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Open AccessArticle

A Variant of Chebyshev’s Method with 3αth-Order of Convergence by Using Fractional Derivatives

1
Institute of Multidisciplinary Mathematics, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
2
Facultat de Matemàtiques, Universitat de València, 46022 València, Spain
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(8), 1017; https://doi.org/10.3390/sym11081017
Received: 19 July 2019 / Revised: 31 July 2019 / Accepted: 3 August 2019 / Published: 6 August 2019
(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)
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Abstract

In this manuscript, we propose several iterative methods for solving nonlinear equations whose common origin is the classical Chebyshev’s method, using fractional derivatives in their iterative expressions. Due to the symmetric duality of left and right derivatives, we work with right-hand side Caputo and Riemann–Liouville fractional derivatives. To increase as much as possible the order of convergence of the iterative scheme, some improvements are made, resulting in one of them being of 3 α -th order. Some numerical examples are provided, along with an study of the dependence on initial estimations on several test problems. This results in a robust performance for values of α close to one and almost any initial estimation. View Full-Text
Keywords: nonlinear equations; Chebyshev’s iterative method; fractional derivative; basin of attraction nonlinear equations; Chebyshev’s iterative method; fractional derivative; basin of attraction
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Cordero, A.; Girona, I.; Torregrosa, J.R. A Variant of Chebyshev’s Method with 3αth-Order of Convergence by Using Fractional Derivatives. Symmetry 2019, 11, 1017.

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