New Iterative Methods for the Solution of Nonlinear Equations

Syed Muhammad Hussain Shah; Muhammad Asim Ullah; Murad Ali Shah

Author(s)

Syed Muhammad Hussain Shah Edward College Peshawar, Mall Road, Peshawar Cantonment, Peshawar 25000, Pakistan

Muhammad Asim Ullah School of Mathematics and physics, Anqing Normal University, Anqing 246003, China

Murad Ali Shah School of Mathematics and physics, Anqing Normal University, Anqing 246003, China

Abstract

s The proposed study develops a new iterative techniques to solve highly nonlinear equations of the form ( ) = 0. Two techniques are employed, one of which involves the second derivative and the other free from the second derivative. The Taylor series is used to expand the function by substituting = - 2 ' ( ')'-( )''( ) and ''( ) = '( + (( )))- '( ) to get the two algorithms. Furthermore, several examples of nonlinear equations are considered to get the numerical solutions through the derived algorithms. The proposed work is compared with some other iterative techniques like Newton's Method (NM), Adomian Method (AM), Newton Raphson Method (NRM), Chebyshev's method (CM), and Halley's Method (HM). The obtained results are highly accurate, and the convergence is faster than the techniques mentioned. The results are presented in tabular form and discussed which validates the current work and shows the impact of this article.

Publication Details

Page(s) 63-74

DOI DOI not available

Published Journal: The Sciencetech, Volume: 5, Issue: 4, Year: 2024

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Keywords

Newton's Methods, Newton Raphson method, Taylor Expansion, Adomian Methods, Halley's Methods, Nonlinear equation, Chebyshev's Method

References

Azad M.,N. M.,Pan R.,K. R. .2023 .The new iteration methods for solving absolute value equations. Applications of Mathematics, 68(1) : 109-122.

Sidi Avram .2006 .Unified treatment of regular falai, Newton-Raphson, Secent, and Steffensen methods for nonlinear equations. Journal of Online Mathematics and Its Applications, : 113.

Ahmed. F. ,Hussain. S. .2012 .New derivative free iterative method for solving nonlinear equations. Academic. Research int. 2, (1) : 117-123.

Behl R.,Cordero A.,Torregrosa J. R.,Saleh Alshomrani A. .2018 .New iterative methods for solving nonlinear problems with one and several unknowns. Mathematics, 6(12) : 296.

Chun C. .2006 .A new iterative method for solving nonlinear equations. Applied mathematics and computation, 178(2) : 415-422.

Chun C. .2008 .Some fourth-order iterative methods for solving nonlinear equations. Applied mathematics and computation, 195(2) : 454-459.

Daftardar-Gejji V.,Jafari H. .2006 .An iterative method for solving nonlinear functional equations. Journal of mathematical analysis and applications, 316(2) : 753-763.

Darvishi M. T.,Barati A. .2007 .A third-order Newton-type method to solve systems of nonlinear equations. Applied Mathematics and Computation, 187(2) : 630-635.

Din. N.A. ,H. .2010 .Numerical Solution of Nonlinear Equation by using Derivative of First order. WASET, : 138-139.

Junjua ,M. U. D. ,Akram S.,Yasmin N.,Zafar F. .2015 .A new Jarratt-type fourth-order method for solving a system of nonlinear equations and applications. Journal of Applied Mathematics, : .

Li. W. ,Chen. H. .2010 .A Unified Framework for the construction of Higher order methods for non-linear Equations, TONMJ. 2. , : 6-11.

Mitlif R. J. .2013 .A new iterative method for solving nonlinear equations. Baghdad Science Journal, 11(4) : 1649-1654.

Nazeer W.,Tanveer M.,Munir M.,Jan A.,Kang S. M. .2016 .Modified Halley's Method for solving nonlinear functions with convergence of order six and efficiency index 1. International Journal of Pure and Applied Mathematics, 111(1) : 55-65.

Noor. M. ,Khan. W. A. .2012 .Fourth - order Iterative Method Free from Second Derivative for solving nonlinear Equations, Appl. , 6(93) : 4617-4625.

Panday S.,Sharma A.,Thangkhenpau G. .2023 .Optimal fourth and eighth-order iterative methods for non-linear equations. Journal of Applied Mathematics and Computing, 69(1) : 953-971.

Raza. M. ,Ahmed. F. .2012 .Fourth order convergent Two step iterative algorithm for solving nonlinear equations. academic. Research.int. 2, (1) : 111-116.

Saeed R. K.,Aziz K. M. .2008 .An iterative method with quartic convergence for solving nonlinear equations. Applied mathematics and computation, 202(2) : 435-440.

Saheya B.,Chen G. Q.,Sui Y. K.,Wu C. Y. .2016 .A new Newtonlike method for solving nonlinear equations. SpringerPlus, 5 : 1-13.

Steffensen J. .1933 .Remarks on iteration. Scandinavian Actuarial Journal, : 64-72.

Thangkhenpau G.,Panday S. .2023 .Optimal eight order derivativefree family of iterative methods for solving nonlinear equations. IAENG Int. J. Comput. Sci, 50 : 335-341.