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The Scientific World Journal
Volume 2014, Article ID 413623, 8 pages
Research Article

Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials

1Department of Mathematics and Statistics, Mississippi State University, MS 39762, USA
2Department of Mathematics and Statistics, Faculty of Science, Prince of Songkla University, Songkhla 90112, Thailand

Received 19 August 2013; Accepted 10 October 2013; Published 12 January 2014

Academic Editors: S. Amat, F. J. Garcia-Pacheco, H. Iiduka, and F. Minhós

Copyright © 2014 S. Mashayekhi 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.


A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.