fractional calculus; fractional operators; set theory; group theory; fractional calculus of sets
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
O
x
,
α
n
(
h
)
{\displaystyle O_{x,\alpha }^{n}(h)}
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:
d
n
d
x
n
{\displaystyle {\frac {d^{n}}{dx^{n}}}}
. Thanks to this notation, L'Hopital was able to inquire in a letter to Leibniz about the interpretation of taking
n
=
1
2
{\displaystyle n={\frac {1}{2}}}
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
α
∈
R
{\displaystyle \alpha \in \mathbb {R} }
. 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:
d
α
d
x
α
.
{\displaystyle {\frac {d^{\alpha }}{dx^{\alpha }}}.}
Fractional operators have various representations, but one of their fundamental properties is that they recover the results of traditional calculus as
α
→
n
{\displaystyle \alpha \to n}
. Considering a scalar function
h
:
R
m
→
R
{\displaystyle h:\mathbb {R} ^{m}\to \mathbb {R} }
and the canonical basis of
R
m
{\displaystyle \mathbb {R} ^{m}}
denoted by
{
e
^
k
}
k
≥
1
{\displaystyle \{{\hat {e}}_{k}\}_{k\geq 1}}
, the following fractional operator of order
α
{\displaystyle \alpha }
is defined using Einstein notation [ 10] :
o
x
α
h
(
x
)
:=
e
^
k
o
k
α
h
(
x
)
.
{\displaystyle o_{x}^{\alpha }h(x):={\hat {e}}_{k}o_{k}^{\alpha }h(x).}
Denoting
∂
k
n
{\displaystyle \partial _{k}^{n}}
as the partial derivative of order
n
{\displaystyle n}
with respect to the
k
{\displaystyle k}
-th component of the vector
x
{\displaystyle x}
, the following set of fractional operators is defined:
O
x
,
α
n
(
h
)
:=
{
o
x
α
:
∃
o
k
α
h
(
x
)
and
lim
α
→
n
o
k
α
h
(
x
)
=
∂
k
n
h
(
x
)
∀
k
≥
1
}
,
{\displaystyle O_{x,\alpha }^{n}(h):=\left\{o_{x}^{\alpha }:\exists o_{k}^{\alpha }h(x){\text{ and }}\lim _{\alpha \to n}o_{k}^{\alpha }h(x)=\partial _{k}^{n}h(x)\ \forall k\geq 1\right\},}
with its complement:
O
x
,
α
n
,
c
(
h
)
:=
{
o
x
α
:
∃
o
k
α
h
(
x
)
∀
k
≥
1
and
lim
α
→
n
o
k
α
h
(
x
)
≠
∂
k
n
h
(
x
)
for at least one
k
≥
1
}
.
{\displaystyle O_{x,\alpha }^{n,c}(h):=\left\{o_{x}^{\alpha }:\exists o_{k}^{\alpha }h(x)\ \forall k\geq 1{\text{ and }}\lim _{\alpha \to n}o_{k}^{\alpha }h(x)\neq \partial _{k}^{n}h(x){\text{ for at least one }}k\geq 1\right\}.}
Consequently, the following set is defined:
O
x
,
α
n
,
u
(
h
)
:=
O
x
,
α
n
(
h
)
∪
O
x
,
α
n
,
c
(
h
)
.
{\displaystyle O_{x,\alpha }^{n,u}(h):=O_{x,\alpha }^{n}(h)\cup O_{x,\alpha }^{n,c}(h).}
Extension to Vectorial Functions
For a function
h
:
Ω
⊂
R
m
→
R
m
{\displaystyle h:\Omega \subset \mathbb {R} ^{m}\to \mathbb {R} ^{m}}
, the set is defined as:
m
O
x
,
α
n
,
u
(
h
)
:=
{
o
x
α
:
o
x
α
∈
O
x
,
α
n
,
u
(
[
h
]
k
)
∀
k
≤
m
}
,
{\displaystyle {}_{m}O_{x,\alpha }^{n,u}(h):=\left\{o_{x}^{\alpha }:o_{x}^{\alpha }\in O_{x,\alpha }^{n,u}([h]_{k})\ \forall k\leq m\right\},}
where
[
h
]
k
:
Ω
⊂
R
m
→
R
{\displaystyle [h]_{k}:\Omega \subset \mathbb {R} ^{m}\to \mathbb {R} }
denotes the
k
{\displaystyle k}
-th component of the function
h
{\displaystyle h}
.
