Open Access

ARTICLE

A Wavelet Method for the Solution of Nonlinear Integral Equations with Singular Kernels

Jizeng Wang^1,2, Lei Zhang¹, Youhe Zhou¹

1 Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education of China, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu, 730000, China.
2 Corresponding author. E-mail: jzwang@lzu.edu.cn

Computer Modeling in Engineering & Sciences 2014, 102(2), 127-148. https://doi.org/10.3970/cmes.2014.102.127

Abstract

In this paper, we propose an efficient wavelet method for numerical solution of nonlinear integral equations with singular kernels. The proposed method is established based on a function approximation algorithm in terms of Coiflet scaling expansion and a special treatment of boundary extension. The adopted Coiflet bases in this algorithm allow each expansion coefficient being explicitly expressed by a single-point sampling of the function, which is crucially important for dealing with nonlinear terms in the equations. In addition, we use the technique of integration by parts to transform the original integral equations with non-smooth or singular kernels into regular ones with smooth kernels. Then, we incorporate the proposed function approximation algorithm into the Galerkin method for the solution of the transformed nonlinear integral equations. Numerical examples show that the proposed wavelet method is much more accurate and efficient than several others.

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Keywords

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