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

Yanping Chen*, Tao TANG

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

179 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)147-167
Number of pages21
JournalMathematics of Computation
Volume79
Issue number269
DOIs
Publication statusPublished - 2010

Scopus Subject Areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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