An exponentially fitted integration scheme for a class of quasilinear singular perturbation problems

Abstract

A class of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations exhibiting boundary layer at the left end of the underlying interval is considered and a new exponentially fitted integration scheme on a uniform mesh is devised for its numerical solution. The devised scheme is obtained by introducing a suitable constant fitting factor in a new three term recurrence relationship derived by the application of outer region solution obtained by asymptotic expansion procedure and a combination of the exact and approximate rules of integration with finite difference approximations of the first derivative. Value of the fitting factor is determined using the theory of singular perturbations and used to take care of rapid changes in the solution. Resulting tridiagonal algebraic system of equations is solved by the Thomas algorithm. Convergence of the scheme is analyzed. Three numerical example problems are solved and computational results are tabulated to show the accuracy and efficiency of the method. It is easily observed that the derived scheme is able to produce accurate results with minimal computational effort for all the values of the mesh size h when the perturbation parameter ε tends to zero. Both the theoretical and numerical analysis of the method reveals that the method is able to produce uniformly convergent results with quadratic convergence rate.