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Open AccessArticle

Multipoint Fractional Iterative Methods with (2α + 1)th-Order of Convergence for Solving Nonlinear Problems

1
Área de Ciencias Básicas y Ambientales, Instituto Tecnológico de Santo Domingo (INTEC), Santo Domingo 10602, Dominican Republic
2
Institute for Multidisciplinary Mathematics, Universitat Politècnica de Valenència, Camino de Vera s/n, 46022 València, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(3), 452; https://doi.org/10.3390/math8030452
Received: 17 February 2020 / Revised: 6 March 2020 / Accepted: 18 March 2020 / Published: 20 March 2020
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems 2020)
In the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence α + 1 and compare it with the existing fractional Newton method with order 2 α . Moreover, we also introduce a multipoint fractional Traub-type method with order 2 α + 1 and compare its performance with that of its first step. Some numerical tests and analysis of the dependence on the initial estimations are made for each case, including a comparison with classical Newton ( α = 1 of the first step of the class) and classical Traub’s scheme ( α = 1 of fractional proposed multipoint method). In this comparison, some cases are found where classical Newton and Traub’s methods do not converge and the proposed methods do, among other advantages. View Full-Text
Keywords: nonlinear equations; fractional derivatives; multistep methods; convergence; stability nonlinear equations; fractional derivatives; multistep methods; convergence; stability
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Candelario, G.; Cordero, A.; Torregrosa, J.R. Multipoint Fractional Iterative Methods with (2α + 1)th-Order of Convergence for Solving Nonlinear Problems. Mathematics 2020, 8, 452.

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