Clenshaw-Curtis-Filon-type methods for highly oscillatory Bessel transforms and applications

We consider a Clenshaw-Curtis-Filon-type method for highly oscillatory Bessel transforms. It is based on a special Hermite interpolation polynomial at the Clenshaw-Curtis points that can be efficiently evaluated using O operations, where N is the number of Clenshaw-Curtis points in the interval of integration. Moreover, we derive corresponding error bounds in terms of the frequency and the approximating polynomial. We then show that this method yields an efficient numerical approximation scheme for a class of Volterra integral equations containing highly oscillatory Bessel kernels (a problem for which standard numerical methods fail), and it also allows the study of the asymptotics of the solutions. Numerical examples are used to illustrate the efficiency and accuracy of the Clenshaw-Curtis-Filon-type method for approximating these highly oscillatory integrals and integral equations.