On the convergence rate of collocation methods for Volterra integral equations with weakly singular oscillatory trigonometric kernels

Abstract

This paper presents efficient collocation methods for linear Volterra integral equations with weakly singular highly oscillatory kernels. The numerical steepest descent method and generalized Gauss–Laguerre rule are utilized to calculate the weakly singular oscillatory integrals, which is the main challenge of the problem. Moreover, we derive the corresponding error estimation formula in terms of the frequency and the step length. This formula reflects, to some extent, the global convergence of the method. However, numerical examples show that the method is very effective and verify the correctness of the theoretical results.