New Parameter-Uniform Discretisations of Singularly Perturbed Volterra Integro-Differential Equations

Abstract

We design and analyse two numerical methods namely a fitted mesh and a fitted operator finite difference methods for
\nsolving singularly perturbed Volterra integro-differential equations. The fitted mesh method we propose is constructed using a finite
\ndifference operator to approximate the derivative part and some suitably chosen quadrature rules for the integral part. To obtain a
\nparameter-uniform convergence, we use a piecewise-uniform mesh of Shishkin type. On the other hand, to construct the fitted operator
\nmethod, the Volterra integro-differential equation is discretised by introducing a fitting factor via the method of integral identity with
\nthe use of exponential basis function along with interpolating quadrature rules [2]. The two methods are analysed for convergence and
\nstability. We show that the two methods are robust with respect to the perturbation parameter. Two numerical examples are solved to
\nshow the applicability of the proposed schemes.