Parameter--Uniform Finite Difference Method for a Singularly Perturbed Linear Dynamical System

A coupled system of two singularly perturbed ordinary differential equations of first order with the prescribed initial values are considered. The leading term of each equation is multiplied by a small positive parameter and the parameters may differ. The solution exhibits overlapping layers. A Shishkin mesh is constructed. A classical finite difference scheme applied on this mesh (which is piecewise uniform) is proved to be uniformly first order accurate in both the parameters. Numerical results are presented in support of the theory.