A compact discretization of the boundary value problems of the nonlinear Fredholm integro-differential equations

In this paper, we propose a  fourth-order compact discretization method   for solving a second-order boundary value problem governed by the nonlinear Fredholm integro-differential equations.  Using an efficient approximate polynomial,  a  compact numerical integration method is first designed. Then by applying the derived numerical integration formulas, the original problem is converted into a nonlinear system of algebraic equations.  It is shown that the proposed method is easy to implement and has the third order of accuracy in the infinity norm. Some  numerical examples are presented to demonstrate its  approximation accuracy and computational efficiency,   as well as to compare the derived results with those  obtained in the literature.