Numerical solutions to Volterra integral equations of the second kind with oscillatory trigonometric kernels

This paper explores quadrature methods for weakly singular Volterra integral equation of the second kind with highly oscillatory trigonometric kernels, and presents piecewise constant and linear collocation methods for approximation of the solution. The evaluation of the highly oscillatory kernels of integral equation will give rise to the computation of oscillatory integrals, which can be done by using efficient Filon-type methods. Moreover, by using asymptotic analysis of the solution, the convergence rates for piecewise constant and linear collocation methods, which show the higher the frequency the more accurate the numerical results. The effectiveness and accuracy of the proposed method are tested by numerical examples.