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).

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