A NUMERICAL SCHEME ON S-MESH FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE SOBOLEV PROBLEMS WITH LARGE TIME DELAY

SOLVING SINGULARLY PERTURBED IBV SOBOLEV PROBLEMS WITH LARGE TIME DELAY USING S-MESH METHOD

Abstract

The aim of this paper is to provide a numerical method for time delay singularly perturbed Sobolev-type equations. First, asymptotic estimates for the Sobolev problem solution with singular perturbation and delay parameters were obtained. This estimate showed that the solution depends on the initial data. On a special piecewise uniform mesh (Shishkin mesh), whose solution converges pointwise independently of the singular perturbation parameter, it is built and studied to solve this issue using the finite difference method. A discrete norm was used to investigate the stability of difference schemes. It is shown that the completely discrete scheme converges with order $%O\left( \tau ^{2}+N_{l}^{-2}\ln ^{2}N_{l}\right) $ in both space and time, independent of the perturbation parameter. Finally, with a test problem and numerical experiments, the theoretical accuracy and computational effectiveness of the proposed methods are further testified.