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.


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How to Cite
CHIYANEH, A. B.; DURU, H.. A NUMERICAL SCHEME ON S-MESH FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE SOBOLEV PROBLEMS WITH LARGE TIME DELAY. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 117, n. 1, apr. 2023. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/1245>. Date accessed: 08 june 2023. doi: https://doi.org/10.26577/JMMCS.2023.v117.i1.08.
Keywords Delayed partial differential equation, Finite difference method, Shishkin mesh, Singular perturbation, Sobolev problem