Numerical solution of the Cauchy problem for Volterra integrodifferential equations with difference kernels

TL;DR

This paper presents a numerical method for solving Volterra integrodifferential equations with difference kernels by transforming the problem into a local system, using exponential sum approximations, and analyzing stability and efficiency.

Contribution

It introduces a novel approach to convert nonlocal problems into local systems via exponential sum approximation, enabling efficient numerical solutions.

Findings

Stable two-level schemes with weights are developed.

The method effectively solves equations with stretching exponential kernels.

Numerical experiments confirm the theoretical stability and accuracy.

Abstract

We consider the problems of the numerical solution of the Cauchy problem for an evolutionary equation with memory when the kernel of the integral term is a difference one. The computational implementation is associated with the need to work with an approximate solution for all previous points in time. In this paper, the considered nonlocal problem is transformed into a local one; a loosely coupled equation system with additional ordinary differential equations is solved. This approach is based on the approximation of the difference kernel by the sum of exponentials. Estimates for the stability of the solution concerning the initial data and the right-hand side for the corresponding Cauchy problem are obtained. Two-level schemes with weights with convenient computational implementation are constructed and investigated. The theoretical consideration is supplemented by the results of the numerical solution of the integrodifferential equation when the kernel is the stretching exponential function.