Local Computational Algorithms for a System of First-Order Equations with Memory Effects

Abstract

We consider the Cauchy problem for a system of first-order integro-differential equationswith difference kernels in a finite-dimensional Hilbert space. This class of equations arises in themathematical modeling of a wide range of nonstationary processes taking into account memoryeffects, including, in particular, the system of Maxwell equations. For the numerical solution,a method of reducing the original nonlocal problem to an equivalent system of local first-orderdifferential equations on the basis of the approximation of kernels by a finite sum of exponentialfunctions is proposed. Two-level operator-difference schemes are proposed, for which the stabilitywith respect to the initial data and the right-hand side is analyzed. The theoretical analysisperformed demonstrates the well-posedness of the approach proposed.