Set
m
M
O
x
,
α
∞
,
u
(
h
)
{\displaystyle {}_{m}MO_{x,\alpha }^{\infty ,u}(h)}
of Fractional Operators
The set of fractional operators considering infinite orders is defined as:
m
M
O
x
,
α
∞
,
u
(
h
)
:=
⋂
k
∈
Z
m
O
x
,
α
k
,
u
(
h
)
,
{\displaystyle {}_{m}MO_{x,\alpha }^{\infty ,u}(h):=\bigcap _{k\in \mathbb {Z} }{}_{m}O_{x,\alpha }^{k,u}(h),}
where under classic Hadamard product [ 11] it holds that:
o
x
0
∘
h
(
x
)
:=
h
(
x
)
∀
o
x
α
∈
m
M
O
x
,
α
∞
,
u
(
h
)
.
{\displaystyle o_{x}^{0}\circ h(x):=h(x)\quad \forall o_{x}^{\alpha }\in {}_{m}MO_{x,\alpha }^{\infty ,u}(h).}
Fractional Matrix Operators
For each operator
o
x
α
{\displaystyle o_{x}^{\alpha }}
, the fractional matrix operator is defined as:
A
α
(
o
x
α
)
=
(
[
A
α
(
o
x
α
)
]
j
k
)
=
(
o
k
α
)
,
{\displaystyle A_{\alpha }(o_{x}^{\alpha })=\left([A_{\alpha }(o_{x}^{\alpha })]_{jk}\right)=\left(o_{k}^{\alpha }\right),}
and for each operator
o
x
α
∈
m
M
O
x
,
α
∞
,
u
(
h
)
{\displaystyle o_{x}^{\alpha }\in {}_{m}MO_{x,\alpha }^{\infty ,u}(h)}
, the following matrix, corresponding to a generalization of the Jacobian matrix [ 12] , can be defined:
A
h
,
α
:=
A
α
(
o
x
α
)
∘
A
α
T
(
h
)
,
{\displaystyle A_{h,\alpha }:=A_{\alpha }(o_{x}^{\alpha })\circ A_{\alpha }^{T}(h),}
where
A
α
(
h
)
:=
(
[
A
α
(
h
)
]
j
k
)
=
(
[
h
]
k
)
{\displaystyle A_{\alpha }(h):=\left([A_{\alpha }(h)]_{jk}\right)=\left([h]_{k}\right)}
.
