Approximate solution of the Cauchy problem for a first-order integrodifferential equation with solution derivative memory

TL;DR

This paper develops a method to approximate solutions for a first-order integrodifferential equation with memory by transforming it into a local system, enabling stable numerical computation and analysis.

Contribution

It introduces a transformation of the integrodifferential equation into a local system using exponential kernel approximation, facilitating stable numerical schemes.

Findings

Proposed a transformation to local equations for memory-dependent problems.

Established stability estimates for the solution based on initial data.

Developed and tested two-level difference schemes for numerical implementation.

Abstract

We consider the Cauchy problem for a first-order evolution equation with memory in a finite-dimensional Hilbert space when the integral term is related to the time derivative of the solution. The main problems of the approximate solution of such nonlocal problems are due to the necessity to work with the approximate solution for all previous time moments. We propose a transformation of the first-order integrodifferential equation to a system of local evolutionary equations. We use the approach known in the theory of Voltaire integral equations with an approximation of the difference kernel by the sum of exponents. We formulate a local problem for a weakly coupled system of equations with additional ordinary differential equations. We have given estimates of the stability of the solution by initial data and the right-hand side for the solution of the corresponding Cauchy problem. The primary attention is paid to constructing and investigating the stability of two-level difference schemes, which are convenient for computational implementation. The numerical solution of a two-dimensional model problem for the evolution equation of the first order, when the Laplace operator conditions the dependence on spatial variables, is presented.