Efficient Higher-Order Iterative Schemes for Enhanced Convergence and Dynamical Analysis in Real-World Applications.

Abstract This paper introduces fourth, fifth, sixth and seventh-order iterative schemes by using a Newton-like method and weight functions. We utilize the Newton method as the initial step, followed by Newton-like methods for the second and third steps. The proposed schemes are numerically tested and analyzed for convergence. The intention is to demonstrate the efficiency and validity of the proposed fourth, fifth, sixth and seventh-order methods. Furthermore, we examine the dynamical behavior by discussing the basin of attraction. The Basin of attraction shows that our schemes produce bigger regions as compared to some existing methods, which makes them a good competitor. The order of convergence of these methods is confirmed theoretically and their computational performance is examined when applied to real-world nonlinear problems encountered in a various field such as blood rheology, fluid mechanics, economics, chemical engineering, and quantum mechanics. Math Subject Classifications 2020: 65H04, 65H05, 90C39.