Geodesic
Uniformly Convergent Numerical Method for Singularly Perturbed Delay Parabolic Differential Equations Arising in Computational Neuroscience
Kragujevac Journal of Mathematics, Tome 46 (2022) no. 1, p. 65

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The motive of this work is to develop $\varepsilon$-uniform numerical method for solving singularly perturbed parabolic delay differential equation with small delay. To approximate the term with the delay, Taylor series expansion is used. The resulting singularly perturbed parabolic differential equation is solved by using non-standard finite difference method in spatial direction and implicit Runge-Kutta method for the resulting system of IVPs in temporal direction. Theoretically the developed method is shown to be accurate of order $O(N^{-1}+(\Delta t)^2)$ by preserving $\varepsilon$-uniform convergence. Two numerical examples are considered to investigate $ \varepsilon$-uniform convergence of the proposed scheme and the result obtained agreed with the theoretical one.
Classification : 65M06, 65M20, 65N06, 65N12
Keywords: delay differential equation, method of line, non-standard finite difference, singular perturbation 
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     author = {Mesfin Mekuria Woldaregay and Gemechis File Duressa},
     title = {Uniformly {Convergent} {Numerical} {Method} for {Singularly} {Perturbed} {Delay} {Parabolic} {Differential} {Equations} {Arising} in {Computational} {Neuroscience}},
     journal = {Kragujevac Journal of Mathematics},
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     publisher = {mathdoc},
     volume = {46},
     number = {1},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KJM_2022_46_1_a5/}
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Mesfin Mekuria Woldaregay; Gemechis File Duressa. Uniformly Convergent Numerical Method for Singularly Perturbed Delay Parabolic Differential Equations Arising in Computational Neuroscience. Kragujevac Journal of Mathematics, Tome 46 (2022) no. 1, p. 65 . http://geodesic.mathdoc.fr/item/KJM_2022_46_1_a5/