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A periodic wavelet method for the second kind of the logarithmic integral equation

Published online by Cambridge University Press:  17 April 2009

Jing Gao  and
Yao-Lin Jiang
Jing Gao
Affiliation:
Department of Mathematical Sciences, Xi'an Jiaotong University, Xi'an 710049, Peoples Republic of China e-mail: jgao@mail.xjtu.edu.cnyljiang@mail.xjtu.edu.cn
Yao-Lin Jiang
Affiliation:
Department of Mathematical Sciences, Xi'an Jiaotong University, Xi'an 710049, Peoples Republic of China e-mail: jgao@mail.xjtu.edu.cnyljiang@mail.xjtu.edu.cn
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A periodic wavelet Galerkin method is presented in this paper to solve a weakly singular integral equations with emphasis on the second kind of Fredholm integral equations. The kernel function, which includes of a smooth part and a log weakly singular part, is discretised by the periodic Daubechies wavelets. The wavelet compression strategy and the hyperbolic cross approximation technique are used to approximate the weakly singular and smooth kernel functions. Meanwhile, the sparse matrix of systems can be correspondingly obtained. The bi-conjugate gradient iterative method is used to solve the resulting algebraic equation systems. Especially, the analytical computational formulae are presented for the log weakly singular kernel. The computational error for the representative matrix is also evaluated. The convergence rate of this algorithm is O (N-p log(N)), where p is the vanishing moment of the periodic Daubechies wavelets. Numerical experiments are provided to illustrate the correctness of the theory presented here.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 3 , December 2007 , pp. 321 - 336
Copyright
Copyright © Australian Mathematical Society 2007

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