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Gemechis File Duressa,Tariku Birabasa Mekonnen Korean Mathematical Society 2023 대한수학회논문집 Vol.38 No.1
This article devises an exponentially fitted method for the numerical solution of two parameter singularly perturbed parabolic boundary value problems. The proposed scheme is able to resolve the two lateral boundary layers of the solution. Error estimates show that the constructed scheme is parameter-uniformly convergent with a quadratic numerical rate of convergence. Some numerical test examples are taken from recently published articles to confirm the theoretical results and demonstrate a good performance of the current scheme.
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.
GELU, FASIKA WONDIMU,DURESSA, GEMECHIS FILE The Korean Society for Computational and Applied M 2022 Journal of applied mathematics & informatics Vol.40 No.1-2
In this study, the non-standard finite difference method for the numerical solution of singularly perturbed parabolic reaction-diffusion subject to Robin boundary conditions has presented. To discretize temporal and spatial variables, we use the implicit Euler and non-standard finite difference method on a uniform mesh, respectively. We proved that the proposed scheme shows uniform convergence in time with first-order and in space with second-order irrespective of the perturbation parameter. We compute three numerical examples to confirm 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.
DABA, IMIRU TAKELE,DURESSA, GEMECHIS FILE The Korean Society for Computational and Applied M 2021 Journal of applied mathematics & informatics Vol.39 No.5
A parameter uniform numerical scheme is proposed for solving singularly perturbed parabolic partial differential-difference convection-diffusion equations with a small delay and advance parameters in reaction terms and spatial variable. Taylor's series expansion is applied to approximate problems with the delay and advance terms. The resulting singularly perturbed parabolic convection-diffusion equation is solved by utilizing the implicit Euler method for the temporal discretization and finite difference method for the spatial discretization on a uniform mesh. The proposed numerical scheme is shown to be an ε-uniformly convergent accurate of the first order in time and second-order in space directions. The efficiency of the scheme is proved by some numerical experiments and by comparing the results with other results. It has been found that the proposed numerical scheme gives a more accurate approximate solution than some available numerical methods in the literature.
Higher order Galerkin finite element method for the generalized diffusion PDE with delay
Gemeda Tolessa Lubo,Gemechis File Duressa 한국전산응용수학회 2022 Journal of applied mathematics & informatics Vol.40 No.3
In this paper, a numerical solution of the generalized diffusion equation with a delay has been obtained by a numerical technique based on the Galerkin finite element method by applying the cubic B-spline basis functions. The time discretization process is carried out using the forward Euler method. The numerical scheme is required to preserve the delay-independent asymptotic stability with an additional restriction on time and spatial step sizes. Both the theoretical and computational rates of convergence of the numerical method have been examined and found to be in agreement. As it can be observed from the numerical results given in tables and graphs, the proposed method approximates the exact solution very well. The accuracy of the numerical scheme is confirmed by computing L_{2} and L_{\infty} error norms.
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.