Modified Hadamard Product
Considering that, in general,
o
x
p
α
∘
o
x
q
α
≠
o
x
(
p
+
q
)
α
{\displaystyle o_{x}^{p\alpha }\circ o_{x}^{q\alpha }\neq o_{x}^{(p+q)\alpha }}
, the following modified Hadamard product is defined:
o
i
,
x
p
α
∘
o
j
,
x
q
α
:=
{
o
i
,
x
p
α
∘
o
j
,
x
q
α
,
if
i
≠
j
(
horizontal type Hadamard product
)
o
i
,
x
(
p
+
q
)
α
,
if
i
=
j
(
vertical type Hadamard product
)
,
{\displaystyle o_{i,x}^{p\alpha }\circ o_{j,x}^{q\alpha }:=\left\{{\begin{array}{cl}o_{i,x}^{p\alpha }\circ o_{j,x}^{q\alpha },&{\text{if }}i\neq j\ ({\text{horizontal type Hadamard product}})\\o_{i,x}^{(p+q)\alpha },&{\text{if }}i=j\ ({\text{vertical type Hadamard product}})\end{array}}\right.,}
with which the following theorem is obtained:
Theorem: Abelian Group of Fractional Matrix Operators
Let
o
x
α
{\displaystyle o_{x}^{\alpha }}
be a fractional operator such that
o
x
α
∈
m
M
O
x
,
α
∞
,
u
(
h
)
{\displaystyle o_{x}^{\alpha }\in {}_{m}MO_{x,\alpha }^{\infty ,u}(h)}
. Considering the modified Hadamard product, the following set of fractional matrix operators is defined:
m
G
(
A
α
(
o
x
α
)
)
:=
{
A
α
∘
r
=
A
α
(
o
x
r
α
)
:
r
∈
Z
and
A
α
∘
r
=
(
[
A
α
∘
r
]
j
k
)
:=
(
o
k
r
α
)
}
,
{\displaystyle {}_{m}G(A_{\alpha }(o_{x}^{\alpha })):=\left\{A_{\alpha }^{\circ r}=A_{\alpha }(o_{x}^{r\alpha }):r\in \mathbb {Z} \ {\text{and}}\ A_{\alpha }^{\circ r}=\left([A_{\alpha }^{\circ r}]_{jk}\right):=\left(o_{k}^{r\alpha }\right)\right\},}
(1)
which corresponds to the Abelian group [ 13] generated by the operator
A
α
(
o
x
α
)
{\displaystyle A_{\alpha }(o_{x}^{\alpha })}
.
Proof
Since the set in equation (1) is defined by applying only the vertical type Hadamard product between its elements, for all
A
α
∘
p
,
A
α
∘
q
∈
m
G
(
A
α
(
o
x
α
)
)
{\displaystyle A_{\alpha }^{\circ p},A_{\alpha }^{\circ q}\in {}_{m}G(A_{\alpha }(o_{x}^{\alpha }))}
it holds that:
A
α
∘
p
∘
A
α
∘
q
=
(
[
A
α
∘
p
]
j
k
)
∘
(
[
A
α
∘
q
]
j
k
)
=
(
o
k
(
p
+
q
)
α
)
=
(
[
A
α
∘
(
p
+
q
)
]
j
k
)
=
A
α
∘
(
p
+
q
)
,
{\displaystyle A_{\alpha }^{\circ p}\circ A_{\alpha }^{\circ q}=\left([A_{\alpha }^{\circ p}]_{jk}\right)\circ \left([A_{\alpha }^{\circ q}]_{jk}\right)=\left(o_{k}^{(p+q)\alpha }\right)=\left([A_{\alpha }^{\circ (p+q)}]_{jk}\right)=A_{\alpha }^{\circ (p+q)},}
with which it is possible to prove that the set (1) satisfies the following properties of an Abelian group:
{
∀
A
α
∘
p
,
A
α
∘
q
,
A
α
∘
r
∈
m
G
(
A
α
(
o
x
α
)
)
,
(
A
α
∘
p
∘
A
α
∘
q
)
∘
A
α
∘
r
=
A
α
∘
p
∘
(
A
α
∘
q
∘
A
α
∘
r
)
∃
A
α
∘
0
∈
m
G
(
A
α
(
o
x
α
)
)
such that
∀
A
α
∘
p
∈
m
G
(
A
α
(
o
x
α
)
)
,
A
α
∘
0
∘
A
α
∘
p
=
A
α
∘
p
∀
A
α
∘
p
∈
m
G
(
A
α
(
o
x
α
)
)
,
∃
A
α
∘
−
p
∈
m
G
(
A
α
(
o
x
α
)
)
such that
A
α
∘
p
∘
A
α
∘
−
p
=
A
α
∘
0
∀
A
α
∘
p
,
A
α
∘
q
∈
m
G
(
A
α
(
o
x
α
)
)
,
A
α
∘
p
∘
A
α
∘
q
=
A
α
∘
q
∘
A
α
∘
p
.
