Extended cubic B-spline collocation method for singularly perturbed parabolic differential-difference equation arising in computational neuroscience.

Abstract

A parameter uniform numerical method is presented for solving singularly perturbed parabolic differential-difference equations with small shift arguments in the reaction terms arising in computational neuroscience. To approximate the terms with the shift arguments, Taylor's series expansion is used. The resulting singularly perturbed parabolic differential equation is solved by applying the implicit Euler method in temporal direction and extended cubic B-spline basis functions consisting of a free parameter λ for the resulting system of ordinary differential equations in the spatial direction. The proposed method is shown to be accurate of order by preserving an ε- uniform convergence. To demonstrate the applicability of the proposed method, two test examples are solved by the method and the numerical results are compared with some existing results. The obtained numerical results agreed with the theoretical results.