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Convergence Analysis of Spectral Galerkin Methods for Volterra Type Integral Equations

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Abstract

This work is to provide spectral and pseudo-spectral Jacobi-Galerkin approaches for the second kind Volterra integral equation. The Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. For some spectral and pseudo-spectral Jacobi-Galerkin methods, a rigorous error analysis in both the infinity and weighted norms is given provided that both the kernel function and the source function are sufficiently smooth. Numerical experiments validate the theoretical prediction.

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Acknowledgements

The first author is supported by the National Science Foundation of China (11171104, 10871066) and the Science and Technology Grant of Guizhou Province (LKS[2010]05). The second and third authors are supported by the FRG Grant of Hong Kong Baptist University and the RGC Grants provided by Research Grant Council of Hong Kong.

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, Guizhou, 550001, China

    Ziqing Xie

  2. Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan, 410081, China

    Ziqing Xie

  3. School of Mathematics, Fuzhou University, Fuzhou, 350002, China

    Xianjuan Li

  4. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

    Tao Tang

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  1. Ziqing Xie
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  2. Xianjuan Li
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  3. Tao Tang
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Correspondence to Tao Tang.

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Xie, Z., Li, X. & Tang, T. Convergence Analysis of Spectral Galerkin Methods for Volterra Type Integral Equations. J Sci Comput 53, 414–434 (2012). https://doi.org/10.1007/s10915-012-9577-8

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  • DOI: https://doi.org/10.1007/s10915-012-9577-8

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