{\displaystyle \left\{{\begin{array}{l}\forall A_{\alpha }^{\circ p},A_{\alpha }^{\circ q},A_{\alpha }^{\circ r}\in {}_{m}G(A_{\alpha }(o_{x}^{\alpha })),\ \left(A_{\alpha }^{\circ p}\circ A_{\alpha }^{\circ q}\right)\circ A_{\alpha }^{\circ r}=A_{\alpha }^{\circ p}\circ \left(A_{\alpha }^{\circ q}\circ A_{\alpha }^{\circ r}\right)\\\exists A_{\alpha }^{\circ 0}\in {}_{m}G(A_{\alpha }(o_{x}^{\alpha }))\ {\text{such that}}\ \forall A_{\alpha }^{\circ p}\in {}_{m}G(A_{\alpha }(o_{x}^{\alpha })),\ A_{\alpha }^{\circ 0}\circ A_{\alpha }^{\circ p}=A_{\alpha }^{\circ p}\\\forall A_{\alpha }^{\circ p}\in {}_{m}G(A_{\alpha }(o_{x}^{\alpha })),\ \exists A_{\alpha }^{\circ -p}\in {}_{m}G(A_{\alpha }(o_{x}^{\alpha }))\ {\text{such that}}\ A_{\alpha }^{\circ p}\circ A_{\alpha }^{\circ -p}=A_{\alpha }^{\circ 0}\\\forall A_{\alpha }^{\circ p},A_{\alpha }^{\circ q}\in {}_{m}G(A_{\alpha }(o_{x}^{\alpha })),\ A_{\alpha }^{\circ p}\circ A_{\alpha }^{\circ q}=A_{\alpha }^{\circ q}\circ A_{\alpha }^{\circ p}\end{array}}\right..}
Set
m
S
x
,
α
n
,
γ
(
h
)
{\displaystyle {}_{m}S_{x,\alpha }^{n,\gamma }(h)}
of Fractional Operators
Let
N
0
{\displaystyle \mathbb {N} _{0}}
be the set
N
∪
{
0
}
{\displaystyle \mathbb {N} \cup \{0\}}
. If
γ
∈
N
0
m
{\displaystyle \gamma \in \mathbb {N} _{0}^{m}}
and
x
∈
R
m
{\displaystyle x\in \mathbb {R} ^{m}}
, then the following multi-index notation can be defined:
{
γ
!
:=
∏
k
=
1
m
[
γ
]
k
!
,
|
γ
|
:=
∑
k
=
1
m
[
γ
]
k
,
x
γ
:=
∏
k
=
1
m
[
x
]
k
[
γ
]
k
∂
γ
∂
x
γ
:=
∂
[
γ
]
1
∂
[
x
]
1
∂
[
γ
]
2
∂
[
x
]
2
⋯
∂
[
γ
]
m
∂
[
x
]
m
.
