On Filon methods for a class of Volterra integral equations with highly oscillatory Bessel kernels

This paper focuses on the convergence of a class of collocation methods for Volterra integral equations of the second kind with highly oscillatory Bessel functions. Compared to existing theoretical results, sharper frequency-related convergence rates of these methods are established by exploring the asymptotic expansions of solutions and solving error equations. Theoretical results in this paper show the direct Filon method and continuous linear collocation method share the same convergence rate. Both of them admit a better convergence rate compared to the piecewise constant collocation method in solving Volterra integral equations with highly oscillatory Bessel kernels. These results are verified by numerical experiments.