Parameter-Uniform Numerical Scheme for Singularly Perturbed Delay Parabolic Reaction Diffusion Equations with Integral Boundary Condition

Wakjira Tolassa Gobena,Gemechis File Duressa,Andrea Scapellato

doi:10.1155/2021/9993644

Wakjira Tolassa Gobena, Gemechis File Duressa + Show 1 more

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https://doi.org/10.1155/2021/9993644

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Abstract

Numerical computation for the class of singularly perturbed delay parabolic reaction diffusion equations with integral boundary condition has been considered. A parameter-uniform numerical method is constructed via the nonstandard finite difference method for the spatial direction, and the backward Euler method for the resulting system of initial value problems in temporal direction is used. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and the rate of convergence for different values of perturbation parameter ε and mesh sizes are tabulated for two model examples. The proposed method is shown to be parameter-uniformly convergent.

Highlights

Perturbed delay differential equations model physical problems for which the evaluation depends on the present state of the system and on the past history
It is known that standard numerical methods for solving singular perturbation problems are unstable and fail to give accurate results when the perturbation parameter is small
E accuracy and convergence of the methods need attention, because the treatment of singular perturbation problems is not trivial, and the solution depends on perturbation parameter and mesh size h [6]. is suggests that the numerical treatment should be improved. e presence of the singular perturbation parameter leads to occurrences of oscillations or divergence in the computed solutions while using standard numerical methods [7]

Summary

Introduction

Perturbed delay differential equations model physical problems for which the evaluation depends on the present state of the system and on the past history. E accuracy and convergence of the methods need attention, because the treatment of singular perturbation problems is not trivial, and the solution depends on perturbation parameter and mesh size h [6]. E presence of the singular perturbation parameter leads to occurrences of oscillations or divergence in the computed solutions while using standard numerical methods [7] To overcome these oscillations or divergence, a large number of mesh points are required as ε goes to zero. A second-order fitted operator finite difference method for reaction diffusion type equation

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