An accelerated numerical computation for singularly perturbed Robin-type parabolic problems with small parameters

ABSTRACT

This study presents a computational method for solving singularly perturbed Robin-type parabolic problem with two perturbation parameters. The fitted operator method in the space direction and the implicit Euler approach in the time direction are used to discretize the problem on a uniform mesh. Furthermore, the new fitted operator method is used to discretize the Robin boundary conditions. The linear order of parameter-uniform convergence has been investigated in both the time and space variables. The application of the Richardson extrapolation approach enhanced the accuracy and rate of convergence of the present method from linear to quadratic order. On the computations of two numerical examples, the theoretical and computational results agreed upon.

KEYWORDS:

• An accelerated numerical computation
• Robin-type problem
• two parameters
• Richardson extrapolation

Public Interest Statement

Asymptotic and numerical solutions to two-parameter singularly perturbed differential equations have received relatively little study in comparison to their one-parameter counterparts. Various numerical approaches for solving singularly perturbed parabolic two-parameters problems with initial-Dirichlet boundary conditions have recently been developed. In this study, a non-standard fitted operator method in the space direction and implicit Euler method in the time direction are used to discretize a singularly perturbed parabolic two-parameters problems with initial-Robin boundary conditions. Furthermore, the new fitted operator method is used to discretize the Robin boundary conditions. The present method with Richardson extrapolation gives more accurate, stable, and uniformly convergent numerical solutions.

Acknowledgements

The authors are thanks the anonymous reviewers for their constructive comments and suggestions aimed at improving the quality of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Funding

The authors received no direct funding for this research work.

Supplementary Material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/27684830.2024.2354536

Additional information

Notes on contributors

Fasika Wondimu Gelu

Fasika Wondimu Gelu received his M.Sc. and Ph.D. degrees from the Department of Mathematics, Jimma University, Ethiopia. Since 2016 he has been a lecturer at Dilla University, Ethiopia. He is the author of (13) thirteen articles published in different national and international journals. The authors of this research work have focused on numerical solution of singularly perturbed problems of ODEs and PDEs with one-parameter and two-parameters.

Gemechis File Duressa

Gemechis File Duressa received his M.Sc. degree from Addis Ababa University, Ethiopia, and a Ph.D. degree from National Institute of Technology, Warangal, India. He is presently working at Jimma University as a full Professor of Mathematics and Vice President, administrative and development, Jimma University, Ethiopia. He is the author of more than 185 research papers published in different national and international journals. He is working as a referee for several reputable international journals. The authors of this research work have focused on numerical solution of singularly perturbed problems of ODEs and PDEs with one-parameter and two-parameters.