Spectral-Galerkin method for second kind VIEs with highly oscillatory kernels of the stationary point

In this paper, we discuss an efficient spectral approach for solving a class of second kind VIEs with highly oscillatory kernels possessing the stationary point. First, we use one variable transform to convert the highly oscillatory problem into the long-time one, and then split the long-time problem into a linear system of integral equations by using a dilation approach. Next on each interval we study the characteristic of the original solution and then propose an efficient spectral-Galerkin method for solving the linear system of integral equations. We prove that the proposed algorithm requires to solve the number O(log(2) omega) of linear algebra equations of order n + 1 and reaches the convergence order O(n(-r)), where r denotes the degree of the regularity of the original solution and omega denotes the wave number. Moreover, we give the approach of computing the highly oscillatory integrals produced in the spectral-Galerkin method. At last, two numerical examples are provided to verify the efficiency of our proposed method.