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Fitted Difference Scheme on a Non-uniform Mesh for Singularly Perturbed Parabolic Reaction–Diffusion with Large Negative Shift and Non-local Boundary Condition

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Abstract

In this paper, we study the numerical solution of singularly perturbed parabolic reaction–diffusion problems with large delay in space, and the right end plane is non-local boundary condition. As the perturbation parameter approaches zero, the solution to this problem exhibits a parabolic boundary layers and an interior layer have been exhibited in the solution domain. To solve these problems, we develop a numerical scheme which combines the cubic spline scheme for the spatial derivatives, and backward difference scheme for the time derivative. To resolve the boundary layers, we use the piecewise uniform Shishkin types mesh (Standard Shishkin mesh, Bakhvalov–Shishkin mesh) for the spatial discretization. To treat the non-local boundary condition,numerical integration method is applied. A priori bounds for the solution and its derivatives of the continuous problem are given, which are necessary to analyze the error. Stability analysis and error estimates are obtained. Some numerical results are considered to support our theoretical result, which shows the \(\varepsilon \)-uniform convergent results.

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Acknowledgements

The authors wish to express their thanks to Jimma University, College of Natural Sciences, for financial support and the authors of the literature for the provided scientific aspects and idea for this work.

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Author notes

  1. Gemechis File Duressa and Lakshmi Sireesha Challa have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Nekemte College of Teacher Education, 88, Nekemte, Oromia, Ethiopia

    Wakjira Tolassa Gobena & Lakshmi Sireesha Challa

  2. Department of Mathematics, College of Natural Science, Jimma University, 378, Jimma, Oromia, Ethiopia

    Wakjira Tolassa Gobena, Gemechis File Duressa & Lakshmi Sireesha Challa

  3. Department of Mathematics, University College of Science, Saifabad Osmania University, CBIT, Hyderabad, India

    Gemechis File Duressa

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  1. Wakjira Tolassa Gobena
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Correspondence to Wakjira Tolassa Gobena.

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Gobena, W.T., Duressa, G.F. & Challa, L.S. Fitted Difference Scheme on a Non-uniform Mesh for Singularly Perturbed Parabolic Reaction–Diffusion with Large Negative Shift and Non-local Boundary Condition. Int. J. Appl. Comput. Math 9, 109 (2023). https://doi.org/10.1007/s40819-023-01553-z

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