A study on the convergence and efficiency of a novel seventh-order iterative method to solve systems of nonlinear equations with electrical engineering applications

In this paper, we present a numerical approach for solving nonlinear differential equations by combining finite difference methods with an efficient three-step iterative method. First, the nonlinear differential equations are discretized and transformed into a system of nonlinear algebraic equations using finite difference approximations. This system is then solved using a novel three-step iterative method designed to enhance computational efficiency. A key feature of the proposed iterative method is that each full iteration requires only one evaluation of the Jacobian matrix of the function and computation of its inverse. This significantly reduces the computational cost, resulting in a higher efficiency index compared to existing techniques. To validate the effectiveness of the method, we provide numerical experiments on several nonlinear differential equations. The results demonstrate the superior accuracy and computational efficiency of the proposed approach, highlighting its advantages over conventional methods.