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Mathematical Problems in Engineering
Volume 2010, Article ID 901587, 21 pages
Research Article

High Accuracy Combination Method for Solving the Systems of Nonlinear Volterra Integral and Integro-Differential Equations with Weakly Singular Kernels of the Second Kind

1College of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2Department of Scientific Computing, The Florida State University, Tallahassee, FL 32310, USA

Received 21 October 2009; Accepted 1 April 2010

Academic Editor: Gradimir V. Milovanović

Copyright © 2010 Lu Pan et al. 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.


This paper presents a high accuracy combination algorithm for solving the systems of nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind. Two quadrature algorithms for solving the systems are discussed, which possess high accuracy order and the asymptotic expansion of the errors. By means of combination algorithm, we may obtain a numerical solution with higher accuracy order than the original two quadrature algorithms. Moreover an a posteriori error estimation for the algorithm is derived. Both of the theory and the numerical examples show that the algorithm is effective and saves storage capacity and computational cost.