Efficient integration for a class of highly oscillatory integrals

This paper presents some quadrature methods for a class of highly oscillatory integrals whose integrands may have singularities at the two endpoints of the interval. One is a Filon-type method based on the asymptotic expansion. The other is a Clenshaw–Curtis–Filon-type method which is based on a special Hermite interpolation polynomial and can be evaluated efficiently in O(N log N) operations, where N + 1 is the number of Clenshaw–Curtis points in the interval of integration. In addition, we derive the corresponding error bound in inverse powers of the frequency ω for the Clenshaw–Curtis–Filon-type method for the class of highly oscillatory integrals. The efficiency and the validity of these methods are testified by both the numerical experiments and the theoretical results.