A cutting point technique for singular perturbation problems

Abstract

In this paper, a cutting point technique is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. The method consists of the following steps: (i) The original singularly perturbed two-point boundary value problem is divided into two problems, namely inner and outer region problems. (ii) A boundary condition in implicit form at the cutting point is derived from the inner region problem. (iii) Then, the outer region problem is solved as a two-pint boundary value problem, and an explicit boundary condition is derived at the cutting point. (iv) Using stretching transformation, a modified inner region problem is constructed and is solved as a two-point boundary value problem using explicit boundary condition at the cutting point. (v) Finally, we combine the solutions of both the problems to obtain an approximate solution to the original problem. The proposed method is iterative on the cutting point. The process is to be repeated for various choices of the cutting point, until the solution differ materially from iteration to iteration. Several linear and nonlinear problems have been solved to demonstrate the applicability of the method.