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Fitted Mesh Method for a Class of Singularly Perturbed Differential-Difference Equations

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

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:[email protected] (Devendra Kumar)
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Abstract

This paper deals with a more general class of singularly perturbed boundary value problem for a differential-difference equations with small shifts. In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter ε and the shifts depend on the small parameter ε has been considered. The fitted-mesh technique is employed to generate a piecewise-uniform mesh, condensed in the neighborhood of the boundary layer. The cubic B-spline basis functions with fitted-mesh are considered in the procedure which yield a tridiagonal system which can be solved efficiently by using any well-known algorithm. The stability and parameter-uniform convergence analysis of the proposed method have been discussed. The method has been shown to have almost second-order parameter-uniform convergence. The effect of small parameters on the boundary layer has also been discussed. To demonstrate the performance of the proposed scheme, 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
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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