A uniformly convergent numerical method on non-uniform mesh for singularly perturbed unsteady Burger–Huxley equation

Purpose The purpose of this paper is to design and analyze a robust numerical method for a coupled system of singularly perturbed parabolic delay partial differential equations (PDEs). Design/methodology/approach Some a priori bounds on the regular and layer parts of the solution and their derivatives are derived. Based on these a priori bounds, appropriate layer adapted meshes of Shishkin and generalized Shishkin types are defined in the spatial direction. After that, the problem is discretized using an implicit Euler scheme on a uniform mesh in the time direction and the central difference scheme on layer adapted meshes of Shishkin and generalized Shishkin types in the spatial direction. Findings The method is proved to be robust convergent of almost second-order in space and first-order in time. Numerical results are presented to support the theoretical error bounds. Originality/value A coupled system of singularly perturbed parabolic delay PDEs is considered and some a priori bounds are derived. A numerical method is developed for the problem, where appropriate layer adapted Shishkin and generalized Shishkin meshes are considered. Error analysis of the method is given for both Shishkin and generalized Shishkin meshes.

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Uniformly convergent numerical method for singularly perturbed 2D delay parabolic convection-diffusion problems on Bakhvalov-Shishkin mesh

International Journal of Mathematical Modelling and Numerical Optimisation ◽

10.1504/ijmmno.2018.10015799 ◽

2018 ◽

Vol 8 (4) ◽

pp. 305

Author(s):

Abhishek Das ◽

Srinivasan Natesan

Keyword(s):

Numerical Method ◽

Shishkin Mesh ◽

Singularly Perturbed ◽

Convection Diffusion ◽

Uniformly Convergent ◽

Diffusion Problems

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Exponentially Fitted Numerical Method for Singularly Perturbed Differential-Difference Equations

International Journal of Differential Equations ◽

10.1155/2020/5768323 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Habtamu Garoma Debela ◽

Solomon Bati Kejela ◽

Ayana Deressa Negassa

Keyword(s):

Numerical Method ◽

Difference Equations ◽

Oscillatory Behavior ◽

Singularly Perturbed ◽

Stability And Convergence ◽

Uniform Mesh ◽

Small Shift ◽

Exponentially Fitted ◽

Maximum Absolute Errors ◽

The Stability

This paper presents a numerical method to solve singularly perturbed differential-difference equations. The solution of this problem exhibits layer or oscillatory behavior depending on the sign of the sum of the coefficients in reaction terms. A fourth-order exponentially fitted numerical scheme on uniform mesh is developed. The stability and convergence of the proposed method have been established. The effect of delay parameter (small shift) on the boundary layer(s) has also been analyzed and depicted in graphs. The applicability of the proposed scheme is validated by implementing it on four model examples. Maximum absolute errors in comparison with the other numerical experiments are tabulated to illustrate the proposed method.

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