Numerical Solution of Singularly Perturbed Boundary Value Problems with Twin Boundary Layers using Exponential Fitted Scheme

S. Rakmaiah, K. Phaneendra


This paper deals with a numerical method with fitted operator difference method for twin (dual) boundary layers singularly perturbed boundary value problems. In this method, Numerov method is extended to the given second order problem having derivative of first order. Using the non standard differences and modified upwind difference for the first order derivatives, the discrete scheme is deduced. A fitting parameter is utilized in the difference scheme, which handles the rapid changes that occur in the boundary layers due to the small perturbation parameter. Tridiagonal solver is implemented to solve the system of the method. Convergence analysis of the deduced method is discussed. Maximum errors in the solution of the model numerical examples are tabulated and comparison is made, to illustrate and support the method. Solutions are depicted graphically to show the layer behaviour.


Singular perturbation problem; Twin layers; Fitting factor; Tridiagonal system

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E. Angel and R. Bellman, Dynamic Programming and Partial Differential Equation, Academic Press, New York (1972), DOI: 10.1007/978-94-009-5209-6_3.

L. Abrahamsson, A priori estimates for solutions of singular perturbations with a turning point, Stud. Appl. Math. 56 (1977), 51 – 69, DOI: 10.1002/sapm197756151.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York (1978).

E. P. Doolan, J. J. H. Miller and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin (1980), DOI: 10.1137/1025146.

P. Farrell, Sufficient conditions for the uniform convergence of a difference scheme for a singularly perturbed turning point problem, SIAM J. Numer. Anal. 25 (1988), 618 – 643, DOI: 10.1137/0725038.

M. K. Kadalbajoo and C. Patidar Kailash, A survey of numerical techniques for solving singularly perturbed ordinary differential equations, Applied Mathematics and Computation 130 (2002), 457 – 510, DOI: 10.1016/S0096-3003(01)00112-6.

M. K. Kadalbajoo and Y. N. Reddy, Asymptotic and numerical analysis of singular perturbation problems: A survey, Applied Mathematics and Computation 30(3) (1989), 223 – 259, DOI: 10.1016/0096-3003(89)90054-4.

J. J. H. Miller, E. O’Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore (1996), DOI: 10.1142/8410.

S. Natesan, J. Jayakumar and J. Vigo-Aguiar, Parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers, Journal of Computational and Applied Mathematics 158 (2003), 121 – 134, DOI: 10.1016/S0377-0427(03)00476-X.

S. Natesan and N. Ramanujam, A computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers, Applied Mathematics and Computation 93 (1998), 259 – 275, DOI: 10.1016/S0096-3003(97)10056-X.

R. E. O’Malley, Introduction to Singular Perturbations, Academic Press, New York, USA (1974).



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