A Chebyshev collocation method for a class of Fredholm integral equations with highly oscillatory kernels

Based on the Filon–Clenshaw–Curtis method for highly oscillatory integrals, and together with the Sommariva’s result (Sommariva, 2013) for Clenshaw–Curtis quadrature rule, we present a Chebyshev collocation method for a class of Fredholm integral equations with highly oscillatory kernels, whose unknown function is assumed to be less oscillatory than the kernel. In the proposed method, the Filon–Clenshaw–Curtis method is used to compute the involved oscillatory integrals, which makes the proposed method very precise. By solving only a small system of linear equations, we can obtain a very satisfactory numerical solution. The performance of the presented method is illustrated by several numerical examples. Compared with the method proposed by Li et al. (2012), this method enjoys a lower computational complexity. Furthermore, numerical examples show that the presented method has a competitive advantage on the accuracy compared with the method in Li et al. (2012).