheme is investigated by analysing a nonlinear eigenvalue problem. The stability conditions are stated and stability regions are described. Some numerical experiments are presented in order to validate theoretical results

Stability of the weighted splitting finite-difference scheme for a two-dimensional parabolic equation with two nonlocal integral conditions

AbstractNonlocal conditions arise in mathematical models of various physical, chemical or biological processes. Therefore, interest in developing computational techniques for the numerical solution of partial differential equations (PDEs) with various types of nonlocal conditions has been growing fast. We construct and analyse a weighted splitting finite-difference scheme for a two-dimensional parabolic equation with nonlocal integral conditions. The main attention is paid to the stability of the method. We apply the stability analysis technique which is based on the investigation of the spectral structure of the transition matrix of a finite-difference scheme. We demonstrate that depending on the parameters of the finite-difference scheme and nonlocal conditions the proposed method can be stable or unstable. The results of numerical experiments with several test problems are also presented and they validate theoretical results

On iterative methods for some elliptic equations with nonlocal conditions

The iterative methods for the solution of the system of the difference equations derived from the elliptic equation with nonlocal conditions are considered. The case of the matrix of the difference equations system being the M-matrix is investigated. Main results for the convergence of the iterative methods are obtained considering the structure of the spectrum of the difference operators with nonlocal conditions. Furthermore, the case when the matrix of the system of difference equations has only positive eigenvalues was investigated. The survey of results on convergence of iterative methods for difference problem with nonlocal condition is also presented

On iterative methods for some elliptic equations with nonlocal conditions

The iterative methods for the solution of the system of the difference equations derived from the elliptic equation with nonlocal conditions are considered. The case of the matrix of the difference equations system being the M-matrix is investigated. Main results for the convergence of the iterative methods are obtained considering the structure of the spectrum of the difference operators with nonlocal conditions. Furthermore, the case when the matrix of the system of difference equations has only positive eigenvalues was investigated. The survey of results on convergence of iterative methods for difference problem with nonlocal condition is also presented. 1The research was partially supported by the Research Council of Lithuania (grant No. MIP-051/2011). 2The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014)