Numerical Solution of the Cauchy Problem for a Second-Order Integro-Differential Equation

• Published in: Differential equations Vol. 58; no. 7; pp. 899 - 907
• Main Author: Vabishchevich, P. N.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-07-2022; Springer; Springer Nature B.V
• Subjects: Backup software; Cauchy problems; Difference and Functional Equations; Differential equations; Evolutionary computation; Hilbert space; Kernels; Linear operators; Mathematical analysis; Mathematics; Mathematics and Statistics; Numerical Methods; Operators (mathematics); Ordinary Differential Equations; Partial Differential Equations; Stability analysis
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S0012266122070047

In a finite-dimensional Hilbert space, we consider the Cauchy problem for a second-order integro-differential evolution equation with memory where the integrand is the product of a difference kernel by a linear operator of the time derivative of the solution. The main difficulties in finding the approximate value of the solution of such nonlocal problems at a given point in time are due to the need to work with approximate values of the solution for all previous points in time. A transformation of the integro-differential equation in question to a system of weakly coupled local evolution equations is proposed. It is based on the approximation of the difference kernel by a sum of exponentials. We state a local problem for a weakly coupled system of equations with additional ordinary differential equations. To solve the corresponding Cauchy problem, stability estimates of the solution with respect to the initial data and the right-hand side are given. The main attention is paid to the construction and stability analysis of three-level difference schemes and their computational implementation.