Solution of nonlinear models in engineering using a new sixteenth order scheme and their basin of attraction

Abstract

The utilization of computers in various engineering and mathematical fields has become increasingly important in recent times. In computational and applied mathematics, computer programs can be used to efficiently process complex data in mere seconds. By utilizing different computer tools and programming languages, computers can manipulate various iterative algorithms to solve nonlinear problems. The convergence rate and computational costs per-iteration are essential factors that determine the effectiveness of an iterative algorithm. This article introduces a new and highly efficient iterative approach of sixteenth order. The authors propose a four-step iterative optimal derivative-free method for solving non-linear equations that arise in engineering application problems. The proposed method achieves an accuracy of order sixteen with only five functional evaluations, and the efficiency index is 1.741. To demonstrate the performance of the method, the authors present several numerical examples, application problems, and a study of dynamics, comparing the method with other available methods of the same order in the literature. Numerical results and dynamics are obtained using computer tools.