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Journal of Applied Mathematics
Volume 2014, Article ID 381908, 8 pages
http://dx.doi.org/10.1155/2014/381908
Research Article

Application of Radial Basis Function Method for Solving Nonlinear Integral Equations

The State Key Laboratory of Transmission Equipment and System Safety and Electrical New Technology, Chongqing University, Chongqing 400044, China

Received 13 July 2014; Revised 28 August 2014; Accepted 30 August 2014; Published 28 October 2014

Academic Editor: Song Cen

Copyright © 2014 Huaiqing Zhang 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.

Abstract

The radial basis function (RBF) method, especially the multiquadric (MQ) function, was proposed for one- and two-dimensional nonlinear integral equations. The unknown function was firstly interpolated by MQ functions and then by forming the nonlinear algebraic equations by the collocation method. Finally, the coefficients of RBFs were determined by Newton’s iteration method and an approximate solution was obtained. In implementation, the Gauss quadrature formula was employed in one-dimensional and two-dimensional regular domain problems, while the quadrature background mesh technique originated in mesh-free methods was introduced for irregular situation. Due to the superior interpolation performance of MQ function, the method can acquire higher accuracy with fewer nodes, so it takes obvious advantage over the Gaussian RBF method which can be revealed from the numerical results.