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A Fractional Order Collocation Method for Second Kind Volterra Integral Equations with Weakly Singular Kernels

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Abstract

In this paper, we develop a fractional order spectral collocation method for solving second kind Volterra integral equations with weakly singular kernels. It is well known that the original solution of second kind Volterra integral equations with weakly singular kernels usually can be split into two parts, the first is the singular part and the second is the smooth part with the assumption that the integer m being its smooth order. On the basis of this characteristic of the solution, we first choose the fractional order Lagrange interpolation function of Chebyshev type as the basis of the approximate space in the collocation method, and then construct a simple quadrature rule to obtain a fully discrete linear system. Consequently, with the help of the Lagrange interpolation approximate theory we establish that the fully discrete approximate equation has a unique solution for sufficiently large n, where \(n+1\) denotes the dimension of the approximate space. Moreover, we prove that the approximate solution arrives at an optimal convergence order \(\mathcal{O}(n^{-m}\log n)\) in the infinite norm and \(\mathcal{O}(n^{-m})\) in the weighted square norm. In addition, we prove that for sufficiently large n, the infinity-norm condition number of the coefficient matrix corresponding to the linear system is \(\mathcal{O}(\log ^2 n)\) and its spectral condition number is \(\mathcal{O}(1)\). Numerical examples are presented to demonstrate the effectiveness of the proposed method.

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Acknowledgements

This work is supported by National Science Foundation of China ( 11671157, 91430104) and National Science Foundation of Shandong Province (ZR2014JL003). The authors thank the referees for very helpful suggestions, which help us improve this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Yanping Chen

  2. School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250014, Shandong, People’s Republic of China

    Haotao Cai

  3. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Haotao Cai

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  1. Haotao Cai
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  2. Yanping Chen
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Correspondence to Yanping Chen.

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Cai, H., Chen, Y. A Fractional Order Collocation Method for Second Kind Volterra Integral Equations with Weakly Singular Kernels. J Sci Comput 75, 970–992 (2018). https://doi.org/10.1007/s10915-017-0568-7

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  • DOI: https://doi.org/10.1007/s10915-017-0568-7

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