A uniformly convergent computational method for singularly perturbed parabolic partial differential equation with integral boundary condition

Document Type : Research Article

Authors

1 Mathematics Department, Adama Raya University, Maichew, Ethiopia

2 Department of Mathematics, Jimma University, Jimma, Ethiopia

3 Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia

4 Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia

Abstract

This paper presents a numerical method for a class of singularly perturbed parabolic partial differential equations with integral boundary conditions (IBC). The solution to the considered problem exhibits pronounced boundary layers on both the left and right sides of the spatial domain. To address this challenging problem, we propose the use of the implicit Euler method for time discretization and a finite difference method on a well-designed piecewise uniform Shishkin mesh for spatial discretization. The integral boundary condition is approximated using Simpson's $\frac{1}{3}$ rule. The presented method demonstrates almost second-order uniform convergence in the discretization of the spatial derivative and first-order convergence in the discretization of the time derivative. To validate the applicability and accuracy of the proposed method, two illustrative examples are employed. The computational results not only accurately reflect the theoretical estimations but also highlight the method's effectiveness in capturing the intricate features of singularly perturbed parabolic partial differential equations with integral boundary conditions.

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Journal of Mathematical Modeling
Volume 12, Issue 1
March 2024
Pages 157-175

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