American Journal of Numerical Analysis
ISSN (Print): 2372-2118 ISSN (Online): 2372-2126 Website: http://www.sciepub.com/journal/ajna Editor-in-chief: Emanuele Galligani
Open Access
Journal Browser
Go
American Journal of Numerical Analysis. 2014, 2(6), 184-189
DOI: 10.12691/ajna-2-6-3
Open AccessArticle

A Fitted Second Order Finite Difference Method for Singular Perturbation Problems Exhibiting Dual Layers

H.S. Prasad1 and Y.N. Reddy2,

1Department of Mathematics, National Institute of Technology, Jamshedpur, INDIA

2Department Mathematics, National Institute of Technology, Warangal, INDIA

Pub. Date: December 29, 2014

Cite this paper:
H.S. Prasad and Y.N. Reddy. A Fitted Second Order Finite Difference Method for Singular Perturbation Problems Exhibiting Dual Layers. American Journal of Numerical Analysis. 2014; 2(6):184-189. doi: 10.12691/ajna-2-6-3

Abstract

In this paper a fitted second-order finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at both end (left and right) points. We have introduced a fitting factor in second-order tri-diagonal finite difference scheme and it is obtained from the theory of singular perturbations. The efficient Thomas algorithm is used to solve the tri-diagonal system. Maximum absolute errors are presented in tables to show the efficiency of the method.

Keywords:
singular perturbation problems Boundary layer dual layer Finite differences fitted method

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers, Mc. Graw-Hill, . 1978.
 
[2]  Doolan, E.P., Miller, J.J.H., Schilders, W.H.A.: Uniform Numerical methods for problems with initial and boundary layers, Boole Press, Dublin. 1980.
 
[3]  Hemker, P.W., Miller, J.J.H. (Editors).: Numerical Analysis of Singular Perturbation Problems, Academic Press, New York, 1978.
 
[4]  Jain, M.K.: Numerical solution of differential equations, 2nd Ed., Wiley Eastern Ltd., New Delhi 1984.
 
[5]  Kevorkian, J., Cole, J.D.: Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981.
 
[6]  Kadalbajoo, M.K., Reddy, Y.N.: Asymptotic and Numerical Analysis of Singular Perturbation Problems: A Survey, Applied Mathematics and Computation, 30: 223-259, 1989.
 
[7]  Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical methods for singular perturbation problems, Error estimates in the maximum norm for linear problems in one and two dimensions, World Scientific Publishing Company Pvt. Ltd. 1996.
 
[8]  Nayfeh, A.H.: Perturbation Methods, Wiley, New York. 1973.
 
[9]  O’ Malley, R.E.: Introduction to Singular Perturbations, Academic Press, New York, 1974.
 
[10]  Phaneendra, K., Pramod Chakravarthy, P., Reddy, Y. N.: A Fitted Numerov Method for Singular Perturbation Problems Exhibiting Twin Layers, Applied Mathematics & Information Sciences – An International Journal, Dixie W Publishing Corporation, U.S. A., 4 (3): 341-352, 2010.
 
[11]  Reddy, Y.N. (1986). Numerical Treatment of Singularly Perturbed Two Point Boundary Value Problems, Ph.D. thesis, IIT, Kanpur, India. 1986.
 
[12]  Reddy Y.N., Pramod Chakravarthy, P. (2004). An exponentially fitted finite difference method for singular perturbation problems, Applied Mathematics and Computation, 154: 83-101 2004.