Some Novel Sixth-Order Iteration Schemes for Computing Zeros of Nonlinear Scalar Equations and Their Applications in Engineering

M A Rehman,Thabet Abdeljawad,Amir Naseem,Ioan Rasa

doi:10.1155/2021/5566379

Abstract

In this paper, we propose two novel iteration schemes for computing zeros of nonlinear equations in one dimension. We develop these iteration schemes with the help of Taylor’s series expansion, generalized Newton-Raphson’s method, and interpolation technique. The convergence analysis of the proposed iteration schemes is discussed. It is established that the newly developed iteration schemes have sixth order of convergence. Several numerical examples have been solved to illustrate the applicability and validity of the suggested schemes. These problems also include some real-life applications associated with the chemical and civil engineering such as adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia, the van der Wall’s equation, and the open channel flow problem whose numerical results prove the better efficiency of these methods as compared to other well-known existing iterative methods of the same kind.

Highlights

The solution of nonlinear scalar equations plays a vital role in many fields of applied sciences such as Engineering, Physics, and Mathematics
In the last few years, a lot of researchers worked on iterative methods with their applications and proposed some new iterative schemes which possesses either a high convergence rate or have less number of functional evaluations per iteration, see [9,10,11,12,13,14,15,16,17,18,19,20,21] and the references therein
The convergence rate of an iterative method can be increased by involving predictor and corrector steps which results multistep iterative methods whereas the number of functional evaluations can be reduced by removing second and higher derivatives in the considered iterative method using different mathematical techniques

Summary

Introduction

The solution of nonlinear scalar equations plays a vital role in many fields of applied sciences such as Engineering, Physics, and Mathematics. The convergence rate of an iterative method can be increased by involving predictor and corrector steps which results multistep iterative methods whereas the number of functional evaluations can be reduced by removing second and higher derivatives in the considered iterative method using different mathematical techniques. In twenty-first century, many mathematicians try to modify the existing methods with less number of functional evaluations per iterations and higher convergence order by applying different techniques such as predictor-corrector technique, finite difference scheme, interpolation technique, Taylor’s series, and quadrature formula etc. In 2007, Noor et al [22] introduced a two-step Halley’s method with sextic convergence and approximated its Journal of Function Spaces second derivative by the utilization of finite difference scheme and suggested a novel second-derivative free iterative algorithm which have fifth convergence order. The proposed iteration schemes have been applied to solve some real life problems along with the arbitrary transcendental and algebraic equations in order to assess its applicability, validity, and accuracy

Main Results

Convergence Analysis

Comparison of Efficiency Index

Numerical Comparisons and Applications

Concluding Remarks