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Fitted Mesh Method for a Class of Singularly PerturbedDifferential-Difference Equations

Part of: Numerical approximation and computational geometry Functional-differential and differential-difference equations

Published online by Cambridge University Press:  10 November 2015

Devendra Kumar
Devendra Kumar*
Affiliation:
Department of Mathematics, Birla Institute of Technology & Science, Pilani, Rajasthan-333031, India
*
*Corresponding author. Email address:dkumar2845@gmail.com (Devendra Kumar)

Abstract

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This paper deals with a more general class of singularly perturbed boundary valueproblem for a differential-difference equations with small shifts. Inparticular, the numerical study for the problems where second order derivativeis multiplied by a small parameter ε and the shifts depend on thesmall parameter ε has been considered. The fitted-mesh technique isemployed to generate a piecewise-uniform mesh, condensed in the neighborhood ofthe boundary layer. The cubic B-spline basis functions with fitted-mesh areconsidered in the procedure which yield a tridiagonal system which can besolved efficiently by using any well-known algorithm. The stability andparameter-uniform convergence analysis of the proposed method have beendiscussed. The method has been shown to have almost second-orderparameter-uniform convergence. The effect of small parameters on the boundarylayer has also been discussed. To demonstrate the performance of the proposedscheme, several numerical experiments have been carried out.

Keywords

Singular perturbationdifferential-difference equationsfitted-meshB-spline collocation methodboundary layer

MSC classification

Secondary: 34K10: Boundary value problems 34K28: Numerical approximation of solutions 65D05: Interpolation

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 4 , November 2015 , pp. 496 - 514
Copyright
Copyright © Global-Science Press 2015 

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