An exponentially fitted special second-order finite difference method for solving singular perturbation problems

Abstract

Singularly perturbed second-order two-point boundary value problems occur very frequently in fluid mechanics and other branches of Applied Mathematics. These problems depend on a small positive parameter in such away that the solution varies rapidly (called boundary layer region) in some part and varies slowly in some other parts. The numerical treatment of singular perturbation problems is far from the trivial because of the boundary layer behavior of the solution. There are a wide variety of techniques for solving singular perturbation problems (cf. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]). In this paper, an exponentially fitted special second-order finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. A fitting factor is introduced in a tri-diagonal finite difference scheme and is obtained from the theory of singular perturbations. Thomas Algorithm is used to solve the system and its stability is investigated. To demonstrate the applicability of the method, we have solved several linear and non-linear problems. From the results, it is observed that the present method approximates the exact solution very well.