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

Authors

Bakulikira C. Iragi, Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South AfricaFollow
Justin B. Munyakazi, Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South AfricaFollow

Author Country (or Countries)

South Africa

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.

Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/journal/100306

Recommended Citation

C. Iragi, Bakulikira and B. Munyakazi, Justin (2018) "New Parameter-Uniform Discretisations of Singularly Perturbed Volterra Integro-Differential Equations," Applied Mathematics & Information Sciences: Vol. 12: Iss. 3, Article 6.
DOI: http://dx.doi.org/10.18576/journal/100306
Available at: https://digitalcommons.aaru.edu.jo/amis/vol12/iss3/6