Clenshaw-Curtis-Filon-type methods for highly oscillatory Bessel transforms and applications

Shuhuang Xiang*, Yeol Je Cho, Haiyong Wang, Hermann BRUNNER

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

62 Citations (Scopus)

Abstract

We consider a Clenshaw-Curtis-Filon-type method for highly oscillatory Bessel transforms. It is based on a special Hermite interpolation polynomial at the Clenshaw-Curtis points that can be efficiently evaluated using O(NlogN) operations, where N is the number of Clenshaw-Curtis points in the interval of integration. Moreover, we derive corresponding error bounds in terms of the frequency and the approximating polynomial. We then show that this method yields an efficient numerical approximation scheme for a class of Volterra integral equations containing highly oscillatory Bessel kernels (a problem for which standard numerical methods fail), and it also allows the study of the asymptotics of the solutions. Numerical examples are used to illustrate the efficiency and accuracy of the Clenshaw-Curtis-Filon-type method for approximating these highly oscillatory integrals and integral equations.

Original languageEnglish
Pages (from-to)1281-1314
Number of pages34
JournalIMA Journal of Numerical Analysis
Volume31
Issue number4
DOIs
Publication statusPublished - Oct 2011

Scopus Subject Areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Bessel transforms
  • Clenshaw-Curtis points
  • Clenshaw-Curtis-Filon quadrature
  • Fast Fourier transform
  • highly oscillatory Volterra integral equations

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