A Locally One-Dimensional Difference Scheme for a Multidimensional Integro-Differential Equation of Parabolic Type of General Form

Abstract

The first boundary value problem for a multidimensional integro-differential equation of parabolic type of general form with variable coefficients is investigated. To solve numerically the multidimensional problem, a locally one-dimensional difference scheme is constructed, the essence of the idea of which is to reduce the transition from layer to layer to sequential solving of a number of one-dimensional problems in each of the coordinate directions. It is shown that the approximation error for the locally one-dimensional scheme is \(O(|h{|}^{2}+{\tau }^{2})\), where \(|h{|}^{2}={h}_{1}^{2},{h}_{2}^{2},...,{h}_{p}^{2}\). Using the method of energy inequalities in the \({L}_{2}-\)norm, for the solution of a locally one-dimensional difference scheme, an a priori estimate is obtained. The obtained estimate implies uniqueness, stability with respect to the right-hand side and initial data, as well as the convergence of the solution of the difference problem to the solution of the original differential problem with a rate equal to the approximation error. In the two-dimensional case (for p = 2), an algorithm for finding the approximate solution of the problem under consideration is constructed and numerical calculations of test examples are carried out, illustrating the theoretical calculations obtained in the work.KeywordsFirst initial-boundary value problemLocally one-dimensional schemeA priori estimateDifference schemeParabolic equationIntegro-differential equationEquation with memory