On the nonlocal boundary value problem for the coupled system

We investigate the nonlocal boundary value problem for the coupled system of Fredholm integro-differential equations in this paper. This analysis differs from other works in that we conduct an empirical investigation into the existence and uniqueness of the solution, as well as investigate the continuous dependence on the initial data. In addition, we apply all results of the analytical study to some examples and find the exact solutions for them using the Adomian decomposition method. Also, we offer a numerical study of this system using the finite difference Simpson's method. This numerical method is based on converting the coupled system into a system of algebraic equations which can be solved together to get the solutions. Some comparisons of numerical solutions are given with exact solutions to show the accuracy of the methods used, in addition to some figures that illustrate this.