Uniformly Convergent Numerical Method for Singularly Perturbed Delay Parabolic Differential Equations Arising in Computational Neuroscience

Abstract

The motive of this work is to develop ε-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 + (∆t) 2 ) by preserving ε-uniform convergence. Two numerical examples are considered to investigate εuniform convergence of the proposed scheme and the result obtained agreed with the theoretical one

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