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

H.S. Prasad¹ and Y.N. Reddy^2,

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

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

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

This 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.