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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 652631, 8 pages
http://dx.doi.org/10.1155/2014/652631
Research Article

Numerical Solutions of a Class of Nonlinear Volterra Integral Equations

Department of Pure and Applied Mathematics, University of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa

Received 10 April 2014; Revised 17 June 2014; Accepted 26 June 2014; Published 9 July 2014

Academic Editor: Chun-Gang Zhu

Copyright © 2014 H. S. Malindzisa and M. Khumalo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider numerical solutions of a class of nonlinear (nonstandard) Volterra integral equations. We first prove the existence and uniqueness of the solution of the Volterra integral equation in the context of the space of continuous functions over a closed interval. We then use one-point collocation methods with a uniform mesh to construct solutions of the nonlinear (nonstandard) VIE and quadrature rules. We conclude that the repeated Simpson's rule gives better solutions when a reasonably large value of the stepsize is used.