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Mathematics of Computation

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Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel

Authors: Yanping Chen and Tao Tang
Journal: Math. Comp. 79 (2010), 147-167
MSC (2000): Primary 35Q99, 35R35, 65M12, 65M70
Published electronically: June 16, 2009
MathSciNet review: 2552221
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Abstract: In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of the second kind with a weakly singular kernel. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval $ [-1,1]$, so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high-order accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, polynomial approximation theory for orthogonal polynomials and operator theory. The spectral rate of convergence for the proposed method is established in the $ L^{\infty}$-norm and the weighted $ L^2$-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

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Additional Information

Yanping Chen
Affiliation: School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Tao Tang
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong –and– Faculty of Science, Beijing University of Aeronautics and Astronautics, Beijing, China

Received by editor(s): March 24, 2008
Received by editor(s) in revised form: February 14, 2009
Published electronically: June 16, 2009
Additional Notes: The first author is supported by Guangdong Provincial “Zhujiang Scholar Award Project”, National Science Foundation of China 10671163, the National Basic Research Program under the Grant 2005CB321703
The second author is supported by Hong Kong Research Grant Council, Natural Science Foundation of China (G10729101), and Ministry of Education of China through a Changjiang Scholar Program.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.