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Extended cubic B‐spline collocation method for singularly perturbed parabolic differential‐difference equation arising in computational neuroscience
Author(s) -
Daba Imiru Takele,
Duressa Gemechis File
Publication year - 2021
Publication title -
international journal for numerical methods in biomedical engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.741
H-Index - 63
eISSN - 2040-7947
pISSN - 2040-7939
DOI - 10.1002/cnm.3418
Subject(s) - mathematics , collocation method , mathematical analysis , ordinary differential equation , differential equation , explicit and implicit methods , collocation (remote sensing) , order of accuracy , taylor series , numerical analysis , euler method , partial differential equation , convergence (economics) , euler's formula , numerical partial differential equations , computer science , machine learning , economic growth , economics
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 O Δ t + h 2 ε + hby 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.