Ishikawa-Collocation Method for Nonlinear Fredholm Equations with Non-Separable Kernels

A fixed point method is developed on a mesh for the solution of nonlinear Fredholm equation. First, the problem is collocated at mesh points and a second order quadrature rule is used to approximate the nonlinear integral. Under the assumption of nonexpansivity of self-map, we construct an Ishikawa iteration to linearize the resulting system and approximate the solution at the mesh points. Four numerical examples are given to verify the accuracy and practicability of the method. The results show that indeed the method converges with second order of accuracy. One important lesson from this study is that the results support the claim, in previous studies, that fixed point iterations can provide reliable means of solving several nonlinear problems. It is recommended to extend this work to functional integral equations using higher order quadrature rules.