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http://dx.doi.org/10.14317/jami.2021.655

UNIFORMLY CONVERGENT NUMERICAL SCHEME FOR A SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS ARISING IN COMPUTATIONAL NEUROSCIENCE  

DABA, IMIRU TAKELE (Department of Mathematics, College of Natural Sciences, Wollega University)
DURESSA, GEMECHIS FILE (Department of Mathematics, College of Natural Sciences, Jimma University)
Publication Information
Journal of applied mathematics & informatics / v.39, no.5_6, 2021 , pp. 655-676 More about this Journal
Abstract
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.
Keywords
Finite difference method; implicit Euler method; differential-difference equations; singular perturbation problem;
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