A novel algorithm for singularly perturbed parabolic differential-difference equations. Issue 1 (31st December 2022)

• Record Type: Journal Article
• Authors: Daba, Imiru Takele; Duressa, Gemechis File
• Editors: Liu, Lishan
• Is Part Of: Research in Mathematics. Volume 9:Issue 1(2022)
• Publication Date: 2022-12-31
• Subjects: Singularly perturbed -- differential-difference equations -- cubic spline in tension method -- implicit Euler method -- uniformly convergent computational scheme
• DOI: 10.1080/27684830.2022.2133211
• Languages: English
• ISSNs: 2768-4830

Abstract

ABSTRACT: In this study, numerical treatment of singularly perturbed parabolic partial differential-difference equations with small mixed shifts arising in computational neuroscience is presented. Due to the simultaneous presence of a singular perturbation parameter and shifts in the problem applying classical numerical methods to solve such a problem on a uniform mesh are unable to provide oscillation-free results unless they are applied with very fine meshes inside the region. The main objective of the work is to develop and analyze an oscillation-free higher order numerical scheme, which plays a vital role in applications. A uniformly convergent computational scheme is proposed using the implicit Euler method in temporal discretization and the cubic spline in tension method in spatial discretization. The stability and convergence analysis of the method are investigated. The scheme is uniformly convergent with linear order of convergence before Richardson extrapolation and second-order convergent after Richardson extrapolation technique. Three test problems are considered to validate the applicability of the scheme. The efficiency of the proposed scheme is demonstrated by comparing some methods available in the literature. It is observed that the proposed scheme provides more accurate results and higher order of convergence than methods available in the literature.