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

Abstract

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(log2⁡ω) 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 ω 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.