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Extended B-Spline Collocation Method for Singularly Perturbed Time Delay Parabolic Differential-Difference Problems

Imiru Takele Daba

https://orcid.org/0000-0002-7822-9679

Department of Mathematics, College of Natural Sciences, Salale University, Fitche 245, Ethiopia

E-mail Address: imirutakele@gmail.com

Corresponding author.

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Nayilot Ejara Kebede

Department of Mathematics, College of Natural Sciences, Salale University, Fitche 245, Ethiopia

E-mail Address: nailot@gmail.com

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, and

Genanew Gofe Gonfa

Department of Mathematics, College of Natural Sciences, Salale University, Fitche 245, Ethiopia

E-mail Address: genagof@gmail.com

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https://doi.org/10.1142/S0219876224500464Cited by:0 (Source: Crossref)

Abstract

The paper presents a robust collocation B-spline numerical scheme for solving singularly perturbed time lag parabolic differential–difference problems. The method uses Taylor’s series expansion for the approximation of retarded terms, the $𝜃$ $𝜃$ -method (when $𝜃 = 1$ $𝜃 = 1$ ) for time discretization, and the extended third-degree B-spline collocation method for spatial discretization in a piecewise uniform Shishkin mesh. The advantage of using an extended cubic B-spline rather than the classical third-degree B-spline method is that it introduces one additional free parameter to control the global shape parameter. The stability and uniform convergence of the proposed scheme are investigated. The method is first-order accurate in $t$ $t$ and almost second-order accurate in $x$ $x$ , surpassing existing literature methods.

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