[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.14317/jami.2011.29.3_4.937

THE METHOD OF ASYMPTOTIC INNER BOUNDARY CONDITION FOR SINGULAR PERTURBATION PROBLEMS

Andargie, Awoke (Department of Mathematics, Bahir dar University)
Reddy, Y.N. (Department of Mathematics, National Institute of Technology)

Publication Information

Journal of applied mathematics & informatics / v.29, no.3_4, 2011 , pp. 937-948 More about this Journal

Abstract

The method of Asymptotic Inner Boundary Condition for Singularly Perturbed Two-Point Boundary value Problems is presented. By using a terminal point, the original second order problem is divided in to two problems namely inner region and outer region problems. The original problem is replaced by an asymptotically equivalent first order problem and using the stretching transformation, the asymptotic inner condition in implicit form at the terminal point is determined from the reduced equation of the original second order problem. The modified inner region problem, using the transformation with implicit boundary conditions is solved and produces a condition for the outer region problem. We used Chawla's fourth order method to solve both the inner and outer region problems. The proposed method is iterative on the terminal point. Some numerical examples are solved to demonstrate the applicability of the method.

Keywords

Singular perturbation problems; Finite Differences; Asymptotic Inner Boundary Condition; Terminal point;

Citations & Related Records

Reference

1	L.Wang: A novel method for a class of nonlinear Singular Perturbation Problems, Applied Mathematics and Computation, 156(2004),847-856. DOI ScienceOn
2	M.Van Dyk:Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964.
3	J.Vigo-aguiar and S.Natesan: A Parallel Boundary Value Technique for Singularly Per- turbed Two-Point Boundary Value Problems, The journal of Supercomputing, Volume 27(2004), No. 2, , pp.195-206. DOI
4	A.H.Nayfeh : Perturbation Methods, Wiley, New York, 1979.
5	E.J.Hinch: Perturbation Methods, Cambridge University Press, 1991.
6	R.E.O'Malley: Introduction to Singular Perturbations, Academic Press, New York, 1974.
7	H.J.Reinhardt: Singular Perturbations of difference methods for linear ordinary differential equations, Applicable Anal., 10(1980), 53-70. DOI ScienceOn
8	M.K.Kadalabajoo and Y.N.Reddy: Numerical Solution of Singular Perturbation Problems via Deviating Arguments, Applied Mathematics and Computation, 21(1987):221- 232. DOI ScienceOn
9	J.Kevorkian and J.D.Cole: Perturbation Methods in Applied Mathematics,Springer-Verlag, New York, 1981.
10	A.H.Nayfeh : Introduction to Perturbation Techniques, Wiley, New York,1981
11	A. Awoke and Y.N.Reddy : Terminal Boundary Condition For Singularly Perturbed Two-Point Boundary Value Problems, Neural, Parallel, and Scientific Computations 16(2008),435 - 448.
12	M.K.Kadalabajoo and Y.N.Reddy: Numerical Solution of Singular Perturbation Problems by a Terminal Boundary Condition, Journal of Computational Physics, Vol.52(1987), N0. 2.
13	P.P.Chakravarthy and Y.N. Reddy: A Cutting point technique for Singular Perturbation Problems, Journal of Mathematical control science and Applications(JMSCA), vol.1(2007), No.1, pp. 39-59.
14	M.M.Chawla: A fourth-order Tridiagonal Finite Difference Method for General Non-Linear Two-Point Boundary Value Problems with Mixed Boundary Conditions, J.Inst.Maths Applies 21(1978), 83-93. DOI
15	R. Bellman: Perturbation Methods in Mathematics, Physics and Engineering, New York, Holt, Rinehart, and Winston, 1964.
16	C.M.Bender and S.A.Orsazag: Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York,1978.