Computation of highly oscillatory Bessel transforms with algebraic singularities

Abstract

In this paper, we consider the Clenshaw-Curtis-Filon method for the highly oscillatory Bessel transform ${\int }_0^1{x}^{\hat{I}}\pm (1-x{)}^{2}f(x)J_{\hat{I}}½(\ddot{I}‰x)dx$, where $f$ is a smooth function on $[0,1]$, and $\hat{I}½\ge 0.$ The method is based on Fast Fourier Transform (FFT) and fast computation of the modified moments. We give a recurrence relation for the modified moments and present an efficient method for the evaluation of modified moments by using recurrence relation. Moreover, the corresponding error bound in inverse powers of $N$ for this method for the integral is presented. Numerical examples are provided to support our analysis and show the efficiency and accuracy of the method.

12 pages, 3 figures