In this paper, we present an initial value technique for solving self-adjoint singularly perturbed linear boundary value problems. The original problem is reduced to its normal form and the reduced problem is converted to first order initial value problems. This replacement is significant from the computational point of view. The classical fourth order Runge-Kutta method is used to solve these initial value problems. This approach to solve singularly perturbed boundary-value problems is numerically very appealing. To demonstrate the applicability of this method, we have applied it on several linear examples with left-end boundary layer and right-end layer. From the numerical results, the method seems accurate and solutions to problems with extremely thin boundary layers are obtained.
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 O’Malley R.E. (1974): Introduction to Singular Perturbations. – New York: Academic Press.
 Nayfeh A.H. (1981): Introduction to Perturbation Techniques. – New York: Wiley.
 Kevorkian J. and Cole J.D. (1981): Perturbation Methods in Applied Mathematics. – New York: Springer-Verlag.
 Bender C.M. and Orszag S.A. (1978): Advanced Mathematical Methods for Scientists and Engineers. – New York: McGraw-Hill.
 Van Dyke M. (1964): Perturbation Methods in Fluid Mechanics. – New York: Academic Press.
 Wasow W. (1965): Asymptotic Expansions for Ordinary Differential Equations. – New York: Inter Science.
 Hemker P.W. (1977): A numerical study of stiff two point boundary problems. – MCT 80, Mathematical Centre, Amsterdam.
 Hemker P.W. and Miller J.J.H. (Eds.) (1979): Numerical Analysis of Singular Perturbation Problems. – New York: Academic Press.
 Doolan E.P., Miller J.J.H. and Schilders W.H.A. (1980): Uniform Numerical Methods for Problems with Initial and Boundary Layers. – Dublin: Boole Press.
 Morton K.W. (1995): Numerical Solution of Convection – Diffusion Problems. – Oxford University Press.
 Gasparo M.G. and Macconi M. (1990): Initial value methods for second-order singularly perturbed boundary-value problems. – Journal of Optimization Theory and Applications, vol.66, pp.197-210.
 Gasparo M.G. and Macconi M. (1989): New initial-value method for singularly perturbed boundary-value problems. – Journal of Optimization Theory and Applications, vol.63, pp.213-224.
 Gasparo M.G. and Macconi M. (1992): Numerical solution of second-order nonlinear singularly perturbed boundary-value problems by initial value methods. – Journal of Optimization Theory and Applications, vol.73, pp.309-327.
 Gasparo M.G. and Macconi M. (1992): Parallel initial-value algorithms for singularly perturbed boundary-value problems. – Journal of Optimization Theory and Applications, vol.73, pp.501-517.
 Reddy Y.N. and PramodChakravarthy P. (2003): Method of Reduction of Order for Solving Singularly Perturbed Two-Point Boundary Value Problems. – Applied Mathematics and Computation, vol.136, pp 27-45.
 Reddy Y.N. and PramodChakravarthy P. (2004): An initial-value approach for solving singularly perturbed two-point boundary value problem.– Applied Mathematics and Computation, vol.155, pp.95-110.
 Mishra H.K., Kumar M. and Singh P. (2009): Initial value technique for self adjoint singular perturbation boundary value problems. – Computational Mathematics and modeling, vol.29, No.2.
 Natesan S. and Ramanujam N. (1998): Initial-value technique for singularly perturbed boundary value problems for second-order ordinary differential equations arising in chemical reactor theory. – Journal of Optimization Theory and Applications, vol.97, No.2, pp.455-470.
 Kadalbajoo M.K. and Reddy Y.N. (1987): An initial value technique for a class of non-linear singular perturbation problems. – Journal of Optimization Theory and Applications, vol.53, pp.395-406.
 Pearson C.E. (1968): On a differential equation of boundary layer type. – J. Math. Phys., vol.47, pp.134-154.