Numerical evaluation of a class of highly oscillatory integrals involving Airy functions

In this paper, we consider the implementation of the Clenshaw–Curtis–Filon-type method for a class of highly oscillatory integrals ∫01xα(1-x)βf(x)Ai(-ωx)dx, where Ai(x)Ai(x) is an Airy function, and α>-1,β>-1α>-1,β>-1. By replacing f   by its Chebyshev interpolation polynomial at the Clenshaw–Curtis points so that the modified moments can be computed by recursive formula based on special functions, an efficient and stable method for this integral is presented. Error analysis for the presented method is given. Moreover, the method shares the property that the larger the ωω, the higher the precision. Theoretical results and numerical examples show that the method is very efficient in obtaining very high precision approximations if ωω is sufficiently large.