A fitted mesh method for a coupled semi-linear system of singularly perturbed initial value problems

In this article, we analyzed a general system of first order singularly perturbed semi-linear equations with distinct perturbation parameters in the unit interval. As boundary layers are expected near the origin in the solution components, variants of piecewise uniform meshes, introduced by Shishkin, are constructed to discretize the unit interval and standard finite difference scheme is used to discretize the equations. Parameter uniform convergence of the composed numerical method is proved. A continuation method is used to compute the numerical solution of the non-linear problem and numerical illustrations are given in support.