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ORIGINAL RESEARCH article

Front. Appl. Math. Stat.
Sec. Numerical Analysis and Scientific Computation
Volume 10 - 2024 | doi: 10.3389/fams.2024.1414899
This article is part of the Research Topic Robust Numerical Schemes for Singularly Perturbed Differential Problems View all articles

Numerical Integration Method for Two-Parameter Singularly Perturbed Time Delay Parabolic Problem

Provisionally accepted
Tariku Birabasa Mekonnen Tariku Birabasa Mekonnen 1*Gemechis File Duressa Gemechis File Duressa 2Shegaye Lema Cheru Shegaye Lema Cheru 1
  • 1 Wollega University, Nekemte, Ethiopia
  • 2 Jimma University, Jimma, Oromia, Ethiopia

The final, formatted version of the article will be published soon.

    This paper presents an (ε, µ)-uniform numerical method for a two parameter singularly perturbed time delayed parabolic problems. The proposed approach is based on a fitted operator finite difference method. The Crank-Nicolson method is employed on a uniform mesh to discretize the time variables initially. Subsequently, the resulting semi-discrete scheme is further discretized in space using Simpson's 1/3rd rule. Finally, the finite difference approximation of the first derivatives is applied. The method is unique in that it is not dependent on delay terms, asymptotic expansions, or fitted meshes. The fitting factor's value, which is used to account for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second order accurate and uniformly convergent. The proposed method's applicability is validated by three model examples, which yielded more accurate results than some other methods found in the literature.

    Keywords: Singularly perturbed problems, Delay parabolic differential equation, numerical integration, Simpson's 1/3rd rule, Parameter uniform

    Received: 09 Apr 2024; Accepted: 28 Jun 2024.

    Copyright: © 2024 Mekonnen, Duressa and Cheru. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

    * Correspondence: Tariku Birabasa Mekonnen, Wollega University, Nekemte, Ethiopia

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