A Parameter-Uniform Finite Difference Method for a Singularly Perturbed Initial Value Problem: a Special Case

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. A Shishkin piecewise-uniform mesh is constructed, which is used, in conjunction with a classical finite difference discretisation, to form a new numerical method for solving this problem. It is proved, in a special case, that the numerical approximations obtained from this method are essentially first order convergent uniformly in all of the parameters. Numerical results are presented in support of the theory.