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.
Highlights
A fractional differential equation is an equation that involves at least one differential operator of order α, with (n − 1) < α ≤ n for some positive integer n, and it is said to be a differential equation of order α if this operator is the highest order in the equation
This paper presents 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 allows generalizing objects of the 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
Summary
A fractional derivative is an operator that generalizes the ordinary derivative, in the sense that if Publisher’s Note: MDPI stays neutral dα dx α with regard to jurisdictional claims in published maps and institutional affiliations. Denoting by ∂nk the partial derivative of order n applied with respect to the k-th component of the vector x and using the previous operator, it is possible to define the following set of fractional operators as follows:. Considering a function h : Rm × R≥0 → R, as well as the vectors α, n ∈ R3 with α = êk [α]k and n = êk [n]k , it is possible to combine the sets (2) and (7) to define new sets of fractional operators related to the theory of differential equations, as shown with the following set: o [α]. It is presented one method to estimate numerically in a region Ω the mean order of convergence of any fractional fixed-point method through the problem of determining the critical points of a scalar function, and it is shown how to construct a hybrid fractional iterative method to determine the critical points of a scalar function
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