{\displaystyle \left\{{\begin{array}{c}{\begin{array}{ccc}\displaystyle \gamma !:=\prod _{k=1}^{m}[\gamma ]_{k}!,&|\gamma |:=\displaystyle \sum _{k=1}^{m}[\gamma ]_{k},&\displaystyle x^{\gamma }:=\prod _{k=1}^{m}[x]_{k}^{[\gamma ]_{k}}\end{array}}\\\displaystyle {\frac {\partial ^{\gamma }}{\partial x^{\gamma }}}:={\frac {\partial ^{[\gamma ]_{1}}}{\partial [x]_{1}}}{\frac {\partial ^{[\gamma ]_{2}}}{\partial [x]_{2}}}\cdots {\frac {\partial ^{[\gamma ]_{m}}}{\partial [x]_{m}}}\end{array}}\right..}
Then, considering a function
h
:
Ω
⊂
R
m
→
R
{\displaystyle h:\Omega \subset \mathbb {R} ^{m}\to \mathbb {R} }
and the fractional operator:
s
x
α
γ
(
o
x
α
)
:=
o
1
α
[
γ
]
1
o
2
α
[
γ
]
2
⋯
o
m
α
[
γ
]
m
,
{\displaystyle s_{x}^{\alpha \gamma }\left(o_{x}^{\alpha }\right):=o_{1}^{\alpha [\gamma ]_{1}}o_{2}^{\alpha [\gamma ]_{2}}\cdots o_{m}^{\alpha [\gamma ]_{m}},}
the following set of fractional operators is defined:
S
x
,
α
n
,
γ
(
h
)
:=
{
s
x
α
γ
=
s
x
α
γ
(
o
x
α
)
:
∃
s
x
α
γ
h
(
x
)
such that
o
x
α
∈
O
x
,
α
s
(
h
)
∀
s
≤
n
2
and
lim
α
→
k
s
x
α
γ
h
(
x
)
=
∂
k
γ
∂
x
k
γ
h
(
x
)
∀
α
,
|
γ
|
≤
n
}
.
{\displaystyle S_{x,\alpha }^{n,\gamma }(h):=\left\{s_{x}^{\alpha \gamma }=s_{x}^{\alpha \gamma }\left(o_{x}^{\alpha }\right)\ :\ \exists s_{x}^{\alpha \gamma }h(x)\ {\text{ such that }}\ o_{x}^{\alpha }\in O_{x,\alpha }^{s}(h)\ \forall s\leq n^{2}\ {\text{ and }}\ \lim _{\alpha \to k}s_{x}^{\alpha \gamma }h(x)={\frac {\partial ^{k\gamma }}{\partial x^{k\gamma }}}h(x)\ \forall \alpha ,|\gamma |\leq n\right\}.}
From which the following results are obtained:
If
s
x
α
γ
∈
S
x
,
α
n
,
γ
(
h
)
⇒
{
lim
α
→
0
s
x
α
γ
h
(
x
)
=
o
1
0
o
2
0
⋯
o
m
0
h
(
x
)
=
h
(
x
)
lim
α
→
1
s
x
α
γ
h
(
x
)
=
o
1
[
γ
]
1
o
2
[
γ
]
2
⋯
o
m
[
γ
]
m
h
(
x
)
=
∂
γ
∂
x
γ
h
(
x
)
∀
|
γ
|
≤
n
lim
α
→
q
s
x
α
γ
h
(
x
)
=
o
1
q
[
γ
]
1
o
2
q
[
γ
]
2
⋯
o
m
q
[
γ
]
m
h
(
x
)
=
∂
q
γ
∂
x
q
γ
h
(
x
)
∀
q
|
γ
|
≤
q
n
lim
α
→
n
s
x
α
γ
h
(
x
)
=
o
1
n
[
γ
]
1
o
2
n
[
γ
]
2
⋯
o
m
n
[
γ
]
m
h
(
x
)
=
∂
n
γ
∂
x
n
γ
h
(
x
)
∀
n
|
γ
|
≤
n
2
,
{\displaystyle {\text{If }}s_{x}^{\alpha \gamma }\in S_{x,\alpha }^{n,\gamma }(h)\ \Rightarrow \ \left\{{\begin{array}{l}\displaystyle \lim _{\alpha \to 0}s_{x}^{\alpha \gamma }h(x)=o_{1}^{0}o_{2}^{0}\cdots o_{m}^{0}h(x)=h(x)\\\displaystyle \lim _{\alpha \to 1}s_{x}^{\alpha \gamma }h(x)=o_{1}^{[\gamma ]_{1}}o_{2}^{[\gamma ]_{2}}\cdots o_{m}^{[\gamma ]_{m}}h(x)={\frac {\partial ^{\gamma }}{\partial x^{\gamma }}}h(x)\ \forall |\gamma |\leq n\\\displaystyle \lim _{\alpha \to q}s_{x}^{\alpha \gamma }h(x)=o_{1}^{q[\gamma ]_{1}}o_{2}^{q[\gamma ]_{2}}\cdots o_{m}^{q[\gamma ]_{m}}h(x)={\frac {\partial ^{q\gamma }}{\partial x^{q\gamma }}}h(x)\ \forall q|\gamma |\leq qn\\\displaystyle \lim _{\alpha \to n}s_{x}^{\alpha \gamma }h(x)=o_{1}^{n[\gamma ]_{1}}o_{2}^{n[\gamma ]_{2}}\cdots o_{m}^{n[\gamma ]_{m}}h(x)={\frac {\partial ^{n\gamma }}{\partial x^{n\gamma }}}h(x)\ \forall n|\gamma |\leq n^{2}\end{array}}\right.,}
