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

Abstract

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.