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

Abstract

The purpose of this article 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. It is constructed and examined to solve this problem using a finite difference technique on a specific piecewise uniform mesh (Shishkin mesh)whose solution converges pointwise independent of the singular perturbation parameter. A discrete norm was used to investigate the stability of difference schemes. It is showed 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.