Extended cubic B‐spline collocation method for singularly perturbed parabolic differential‐difference equation arising in computational neuroscience,International Journal for Numerical Methods in Biomedical Engineering

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Extended cubic B‐spline collocation method for singularly perturbed parabolic differential‐difference equation arising in computational neuroscience
International Journal for Numerical Methods in Biomedical Engineering ( IF 2.2 ) Pub Date : 2020-11-22 , DOI: 10.1002/cnm.3418
Imiru Takele Daba ₁ , Gemechis File Duressa ₂

Affiliation

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

O (Δ t + \frac{h^{2}}{ε + h})

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.

中文翻译：

计算神经科学中奇异微扰抛物线微分方程的扩展三次B样条搭配方法

提出了一种参数统一数值方法，用于求解计算神经科学中出现的反应项中具有小位移参数的奇异扰动抛物线微分差分方程。为了用移位参数来近似这些项，使用了泰勒级数展开式。通过在时间方向应用隐式欧拉方法和扩展三次 B 样条基函数求解得到的奇异微扰抛物线微分方程，该函数由一个自由参数λ组成，用于生成的常微分方程系统在空间方向。所提出的方法被证明是准确的