This paper presents findings related to a class of second-order nonlinear differential systems with mixed boundary conditions. By employing Green’s function in combination with the theory of operator equations, the original boundary value problem is reformulated into a fixed-point problem involving a nonlinear operator that depends on the right-hand side functions. Based on this formulation, we establish theoretical results ensuring the existence and uniqueness of the solution, and we introduce a continuous iterative process to approximate it. To enhance computational accuracy, we utilize high-order finite difference schemes for derivative approximation, together with precise integration techniques. As a result, we propose a discrete iterative method that achieves eighth-order accuracy. The validity and efficiency of the proposed approach are demonstrated through comprehensive numerical experiments.