Spectral methods for weakly singular Volterra integral equations with smooth solutions

Authors:
Yanping Chen;Tao Tang
Affiliations:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong & Faculty of Science, Beijing University of Aeronautics and Astronautics, Beijing, China
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

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Abstract

We propose and analyze a spectral Jacobi-collocation approximation for the linear Volterra integral equations (VIEs) of the second kind with weakly singular kernels. In this work, we consider the case when the underlying solutions of the VIEs are sufficiently smooth. In this case, we provide a rigorous error analysis for the proposed method, which shows that the numerical errors decay exponentially in the infinity norm and weighted Sobolev space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.