Approximate solution of Abel integral equation

Authors:
Li Huang;Yong Huang;Xian-Fang Li
Affiliations:
Department of Mathematics, Hunan University of Technology, Metallurgy Campus, Zhuzhou, Hunan 412000, China;Institute of Mechanics and Sensor Technology, School of Civil Engineering and Architecture, Central South University, Changsha, Hunan 410083, China;Institute of Mechanics and Sensor Technology, School of Civil Engineering and Architecture, Central South University, Changsha, Hunan 410083, China
Venue:
Computers & Mathematics with Applications
Year:
2008

Citing 2
Cited 2

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Abstract

This paper presents a new, stable, approximate inversion of Abel integral equation. By using the Taylor expansion of the unknown function, Abel equation is approximately transformed to a system of linear equations for the unknown function together with its derivatives. A desired solution can be determined by solving the resulting system according to Cramer's rule. This method gives a simple and closed form of approximate Abel inversion, which can be performed by symbolic computation. The nth-order approximation is exact for a polynomial of degree up to n. Abel integral equation is approximately expressed in terms of integrals of input data; so the suggested approach is stable for experimental data with random noise. An error analysis of this approach is given. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.