A novel hybrid iterative method for applied mathematical models with time-efficiency

Abstract

Non-linear phenomena appear in many fields of engineering and science. Research on numerical methods is continually extending with the improvement of the latest computing tools. In today's computational field, one requires maximum achievement in a minimum amount of time. Therefore, there is a need to modify the Newton-type method to achieve higher-order convergence to solve non-linear equations. While the modified methods are expected to be higher-order convergent, the minor computational information and the maximum time efficiency are also important factors. We propose a three-step hybrid iterative method having a non-linear nature. Per iteration, the method requires three function evaluations and three first-order derivatives. The method is theoretically proven to be tenth-order convergent. The mathematical results of the proposed strategy to solve models from fluid dynamics, electric field, and real gases demonstrated better performance. In light of error analysis, computational productivity, and CPU times, the proposed method's performance is compared to the famous Newton and a recently proposed tenth-order method.

A Modified Hybrid Method For Solving Non-Linear Equations With Computational Efficiency

“…The purpose of presently used method tries to decrease function evaluation costs as much as feasible while enhancing convergence order. Such types of estimations are taken by so many researchers [2,3,6,8,9,11,16,18,21,23]. Hence, after merging Eq n : (8) and Eq n : (9), We acquire the proposed method shown below:…”

An Improved Blended Numerical Root-Solver for Nonlinear Equations

“…The main objective of this study is to introduce a novel and effective multi-step approach, namely a three-step methodology, for addressing approximate the solution of nonlinear problems. The proposed procedure starts with the Newton step, which is a technique that many scholars use including [1,4,5,9,10,13,24] and [2]. Furthermore, the suggested three-step method is proved to have sixth-order convergence by utilizing only five evaluations.…”