An Efficient Spectral-Galerkin Method for Second Kind Weakly Singular VIEs with Highly Oscillatory Kernels

In this paper, we construct an efficient spectral Galerkin method to deal with the classical second kind linear VIEs with weakly singular and highly oscillatory kernel. We first study the oscillation and singularity of the exact solution and then based on those results, we propose an efficient fully discrete spectral Galerkin method. The proposed algorithm reaches an optimal convergence order without the influence of the wave number. At last, two numerical examples are provided to verify the efficiency of our proposed method.