Quadrature rules and asymptotic expansions for two classes of oscillatory Bessel integrals with singularities of algebraic or logarithmic type

Abstract

In this paper we mainly focus on the quadrature rules and asymptotic expansions for two classes of highly oscillatory Bessel integrals with algebraic or logarithmic singularities. Firstly, by two transformations, we transfer them into the standard types on [−1,1], and derive two useful asymptotic expansions in inverse powers of the frequency ω. Then, based on the two asymptotic expansions, two methods are presented, respectively. One is the so-called Filon-type method. The other is the more efficient Clenshaw-Curtis–Filon-type method, which can be implemented in O(Nlog⁡N) operations, based on Fast Fourier Transform (FFT) and fast computation of the modified moments. Here, through large amount of calculation and analysis, we can construct two important recurrence relations for computing the modified moments accurately, based on the Bessel's equation and some properties of the Chebyshev polynomials. In particular, we also provide error analysis for these methods in inverse powers of the frequency ω. Furthermore, we prove directly from the presented error bounds that these methods share the advantageous property, that the larger the values of the frequency ω, the higher the accuracy. The efficiency and accuracy of the proposed methods are illustrated by numerical examples.