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

Part of: Optics, electromagnetic theory - General

Published online by Cambridge University Press:  28 November 2017

Xiong Liu  and
Yanping Chen
Xiong Liu*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China
*
*Corresponding author. Emails:yanpingchen@scnu.edu.cn (Y. P. Chen), liuxiong980211@163.com (X. Liu)
*Corresponding author. Emails:yanpingchen@scnu.edu.cn (Y. P. Chen), liuxiong980211@163.com (X. Liu)
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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 methodVolterra integral equationsspectral rate of convergenceHölder continuity

MSC classification

Secondary: 78A48: Composite media; random media
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 6 , December 2017 , pp. 1506 - 1524
Copyright
Copyright © Global-Science Press 2017 

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