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

**Source**: https://www.maths.tcd.ie/report_series/abstracts/tcdm0811.html
**Parent**: https://www.maths.tcd.ie/research/papers/

**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 not necessarily equal. 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.