On the numerical solution of highly oscillatory Fredholm integral equations using a generalized quadrature method

In this paper, a numerical method is presented for solving Fredholm integral equations with highly oscillatory kernels. The proposed method combined piecewise collocation with a generalized quadrature rule in a uniform mesh. Due to the oscillatory nature of the kernels of integral equation, the discretized collocation equations required the evaluation of oscillatory integrals, which were computed using an efficient generalized quadrature rule. Convergence was analyzed in terms of both asymptotic and classical accuracy. The method's practical performance and reliability were showcased with two numerical examples.