Computational methods for solving two-parameter singularly perturbed boundary value problems for second-order ordinary differential equations

Abstract

Singularly perturbed two-point boundary value problems (SPBVPs) for second-order ordinary differential equations (ODEs) with two small parameters multiplying the derivatives are considered. Computational methods are suggested in this paper to solve this type of problems. One of these methods is fitted operator method (FOM) and the other is bondary value technique (BVT). In FOM, we suggest an exponentially fitted finite difference (EFFD) scheme throughout the domain of definition of the differential equation and the error estimates are derived. In BVT, the domain of definition of the differential equation is divided into three subintervals called inner and outer regions. Then we solve the two-parameter SPBVP (TPSPBVP) over these regions as two-point boundary value problems. An exponentially fitted finite difference scheme is used in the inner regions and a classical finite difference scheme in the outer region. The data for boundary conditions at the transition points are obtained using zero-order asymptotic expansion approximation of the solution of the problem. The BVT is well suited for parallel computing. Numerical examples are given to illustrate the method.