Numerical integration of oscillatory Airy integrals with singularities on an infinite interval

Abstract

This work is devoted to the quadrature rules and asymptotic expansions for two classes of highly oscillatory Airy integrals on an infinite interval. We first derive two important asymptotic expansions in inverse powers of the frequency ω . Then, based on structure characteristics of the two asymptotic expansions in inverse powers of the frequency ω , both the so-called Filon-type method and the more efficient Clenshaw–Curtis–Filon-type method are introduced and analyzed. The required moments in the former can be explicitly expressed by the Meijer G-functions. The latter can be implemented in O ( N log N ) operations, based on fast Fourier transform (FFT) and fast computation of the modified moments. Here, we can construct two useful recurrence relations for computing the required modified moments accurately, with the help of the Airy’s equation and some properties of the Chebyshev polynomials. Particularly, we also provide their error analyses in inverse powers of the frequency ω . Furthermore, the presented error analysis shows the advantageous property that the accuracy improves greatly as ω increases. Numerical examples are provided to illustrate the efficiency and accuracy of the proposed methods.