Efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels

In this paper, we present a special Clenshaw–Curtis Filon (CCF) type scheme for approximation of highly oscillatory integrals with weak and hypersingular kernels. The non-oscillatory and nonsingular part of the integrand is replaced by a special Hermite interpolation polynomial. Error bounds with respect to the frequency k and the number of the Clenshaw–Curtis points N are considered. The overall computational complexity for the scheme is O(Nlog (N)). Numerical experiments support the theoretical analysis.