Fourth- and fifth-order iterative schemes for nonlinear equations in coupled systems: A novel Adomian decomposition approach

In the fields of numerical analysis and applied science, approximating the roots of nonlinear equations is a fundamental and intriguing challenge. With the rapid advancement of computing power, solving nonlinear equations using numerical techniques has become increasingly important.Numerical methods for nonlinear equations play a critical role in many areas of research and industry, enabling scientists and engineers to model and understand complex systems and make accurate predictions about their behavior. This paper aims to propose novel fourth- and fifth-order iterative schemes for approximating solutions to nonlinear equations in coupled systems using Adomian decomposition methods. The proposed method's convergence is examined and numerical examples are provided to demonstrate the effectiveness of the new schemes. We compare these iterative techniques with some previous schemes in the literature, and our results show that the new schemes are more efficient. Our findings represent a significant improvement over previously reported results. Polynomiography is an important tool for visualizing the roots of complex polynomials. It is widely used by researchers, mathematicians and engineers, as it provides a way to visualize complex equations understand their behavior. Our proposed method is capable of generating polynomiographs of complex polynomials, revealing interesting patterns that provide clear visual representations of the roots of complex polynomials.