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Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel

Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel
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Year:    2017

Author:    Xiong Liu, Yanping Chen

Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 6 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2016-0049

Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 6 : pp. 1506–1524

Published online:    2017-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    19

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

Author Details

Xiong Liu

Yanping Chen

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