On graded meshes for weakly singular Volterra integral equations with oscillatory trigonometric kernels

We compare the collocation methods on graded meshes with that on uniform meshes for the solution of the weakly singular Volterra integral equation of the second kind with oscillatory trigonometric kernels. Due to, in general, unbounded derivatives at the left endpoint of the interval of integration, we should approximate the solution of the integral equation by collocation methods on graded meshes. However, we show that this non-smooth behavior of the solution has little effect on the approximate solution when the kernels of integral equation are highly oscillatory. Numerical examples are given to confirm the proposed results.