Iterative methods for nonlinear equations in a coupled system: Using modified homotopy perturbation

Nonlinear equations arise in various branches of applied sciences where the real-world problems cannot be formulated accurately in linear forms. It is a fascinating and thrilling task to develop numerical methods for estimating the roots of nonlinear equations. In this work, new sixth and seventh-order iterative methods for estimating the solutions of nonlinear equations in coupled systems are developed using modified Homotopy perturbation techniques. The convergence of the proposed methods is examined. The recently proposed iterative methods are supported by numerical examples. Computer comparisons with some previous schemes in the literature have demonstrated the effectiveness of these new methods. They also show a lower computational cost and a quicker convergence rate. Researchers, mathematicians, and engineers use polynomiography extensively as a tool to visualize and understand the behavior of complex equations. Our suggested techniques generate polynomiographs of complex polynomials that clearly display the roots.