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Discontinuity, Nonlinearity, and Complexity 13(1) (2024) 133--142 | DOI:10.5890/DNC.2024.03.010

Dumitru Baleanu$^{1,2,3}$, Ali S. Alshomrani$^{4}$, Sania Qureshi$^{5,6}$, Amanullah Soomro$^{5}$

$^{1}$ Department of Mathematics, Cankaya University, Öǧretmenler Cad. 1406530, Ankara, Turkey

$^{2}$ Institute of Space Sciences, Magurele, Bucharest, Romania

$^{3}$ Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

$^{4}$ Department of Mathematics, King Abdul Aziz University, Jeddah, Saudi Arabia

$^{5}$ Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro -- 76062, Pakistan

$^{6}$ Department of Mathematics, Near East University TRNC, Mersin 10, Turkey

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Abstract

The approach of solving nonlinear models with numerical techniques is on the rise owing to the omnipresence of the models in several scientific fields. This paper developed an optimal variant of Halley's method without memory of order five for solving nonlinear equations $w(x)=0$. The technique is one-step with five function evaluations required in each iteration and has an efficiency index of $1.38$. The idea of basins of attraction to study the suggested technique's influence on the initial estimation is considered that reveals stable nature. This is also supported by various numerical examples that show how the proposed approach performs compared to other existing techniques. For examples considered, such as Vander Waals' equation and continuously stirred tank reactors, the proposed method without memory arrives at approximations to the roots with fewer iterations and better accuracy. Convergence analysis is also discussed to prove the fifth-order accuracy and complex dynamics is discussed via polynomiographs.

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