fractional calculus; fractional operators; set theory; group theory; fractional calculus of sets
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The Fractional Calculus of Sets (FCS), first introduced in the article titled "Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods" [1], is a methodology derived from fractional calculus[2]. The primary concept behind FCS is the characterization of fractional calculus elements using sets due to the plethora of fractional operators available.[3][4][5] This methodology originated from the development of the Fractional Newton-Raphson method[6] and subsequent related works [7][8][9]
.
Set of Fractional Operators
Fractional calculus, a branch of mathematics dealing with derivatives of non-integer order, emerged nearly simultaneously with traditional calculus. This emergence was partly due to Leibniz's notation for derivatives of integer order: . Thanks to this notation, L'Hopital was able to inquire in a letter to Leibniz about the interpretation of taking in a derivative. At that moment, Leibniz couldn't provide a physical or geometric interpretation for this question, so he simply replied to L'Hopital in a letter that "... is an apparent paradox from which, one day, useful consequences will be drawn".
The name "fractional calculus" originates from a historical question, as this branch of mathematical analysis studies derivatives and integrals of a certain order . Currently, fractional calculus lacks a unified definition of what constitutes a fractional derivative. Consequently, when the explicit form of a fractional derivative is unnecessary, it is typically denoted as follows:
Fractional operators have various representations, but one of their fundamental properties is that they recover the results of traditional calculus as . Considering a scalar function and the canonical basis of denoted by , the following fractional operator of order is defined using Einstein notation[10]:
Denoting as the partial derivative of order with respect to the -th component of the vector , the following set of fractional operators is defined:
with its complement:
Consequently, the following set is defined:
Extension to Vectorial Functions
For a function , the set is defined as:
where denotes the -th component of the function .
Set of Fractional Operators
The set of fractional operators considering infinite orders is defined as:
^Torres-Hernandez, A.; Brambila-Paz, F.; Ramirez-Melendez, R. (2022). "Sets of Fractional Operators and Some of Their Applications". Operator Theory - Recent Advances, New Perspectives and Applications.
^[2] Einstein summation for multidimensional arrays