On the convergence rates of Filon methods for the solution of a Volterra integral equation with a highly oscillatory Bessel kernel

In this paper, based on the asymptotic property of the solution, we derive the corresponding convergence rates in terms of the frequency for the direct-Filon and linear continuous collocation methods, which solves an open problem in Brunner (2010) [1]. Numerical tests verify that the asymptotic orders obtained are optimal.