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A Legendre-Galerkin method for solving general Volterra functional integral equations

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Abstract

We propose in this paper a fully discrete Legendre-Galerkin method for solving general Volterra functional integral equations. The focus of this paper is the stability analysis of this method. Based on this stability result, we prove that the approximation equation has a unique solution, and then show that the Legendre-Galerkin method gives the optimal convergence order \(\mathcal {O}(n^{-m})\), where m denotes the degree of the regularity of the exact solution and n+1 denotes the dimensional number of the approximation space. Moreover, we establish that the spectral condition constant of the coefficient matrix relative to the corresponding linear system is uniformly bounded for sufficiently large n. Finally, we use numerical examples to confirm the theoretical prediction.

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Authors and Affiliations

  1. School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, Shandong, 250014, China

    Haotao Cai & Jiafei Qi

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  1. Haotao Cai
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  2. Jiafei Qi
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Correspondence to Haotao Cai.

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Cai, H., Qi, J. A Legendre-Galerkin method for solving general Volterra functional integral equations. Numer Algor 73, 1159–1180 (2016). https://doi.org/10.1007/s11075-016-0134-7

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  • DOI: https://doi.org/10.1007/s11075-016-0134-7

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