Stable Numerical Method for Singularly Perturbed Boundary Value Problems with Two Small Parameters

Abstract

Stable numerical method for singularly perturbed boundary value problem with two small positive parameters is presented. Given problem is converted into asymptotically equivalent boundary value problem. Then, using the finite difference approximations, the obtained differential equation is transformed to a three-term recurrence relation. The stability and convergence of the method have been established. To validate the applicability of the proposed method, three examples have been considered and solved for different values of perturbation parameters. Both theoretical error bounds and numerical rate of convergence have been established for the method. The numerical results have been presented in tables and graphs, as it can be observed from the numerical results, the present method approximates the exact solution very well. Moreover, the present method gives better results than some existing numerical methods reported in the literature.

Keywords: Boundary value problems; Singular perturbation; Stability and convergence analysis; Stable numerical method