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:
For each operator , the fractional matrix operator is defined as:
and for each operator , the following matrix, corresponding to a generalization of the
Jacobian matrix[12], can be defined:
where .
Modified Hadamard Product
Considering that, in general, , the following modified Hadamard product is defined:
with which the following theorem is obtained:
Theorem: Abelian Group of Fractional Matrix Operators
Let be a fractional operator such that . Considering the modified Hadamard product, the following set of fractional matrix operators is defined:
which corresponds to the
Abelian group[13] generated by the operator .
Proof
Since the set in equation (1) is defined by applying only the vertical type Hadamard product between its elements, for all it holds that:
with which it is possible to prove that the set (1) satisfies the following properties of an Abelian group:
Set of Fractional Operators
Let be the set . If and , then the following
multi-index notation can be defined:
Then, considering a function and the fractional operator:
the following set of fractional operators is defined:
From which the following results are obtained:
As a consequence, considering a function , the following set of fractional operators is defined:
Set of Fractional Operators
Considering a function and the following set of fractional operators:
Then, taking a ball , it is possible to define the following set of fractional operators:
which allows generalizing the expansion in
Taylor series of a vector-valued function in multi-index notation. As a consequence, the following result can be obtained:
Fractional Newton-Raphson Method
Let be a function with a point such that . Then, for some and a fractional operator , it is possible to define a type of linear approximation of the function around as follows:
which can be expressed more compactly as:
where denotes a square matrix. On the other hand, as and given that , the following is inferred:
As a consequence, defining the matrix:
the following fractional iterative method can be defined:
^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.
fractional calculus; fractional operators; set theory; group theory; fractional calculus of sets
Review waiting, please be patient.
This may take 4 months or more, since drafts are reviewed in no specific order. There are 2,677 pending submissions
waiting for review.
If the submission is accepted, then this page will be moved into the article space.
If the submission is declined, then the reason will be posted here.
In the meantime, you can continue to improve this submission by editing normally.
Where to get help
If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews.
If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the
talk page of a
relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed.
To improve your odds of a faster review, tag your draft with relevant
WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags.
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:
For each operator , the fractional matrix operator is defined as:
and for each operator , the following matrix, corresponding to a generalization of the
Jacobian matrix[12], can be defined:
where .
Modified Hadamard Product
Considering that, in general, , the following modified Hadamard product is defined:
with which the following theorem is obtained:
Theorem: Abelian Group of Fractional Matrix Operators
Let be a fractional operator such that . Considering the modified Hadamard product, the following set of fractional matrix operators is defined:
which corresponds to the
Abelian group[13] generated by the operator .
Proof
Since the set in equation (1) is defined by applying only the vertical type Hadamard product between its elements, for all it holds that:
with which it is possible to prove that the set (1) satisfies the following properties of an Abelian group:
Set of Fractional Operators
Let be the set . If and , then the following
multi-index notation can be defined:
Then, considering a function and the fractional operator:
the following set of fractional operators is defined:
From which the following results are obtained:
As a consequence, considering a function , the following set of fractional operators is defined:
Set of Fractional Operators
Considering a function and the following set of fractional operators:
Then, taking a ball , it is possible to define the following set of fractional operators:
which allows generalizing the expansion in
Taylor series of a vector-valued function in multi-index notation. As a consequence, the following result can be obtained:
Fractional Newton-Raphson Method
Let be a function with a point such that . Then, for some and a fractional operator , it is possible to define a type of linear approximation of the function around as follows:
which can be expressed more compactly as:
where denotes a square matrix. On the other hand, as and given that , the following is inferred:
As a consequence, defining the matrix:
the following fractional iterative method can be defined:
^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.