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

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**Authors:**M. M. WOLDAREGAY AND G. F. DURESSA

**DOI:**10.46793/KgJMat2201.065W

**Abstract:**

The motive of this work is to develop ????-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}+ (Δt)

^{2}) by preserving ????-uniform convergence. Two numerical examples are considered to investigate ????-uniform convergence of the proposed scheme and the result obtained agreed with the theoretical one.

**Keywords:**

Delay diﬀerential equation, method of line, non-standard ﬁnite diﬀerence, singular perturbation.

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