As a consequence, considering a function
h
:
Ω
⊂
R
m
→
R
m
{\displaystyle h:\Omega \subset \mathbb {R} ^{m}\to \mathbb {R} ^{m}}
, the following set of fractional operators is defined:
m
S
x
,
α
n
,
γ
(
h
)
:=
{
s
x
α
γ
:
s
x
α
γ
∈
S
x
,
α
n
,
γ
(
[
h
]
k
)
∀
k
≤
m
}
.
{\displaystyle {}_{m}S_{x,\alpha }^{n,\gamma }(h):=\left\{s_{x}^{\alpha \gamma }\ :\ s_{x}^{\alpha \gamma }\in S_{x,\alpha }^{n,\gamma }\left([h]_{k}\right)\ \forall k\leq m\right\}.}
References
^ Torres-Hernandez, A.; Brambila-Paz, F. (December 29, 2021). "Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods" . Fractal and Fractional . 5 (4): 240. doi :10.3390/fractalfract5040240 .
^ Applications of fractional calculus in physics
^ de Oliveira, Edmundo Capelas; Tenreiro Machado, José António (June 10, 2014). "A Review of Definitions for Fractional Derivatives and Integral" . Mathematical Problems in Engineering . 2014 : e238459. doi :10.1155/2014/238459 .
^ Sales Teodoro, G.; Tenreiro Machado, J.A.; Capelas de Oliveira, E. (July 29, 2019). "A review of definitions of fractional derivatives and other operators" . Journal of Computational Physics . 388 : 195–208. Bibcode :2019JCoPh.388..195S . doi :10.1016/j.jcp.2019.03.008 .
^ Valério, Duarte; Ortigueira, Manuel D.; Lopes, António M. (January 29, 2022). "How Many Fractional Derivatives Are There?" . Mathematics . 10 (5): 737. doi :10.3390/math10050737 .
^ Torres-Hernandez, A.; Brambila-Paz, F. (2021). "Fractional Newton-Raphson Method" . Applied Mathematics and Sciences an International Journal (Mathsj) . 8 : 1–13. doi :10.5121/mathsj.2021.8101 .
^ Torres-Hernandez, A.; Brambila-Paz, F.; Montufar-Chaveznava, R. (September 29, 2022). "Acceleration of the order of convergence of a family of fractional fixed-point methods and its implementation in the solution of a nonlinear algebraic system related to hybrid solar receivers" . Applied Mathematics and Computation . 429 : 127231. arXiv :2109.03152 . doi :10.1016/j.amc.2022.127231 .
^ Torres-Hernandez, A. (2022). "Code of a multidimensional fractional quasi-Newton method with an order of convergence at least quadratic using recursive programming" . Applied Mathematics and Sciences an International Journal (MathSJ) . 9 : 17–24. doi :10.5121/mathsj.2022.9103 .
^ 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 .
^ Einstein summation for multidimensional arrays
^ The hadamard product
^ Jacobians of matrix transformation and functions of matrix arguments
^ Abelian groups