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Volume 26, Issue 6
On Spectral Methods for Volterra Integral Equations and the Convergence Analysis

Tao Tang, Xiang Xu & Jin Cheng

J. Comp. Math., 26 (2008), pp. 825-837.

Published online: 2008-12

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  • Abstract

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach (with theoretical justification) for the Volterra type equations.

  • Keywords

Legendre-spectral method, Second kind Volterra integral equation, Convergence analysis.

  • AMS Subject Headings

35Q99, 35R35, 65M12, 65M70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-26-825, author = {Tao Tang, Xiang Xu and Jin Cheng}, title = {On Spectral Methods for Volterra Integral Equations and the Convergence Analysis}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {6}, pages = {825--837}, abstract = {

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach (with theoretical justification) for the Volterra type equations.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8662.html} }
TY - JOUR T1 - On Spectral Methods for Volterra Integral Equations and the Convergence Analysis AU - Tao Tang, Xiang Xu & Jin Cheng JO - Journal of Computational Mathematics VL - 6 SP - 825 EP - 837 PY - 2008 DA - 2008/12 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8662.html KW - Legendre-spectral method, Second kind Volterra integral equation, Convergence analysis. AB -

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach (with theoretical justification) for the Volterra type equations.

Tao Tang, Xiang Xu and Jin Cheng. (2008). On Spectral Methods for Volterra Integral Equations and the Convergence Analysis. Journal of Computational Mathematics. 26 (6). 825-837. doi:
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