Meshless methods for one-dimensional oscillatory Fredholm integral equations

In this paper, efficient and simple algorithms based on Levin's quadrature theory and our earlier work involving local radial basis function (RBF) and Chebyshev differentiation matrices, are adopted for numerical solution of one-dimensional highly oscillatory Fredholm integral equations. This work is focused on the comparative performance of local RBF meshless and pseudospectral procedures. We have tested the proposed methods on phase functions with and without stationary phase point(s), both on uniform and Chebyshev grid points. The proposed procedures are shown accurate and efficient, and therefore provide a reliable platform for the numerical solution of integral equations. From the numerical results, we draw some conclusions about accuracy, efficiency and robustness of the proposed approaches.