An Efficient Quadrature Rule for Highly Oscillatory Integrals with Airy Function

In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form integral (b)(0) x(alpha)f(x)Ai(-omega x)dx over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when omega >> 1. The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [0,1] and [1,b]. For integrals over the interval [0,1], we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [1,b], we transform the Airy function into the first kind of Bessel function. By applying Cauchy's integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [0,+infinity), which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method.