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Volume 9, Issue 6
Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel

Xiong Liu & Yanping Chen

DOI: 10.4208/aamm.OA-2016-0049

Adv. Appl. Math. Mech., 9 (2017), pp. 1506-1524.

Published online: 2017-09

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  • Abstract

In this paper, a Chebyshev-collocation spectral method is developed for Volterra integral equations (VIEs) of second kind with weakly singular kernel. We first change the equation into an equivalent VIE so that the solution of the new equation possesses better regularity. The integral term in the resulting VIE is approximated by Gauss quadrature formulas using the Chebyshev collocation points. The convergence analysis of this method is based on the Lebesgue constant for the Lagrange interpolation polynomials, approximation theory for orthogonal polynomials, and the operator theory. The spectral rate of convergence for the proposed method is established in the L∞-norm and weighted L2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

  • Keywords

Chebyshev collocation method, Volterra integral equations, spectral rate of convergence, Hölder continuity.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-9-1506, author = {Liu , Xiong and Chen , Yanping}, title = {Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2017}, volume = {9}, number = {6}, pages = {1506--1524}, abstract = {

In this paper, a Chebyshev-collocation spectral method is developed for Volterra integral equations (VIEs) of second kind with weakly singular kernel. We first change the equation into an equivalent VIE so that the solution of the new equation possesses better regularity. The integral term in the resulting VIE is approximated by Gauss quadrature formulas using the Chebyshev collocation points. The convergence analysis of this method is based on the Lebesgue constant for the Lagrange interpolation polynomials, approximation theory for orthogonal polynomials, and the operator theory. The spectral rate of convergence for the proposed method is established in the L∞-norm and weighted L2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2016-0049}, url = {http://global-sci.org/intro/article_detail/aamm/10190.html} }
TY - JOUR T1 - Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel AU - Liu , Xiong AU - Chen , Yanping JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1506 EP - 1524 PY - 2017 DA - 2017/09 SN - 9 DO - http://doi.org/10.4208/aamm.OA-2016-0049 UR - https://global-sci.org/intro/article_detail/aamm/10190.html KW - Chebyshev collocation method, Volterra integral equations, spectral rate of convergence, Hölder continuity. AB -

In this paper, a Chebyshev-collocation spectral method is developed for Volterra integral equations (VIEs) of second kind with weakly singular kernel. We first change the equation into an equivalent VIE so that the solution of the new equation possesses better regularity. The integral term in the resulting VIE is approximated by Gauss quadrature formulas using the Chebyshev collocation points. The convergence analysis of this method is based on the Lebesgue constant for the Lagrange interpolation polynomials, approximation theory for orthogonal polynomials, and the operator theory. The spectral rate of convergence for the proposed method is established in the L∞-norm and weighted L2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

Liu , Xiong and Chen , Yanping. (2017). Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel. Advances in Applied Mathematics and Mechanics. 9 (6). 1506-1524. doi:10.4208/aamm.OA-2016-0049
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