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

Candelario, Giro; Cordero, Alicia; Torregrosa, Juan R.

doi:10.3390/math8030452

Cited by 26 publications

(22 citation statements)

References 11 publications

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“…Next, from the definition of radius Lipschitz given in the inequality (9) and using the inequality (21), it can written as…”

Section: Local Convergence Of Newton Type Methods (3)mentioning

confidence: 99%

“…Corollary 5. Suppose that x * satisfies t(x * ) = 0, t has a continuous derivative in V(x * , r), [t (x * )] −1 exists and [t (x * )] −1 t satisfies (9), (11) with L(u) = cau a−1 and L 0 (u) = c 0 au a−1 i.e.,…”

Section: Convergence Under Weak L-averagementioning

confidence: 99%

“…Numerous researchers studied the local convergence analysis for Newton-type, Jarratt-type, Weerakoon-type, etc. in Banach space setting in the articles [8][9][10][11][12][13][14] and reference therein. In most of articles, the local convergence have been discussed using the hypotheses of Lipschitz, Hölder or w-continuity conditions but sometimes, we will come across that the nonlinear problems do not fulfilled any of these three conditions which limits the applicability of nonlinear equations, but satisfy the generalized Lipschitz condition.…”

Section: Introductionmentioning

confidence: 99%

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On the Local Convergence of Two-Step Newton Type Method in Banach Spaces under Generalized Lipschitz Conditions

et al. 2021

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The motive of this paper is to discuss the local convergence of a two-step Newton-type method of convergence rate three for solving nonlinear equations in Banach spaces. It is assumed that the first order derivative of nonlinear operator satisfies the generalized Lipschitz i.e., L-average condition. Also, some results on convergence of the same method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L-average particularly it is assumed that L is positive integrable function but not necessarily non-decreasing. Our new idea gives a tighter convergence analysis without new conditions. The proposed technique is useful in expanding the applicability of iterative methods. Useful examples justify the theoretical conclusions.

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“…Next, from the definition of radius Lipschitz given in the inequality (9) and using the inequality (21), it can written as…”

Section: Local Convergence Of Newton Type Methods (3)mentioning

confidence: 99%

Section: Convergence Under Weak L-averagementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Local Convergence of Two-Step Newton Type Method in Banach Spaces under Generalized Lipschitz Conditions

et al. 2021

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“…In this section we study the stability of conformable vectorial Newton's method tested above. In that sense, we analyze the dependence on initial estimates by means of convergence planes, which is defined in [14], and used in [3][4][5].…”

Section: Numerical Stabilitymentioning

confidence: 99%

Generalized conformable fractional Newton-type method for solving nonlinear systems

Candelario

Cordero

Torregrosa

et al. 2022

Preprint

Self Cite

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In a recent paper, a conformable fractional Newton-type method was proposed for solving nonlinear equations. This method involves a lower computational cost compared to others fractional iterative methods. Indeed, the theoretical order of convergence is held in practice, and it presents a better numerical behaviour than fractional Newton-type methods formerly proposed, even compared to classical Newton-Raphson method. In this work, we design a generalization of this method for solving nonlinear systems by using a new conformable fractional Jacobian matrix, and a suitable conformable Taylor power series; and it is compared with classical Newton’s scheme. The necessary concepts and results are stated in order to design this method. Convergence analysis is made and a quadratic order of convergence is obtained, as in classical Newton's method. Numerical tests are made, and the Approximated Computational Order ofo Convergence (ACOC) supports the theory. Also, the proposed scheme shows good stability properties observed by means of convergence planes.

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“…To finish this section, it is necessary to mention that the applications of fractional operators have spread to different fields of science, such as finance [1,2], economics [3], number theory through the Riemann zeta function [4,5] and in engineering with the study of the manufacture of hybrid solar receivers [6,7]. It should be mentioned that there is also a growing interest in fractional operators and their properties for the solution of nonlinear algebraic systems [7][8][9][10][11][12][13][14][15][16][17][18], which is a classical problem in mathematics, physics and engineering that consists of finding the set of zeros of a function f :…”

Section: Introductionmentioning

confidence: 99%

Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods

Torres-Hernandez

Brambila-Paz

2021

Fractal Fract

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Considering the large number of fractional operators that exist, and since it does not seem that their number will stop increasing soon at the time of writing this paper, it is presented for the first time, as far as the authors know, a simple and compact method to work the fractional calculus through the classification of fractional operators using sets. This new method of working with fractional operators, which may be called fractional calculus of sets, allows generalizing objects of conventional calculus, such as tensor operators, the Taylor series of a vector-valued function, and the fixed-point method, in several variables, which allows generating the method known as the fractional fixed-point method. Furthermore, it is also shown that each fractional fixed-point method that generates a convergent sequence has the ability to generate an uncountable family of fractional fixed-point methods that generate convergent sequences. So, it is presented a method to estimate numerically in a region Ω the mean order of convergence of any fractional fixed-point method, and it is shown how to construct a hybrid fractional iterative method to determine the critical points of a scalar function. Finally, considering that the proposed method to classify fractional operators through sets allows generalizing the existing results of the fractional calculus, some examples are shown of how to define families of fractional operators that satisfy some property to ensure the validity of the results to be generalized.

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

Cited by 26 publications

References 11 publications

On the Local Convergence of Two-Step Newton Type Method in Banach Spaces under Generalized Lipschitz Conditions

On the Local Convergence of Two-Step Newton Type Method in Banach Spaces under Generalized Lipschitz Conditions

Generalized conformable fractional Newton-type method for solving nonlinear systems

Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods

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