New parameter-uniform discretisations of singularly perturbed Volterra integro-differential equations
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Date
2018
Journal Title
Journal ISSN
Volume Title
Publisher
Natural Sciences Publishing
Abstract
We design and analyse two numerical methods namely a fitted mesh and a fitted operator finite difference methods for
solving singularly perturbed Volterra integro-differential equations. The fitted mesh method we propose is constructed using a finite
difference operator to approximate the derivative part and some suitably chosen quadrature rules for the integral part. To obtain a
parameter-uniform convergence, we use a piecewise-uniform mesh of Shishkin type. On the other hand, to construct the fitted operator
method, the Volterra integro-differential equation is discretised by introducing a fitting factor via the method of integral identity with
the use of exponential basis function along with interpolating quadrature rules [2]. The two methods are analysed for convergence and
stability. We show that the two methods are robust with respect to the perturbation parameter. Two numerical examples are solved to
show the applicability of the proposed schemes.
Description
Keywords
Singularly perturbed problems, Volterra integro-differential equations, Boundary layer, Finite difference schemes, Uniform convergence
Citation
Iragi, Bakulikira C. & Munyakazi, Justin B. (2018). New parameter-uniform discretisations of singularly perturbed Volterra integro-differential equations. Applied Mathematics & Information Sciences, 12(3), 517-527. http://dx.doi.org/10.18576/amis/120306