A robust fitted finite difference method for semi-linear two-parameter singularly perturbed pdes

Abstract

This article presents a novel numerical scheme for solving nonlinear two-parameter singularly perturbed initial-boundary value problems. The proposed method combines an explicit forward Euler discretization for the temporal derivative with a fitted operator finite difference technique in the spatial domain. To handle the nonlinearity, Newton’s quasilinearization technique is employed. A comprehensive error analysis establishes that the developed scheme is first-order convergent uniformly in the perturbation parameters. The efficacy of the method is validated through two numerical examples, implemented in Python. The results, presented in tables and figures, demonstrate the method’s robustness and accuracy in resolving boundary layers for a wide range of perturbation parameters.