Department of Mathematics

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Browsing by Author "Dinka, Tekle Gemechu"

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    A fitted parameter convergent finite difference scheme for two-parameter singularly perturbed parabolic differential equations
    (Tamkang University, 2025) Munyakazi, Justin; Mohye, Mekashaw Ali; Dinka, Tekle Gemechu
    The objective of this paper is to develop a numerical scheme that is uniform in its parameters for a specific type of time-dependent parabolic problem with two perturbation parameters. The existence of these two parameters in the terms with the highest-order derivatives results in the formation of boundary layer(s) in the solution of such problems. Solving these model problems using classical methods does not yield satisfactory results due to the layer behavior. Therefore, nonstandard finite difference schemes have been developed as a means to obtain numerical solutions for these problems. To develop the scheme, we employ the Crank-Nicolson discretization on a uniform time mesh and apply a fitted operator method with a uniform spatial mesh. We have established the stability and convergence of the proposed scheme. The proposed scheme exhibits uniform convergence of second order in the temporal direction and first order in the spatial direction. However, temporal mesh refinements is employed to enhance the order to two in both directions.. Model examples are provided to validate the practicality of the proposed numerical scheme.
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    A new parameter-convergent nonstandard finite difference method for two-parameter singularly perturbed problems
    (Springer Nature, 2025) Munyakazi, Justin B; Mohye, Mekashaw Ali; Dinka, Tekle Gemechu
    This article focuses on the numerical solution of a time-dependent parabolic problem that exhibits singular perturbations and involves two perturbation parameters. To address this problem, a fitted mesh finite difference method is developed. In numerical discretization, the implicit Crank-Nicolson technique is employed to discretize the time derivative using a uniform mesh. As for the spatial derivative, a hybrid finite difference scheme known as the adaptive fitted mesh of the Shishkin type is utilized. The study also includes a discussion on a priori bounds for the continuous solution and its derivatives. The proposed method is proven to be uniformly convergent of order two in both time and space. Theoretical analysis and simulations on various test examples confirm the scheme’s accuracy and convergence properties