Abstract
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.
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Translated by V. Potapchouck
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Vabishchevich, P.N. Numerical Solution of the Cauchy Problem for a Second-Order Integro-Differential Equation. Diff Equat 58, 899–907 (2022). https://doi.org/10.1134/S0012266122070047
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DOI: https://doi.org/10.1134/S0012266122070047