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Uniformly convergent higher-order finite difference scheme for singularly perturbed parabolic problems with non-smooth data

Bullo Tesfaye Aga, Degla Guy Aymard, Duressa Gemechis File

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 1

5-16

EN

A uniformly convergent higher-order finite difference scheme is constructed and analyzed for solving singularly perturbed parabolic problems with non-smooth data. This scheme involves an average non-standard finite difference with the Richardson extrapolation method for space variables and second-order finite difference approximation for time direction on uniform meshes. The scheme is shown to be second-order convergent in both temporal and spatial directions. Further, the scheme is proven to be uniformly convergent and also confirmed by numerical experiments. Wide numerical experiments are conducted to support the theoretical results and to demonstrate its accuracy. Concisely, the present scheme is stable, convergent, and more accurate than existing methods in the literatur

2

Robust numerical method for singularly perturbed differential equations with large delay

Abdulla Murad Ibrahim, Duressa Gemechis File, Debela Habtamu Garoma

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

576--589

EN

In this paper, a singularly perturbed differential equation with a large delay is considered. The considered problem contains a large delay parameter on the reaction term. The solution of the problem exhibits the interior layer due to the delay parameter and the strong right boundary layer due to the small perturbation parameter ε. The resulting singularly perturbed problem is solved using the fitted non-polynomial spline method. The stability and parameter uniform convergence of the proposed method is proved. To validate the applicability of the scheme, two model problems of the variable coefficient are considered for numerical experimentation.

3

Exponential spline method for singularly perturbed third-order boundary value problems

Wakjira Yohannis Alemayehu, Duressa Gemechis File

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

360--372

EN

The exponential spline function is presented to find the numerical solution of third-order singularly perturbed boundary value problems. Convergence analysis of the method is briefly discussed, and it is shown to be sixth order convergence. To validate the applicability of the method, some model problems are solved for different values of the perturbation parameter, and the numerical results are presented both in tables and graphs. Furthermore, the present method gives more accurate solution than some methods existing in the literature.