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WOLDAREGAY, MESFIN MEKURIA,DURESSA, GEMECHIS FILE The Korean Society for Computational and Applied M 2021 Journal of applied mathematics & informatics Vol.39 No.5
In this paper, numerical treatment of singularly perturbed parabolic delay differential equations is considered. The considered problem have small delay on the spatial variable of the reaction term. To treat the delay term, Taylor series approximation is applied. The resulting singularly perturbed parabolic PDEs is solved using Crank Nicolson method in temporal direction with non-standard finite difference method in spatial direction. A detail stability and convergence analysis of the scheme is given. We proved the uniform convergence of the scheme with order of convergence O(N<sup>-1</sup> + (∆t)<sup>2</sup>), where N is the number of mesh points in spatial discretization and ∆t is mesh length in temporal discretization. Two test examples are used to validate the theoretical results of the scheme.
ANGASU, MERGA AMARA,DURESSA, GEMECHIS FILE,WOLDAREGAY, MESFIN MEKURIA The Korean Society for Computational and Applied M 2021 Journal of applied mathematics & informatics Vol.39 No.3
This paper deals with numerical treatment of singularly perturbed differential equations involving small delays. The highest order derivative in the equation is multiplied by a perturbation parameter 𝜀 taking arbitrary values in the interval (0, 1]. For small 𝜀, the problem involves a boundary layer of width O(𝜀), where the solution changes by a finite value, while its derivative grows unboundedly as 𝜀 tends to zero. The considered problem contains delay on the convection and reaction terms. The terms with the delays are approximated using Taylor series approximations resulting to asymptotically equivalent singularly perturbed BVPs. Inducing exponential fitting factor for the term containing the singular perturbation parameter and using central finite difference for the derivative terms, numerical scheme is developed. The stability and uniform convergence of difference schemes are studied. Using a priori estimates we show the convergence of the scheme in maximum norm. The scheme converges with second order of convergence for the case 𝜀 = O(N<sup>-1</sup>) and for the case 𝜀 ≪ N<sup>-1</sup>, the scheme converge uniformly with first order of convergence, where N is number of mesh intervals in the domain discretization. We compare the accuracy of the developed scheme with the results in the literature. It is found that the proposed scheme gives accurate result than the one in the literatures.
Uniformly Convergent Numerical Method for Singularly Perturbed Convection-Diffusion Problems
Derartu Ayansa Turuna,Mesfin Mekuria Woldaregay,Gemechis File Duressa 경북대학교 자연과학대학 수학과 2020 Kyungpook mathematical journal Vol.60 No.3
A uniformly convergent numerical method is developed for solving singularly perturbed 1-D parabolic convection-diffusion problems. The developed method applies a non-standard finite difference method for the spatial derivative discretization and uses the implicit Runge-Kutta method for the semi-discrete scheme. The convergence of the method is analyzed, and it is shown to be first order convergent. To validate the applicability of the proposed method two model examples are considered and solved for different perturbation parameters and mesh sizes. The numerical and experimental results agree well with the theoretical findings.
Anilay, Worku Tilahun,Duressa, Gemechis File,Woldaregay, Mesfin Mekuria Department of Mathematics 2021 Kyungpook mathematical journal Vol.61 No.3
In this paper, a uniformly convergent numerical scheme is designed for solving singularly perturbed reaction-diffusion problems. The problem is converted to an equivalent weak form and then a Galerkin finite element method is used on a piecewise uniform Shishkin mesh with linear basis functions. The convergence of the developed scheme is proved and it is shown to be almost fourth order uniformly convergent in the maximum norm. To exhibit the applicability of the scheme, model examples are considered and solved for different values of a singular perturbation parameter ε and mesh elements. The proposed scheme approximates the exact solution very well.