An Initial Value Method for Solving Singularly Perturbed Boundary Value Problems Using Adaptive Grids

Abstract

In this paper, an initial value technique is presented to solve singularly perturbed two-point boundary value problems. Using the basic idea of the well known Wentzel – Kramers – Brillouin (WKB) method, an approximation due to asymptotic expansion of the solution of the problem is constructed. The original problem is reduced to a combination of an initial value problem and a terminal value problem. The terminal value problem is solved by the trapezoidal method and then the initial value problem is solved by the backward Euler method on an appropriate nonuniform mesh constructed adaptively by equidistributing a positive monitor function based on the solution. An error estimate is derived, and numerical experiments are conducted to illustrate the efficiency of the proposed method.