Volume 8, Issue 4
Fitted Mesh Method for a Class of Singularly Perturbed Differential-Difference Equations

Devendra Kumar

Numer. Math. Theor. Meth. Appl., 8 (2015), pp. 496-514.

Published online: 2015-08

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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.

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@Article{NMTMA-8-496, author = {}, title = { Fitted Mesh Method for a Class of Singularly Perturbed Differential-Difference Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2015}, volume = {8}, number = {4}, pages = {496--514}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2015.my14005}, url = {http://global-sci.org/intro/article_detail/nmtma/12420.html} }
TY - JOUR T1 - Fitted Mesh Method for a Class of Singularly Perturbed Differential-Difference Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 496 EP - 514 PY - 2015 DA - 2015/08 SN - 8 DO - http://doi.org/10.4208/nmtma.2015.my14005 UR - https://global-sci.org/intro/article_detail/nmtma/12420.html KW - AB -

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.

Devendra Kumar. (2020). Fitted Mesh Method for a Class of Singularly Perturbed Differential-Difference Equations. Numerical Mathematics: Theory, Methods and Applications. 8 (4). 496-514. doi:10.4208/nmtma.2015.my14005
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