Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

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

Language

English

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. © 2010 The author.

Keywords

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

Publication Date

2011

Source Publication Title

IMA Journal of Numerical Analysis

Volume

31

Issue

4

Start Page

1281

End Page

1314

Publisher

Oxford University Press

DOI

10.1093/imanum/drq035

Link to Publisher's Edition

http://dx.doi.org/10.1093/imanum/drq035

ISSN (print)

02724979

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