Abstract. A system of Fredholm integro differential equations involving first and second derivatives is addressed using the Successive Approximation Method (SAM) combined with an iterative algorithm. Starting from an initial estimate, the integral term evaluated using previous iterates is treated as a known forcing function in each iteration. This transforms the original integro differential system into a sequence of linear differential systems, which are then solved under given boundary or initial conditions. Convergence and uniqueness of the solution are rigorously established, and the resulting approximate solution is shown to converge the exact one as the iteration proceeds. Numerical examples demonstrate that SAM produces accurate approximations with controlled error and good computational efficiency.
Keywords: System of Fredholm , integro-differential equations ,The Successive Approximation Method
