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Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel

Language

English

Abstract

In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of the second kind with a weakly singular kernel. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [-1, 1], so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high-order accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, polynomial approximation theory for orthogonal polynomials and operator theory. The spectral rate of convergence for the proposed method is established in the L∞-norm and the weighted L2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method. © 2009 American Mathematical Society.

Publication Date

2010

Source Publication Title

Mathematics of Computation

Volume

79

Issue

269

Start Page

147

End Page

167

Publisher

American Mathematical Society

ISSN (print)

00255718

ISSN (electronic)

10886842

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