ε-Uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs

We consider some problems arising from singularly perturbed general differential difference equations. First we construct (in a new way) and analyze a “fitted operator finite difference method (FOFDM)” which is first order ε-uniformly convergent. With the aim of having just one function evaluation at each step, attempts have been made to derive a higher order method via Shishkin mesh to which we refer as the “fitted mesh finite difference method (FMFDM)”. This FMFDM is a direct method and ε  -uniformly convergent with the nodal error as O(n-2ln2n)O(n-2ln2n) which is an improvement over the existing direct methods (i.e., those which do not use any acceleration of convergence techniques, e.g., Richardson’s extrapolation or defect correction, etc.) for such problems on a mesh of Shishkin type that lead the error as O(n-1lnn)O(n-1lnn) where n denotes the total number of sub-intervals of [0, 1]. Comparative numerical results are presented in support of the theory.