Clenshaw-Curtis-type quadrature rule for hypersingular integrals with highly oscillatory kernels

The Clenshaw-Curtis-type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g(x). Based on the fast Hermite interpolation, this paper provides a stable recurrence relation for these modified moments. Convergence rates with respect to the frequency k and the number of interpolation points N are considered. These theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples.