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

Abstract

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.

Sets of Fractional Operators and Some of Their Applications

“…which may be interpreted as the set of fractional fixed-point methods that define a convergent sequence x i fg i ≥ 1 to some value ξ α ∈ B ξ; δ ðÞ . So, denoting by card Á ðÞthe cardinality of a set, under certain conditions it is possible to prove the following result (see reference [24], proof of Theorem 2):…”

“…It is necessary to mention that, for practical purposes, it may be defined that if a fractional iterative method Φ fulfills the properties of the Corollary 1.8 and uses the function (81), it may be called a fractional iterative method accelerated. Finally, it is necessary to mention that fractional iterative methods may be defined in the complex space [24], that is, Φ α, x ðÞ : α ∈ n and x ∈  n fg :…”

“…Before continuing, it is necessary to mention that due to the large number of fractional operators that may exist [1][2][3][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], some sets must be defined to fully characterize elements of fractional calculus. It is worth mentioning that characterizing elements of fractional calculus through sets is the main idea behind of the methodology known as fractional calculus of sets [24,25]. So, considering a scalar function h :  m !…”

“…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 [27,28], economics [29], number theory through the Riemann zeta function [30,31], and in engineering with the study for the manufacture of hybrid solar receivers [32,33]. It is worth mentioning that there exists also a growing interest in fractional operators and their properties for solving nonlinear algebraic systems [24,[34][35][36][37][38][39][40][41], which is a classical problem in mathematics, physics and engineering, which consists of finding the set of zeros of a function f :…”

Proposal for Use of the Fractional Derivative of Radial Functions in Interpolation Problems

“…Fractional operators have various representations, but one of their fundamental properties is that they recover the results of conventional calculus when α → n. Before continuing, it is worth mentioning that due to the large number of fractional operators that exist [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56], it seems that the most natural way to fully characterize the elements of fractional calculus is by using sets, which is the main idea behind the methodology known as fractional calculus of sets [57][58][59][60], whose seed of origin is the fractional Newton-Raphson method [24]. Therefore, considering a scalar function h : R m → R and the canonical basis of R m denoted by { êk } k≥1 , it is feasible to define the following fractional operator of order α using Einstein's notation:…”