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.