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.

In this paper, we introduced a numerical approach for estimating the solutions of nonlinear Fredholm integral equations in the complex plane. The main problem was transformed into a novel integral equation, which simplified the computation of integrals derived from the discretization technique. The combination of the standard collocation method with periodic quasi-wavelets, as well as their fundamental properties, was utilized to convert the solution of the newly formulated integral equation into a nonlinear complex system of algebraic equations. The convergence properties of the scheme were also presented. Finally, several numerical examples were provided to demonstrate the efficiency and precision of our proposed approach, which also confirmed its superiority over polynomial collocation methods.