Numerical solution of initial-boundary value problems for a multi-dimensional pseudoparabolic equation

We consider initial boundary value problems for a multi-dimensional pseudoparabolic equation with Dirichlet boundary conditions of a special form. For an approximate solution of the considered problems, the multi-dimensional pseudoparabolic equation is reduced to an integro-differential equation with a small parameter. It is shown that as the small parameter tends to zero, the solution of the corresponding modified problem converges to the solution of the original problem. For each of the problems we construct a locally one-dimensional difference scheme following A.A. Samarskii. The main idea is to reduce the transition from a layer to a layer to the sequential solving a number of one-dimensional problems in each of the coordinate directions. Using the maximum principle, we obtain apriori estimates, which imply the uniqueness, stability, and convergence of the solution of a locally one-dimensional difference scheme in the uniform metric. We construct an algorithm for numerical solving of the modified problem with conditions of a special form.