In mathematics , the Caputo fractional derivative , also called Caputo-type fractional derivative, is a generalization of derivatives for non-integer orders named after Michele Caputo. Caputo first defined this form of fractional derivative in 1967.[ 1]
The Caputo fractional derivative is motivated from the Riemann–Liouville fractional integral . Let
f
{\textstyle f}
be continuous on
(
0
,
∞
)
{\displaystyle \left(0,\,\infty \right)}
, then the Riemann–Liouville fractional integral
RL
I
{\textstyle {^{\text{RL}}\operatorname {I} }}
states that
0
RL
I
x
α
[
f
(
x
)
]
=
1
Γ
(
−
α
)
⋅
∫
0
x
f
(
t
)
(
x
−
t
)
1
−
α
d
t
{\displaystyle {_{0}^{\text{RL}}\operatorname {I} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f\left(t\right)}{\left(x-t\right)^{1-\alpha }}}\,\operatorname {d} t}
where
Γ
(
⋅
)
{\textstyle \Gamma \left(\cdot \right)}
is the Gamma function .
Let's define
D
x
α
:=
d
α
d
x
α
{\textstyle \operatorname {D} _{x}^{\alpha }:={\frac {\operatorname {d} ^{\alpha }}{\operatorname {d} x^{\alpha }}}}
, say that
D
x
α
D
x
β
=
D
x
α
+
β
{\textstyle \operatorname {D} _{x}^{\alpha }\operatorname {D} _{x}^{\beta }=\operatorname {D} _{x}^{\alpha +\beta }}
and that
D
x
α
=
RL
I
x
−
α
{\textstyle \operatorname {D} _{x}^{\alpha }={^{\text{RL}}\operatorname {I} _{x}^{-\alpha }}}
applies. If
α
=
m
+
z
∈
R
∧
m
∈
N
0
∧
0
<
z
<
1
{\textstyle \alpha =m+z\in \mathbb {R} \wedge m\in \mathbb {N} _{0}\wedge 0<z<1}
then we could say
D
x
α
=
D
x
m
+
z
=
D
x
z
+
m
=
D
x
z
−
1
+
1
+
m
=
D
x
z
−
1
D
x
1
+
m
=
RL
I
x
1
−
z
D
x
1
+
m
{\textstyle \operatorname {D} _{x}^{\alpha }=\operatorname {D} _{x}^{m+z}=\operatorname {D} _{x}^{z+m}=\operatorname {D} _{x}^{z-1+1+m}=\operatorname {D} _{x}^{z-1}\operatorname {D} _{x}^{1+m}={^{\text{RL}}\operatorname {I} }_{x}^{1-z}\operatorname {D} _{x}^{1+m}}
. So if
f
{\displaystyle f}
is also
C
m
(
0
,
∞
)
{\displaystyle C^{m}\left(0,\,\infty \right)}
, then
D
x
m
+
z
[
f
(
x
)
]
=
1
Γ
(
1
−
z
)
⋅
∫
0
x
f
(
1
+
m
)
(
t
)
(
x
−
t
)
z
d
t
.
{\displaystyle {\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(1+m\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t.}
This is known as the Caputo-type fractional derivative, often written as
C
D
x
α
{\textstyle {^{\text{C}}\operatorname {D} }_{x}^{\alpha }}
.
The first definition of the Caputo-type fractional derivative was given by Caputo as:
C
D
x
m
+
z
[
f
(
x
)
]
=
1
Γ
(
1
−
z
)
⋅
∫
0
x
f
(
m
+
1
)
(
t
)
(
x
−
t
)
z
d
t
{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(m+1\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t}
where
C
m
(
0
,
∞
)
{\displaystyle C^{m}\left(0,\,\infty \right)}
and
m
∈
N
0
∧
0
<
z
<
1
{\textstyle m\in \mathbb {N} _{0}\wedge 0<z<1}
.[ 2]
A popular equivalent definition is:
C
D
x
α
[
f
(
x
)
]
=
1
Γ
(
⌈
α
⌉
−
α
)
⋅
∫
0
x
f
(
⌈
α
⌉
)
(
t
)
(
x
−
t
)
α
+
1
−
⌈
α
⌉
d
t
{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}
where
α
∈
R
>
0
∖
N
{\textstyle \alpha \in \mathbb {R} _{>0}\setminus \mathbb {N} }
and
⌈
⋅
⌉
{\textstyle \left\lceil \cdot \right\rceil }
is the ceiling function . This can be derived by substituting
α
=
m
+
z
{\textstyle \alpha =m+z}
so that
⌈
α
⌉
=
m
+
1
{\textstyle \left\lceil \alpha \right\rceil =m+1}
would apply and
⌈
α
⌉
+
z
=
α
+
1
{\textstyle \left\lceil \alpha \right\rceil +z=\alpha +1}
follows.[ 3]
Another popular equivalent definition is given by:
C
D
x
α
[
f
(
x
)
]
=
1
Γ
(
n
−
α
)
⋅
∫
0
x
f
(
n
)
(
t
)
(
x
−
t
)
α
+
1
−
n
d
t
{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(n-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(n\right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-n}}}\,\operatorname {d} t}
where
n
−
1
<
α
<
n
∈
N
.
{\textstyle n-1<\alpha <n\in \mathbb {N} .}
.
The problem with these definitions is that they only allow arguments in
(
0
,
∞
)
{\textstyle \left(0,\,\infty \right)}
. This can be fixed by replacing the lower integral limit with
a
{\textstyle a}
:
a
C
D
x
α
[
f
(
x
)
]
=
1
Γ
(
⌈
α
⌉
−
α
)
⋅
∫
a
x
f
(
⌈
α
⌉
)
(
t
)
(
x
−
t
)
α
+
1
−
⌈
α
⌉
d
t
{\textstyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}
. The new domain is
(
a
,
∞
)
{\textstyle \left(a,\,\infty \right)}
.[ 4]
Properties and theorems
edit
Caputo fractional derivative of some functions
edit
The Caputo fractional derivative of a constant
c
{\displaystyle c}
is given by:
a
C
D
x
α
[
c
]
=
1
Γ
(
⌈
α
⌉
−
α
)
⋅
∫
a
x
D
t
⌈
α
⌉
[
c
]
(
x
−
t
)
α
+
1
−
⌈
α
⌉
d
t
=
1
Γ
(
⌈
α
⌉
−
α
)
⋅
∫
a
x
0
(
x
−
t
)
α
+
1
−
⌈
α
⌉
d
t
a
C
D
x
α
[
c
]
=
0
{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {\operatorname {D} _{t}^{\left\lceil \alpha \right\rceil }\left[c\right]}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {0}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&=0\end{aligned}}}
The Caputo fractional derivative of a power function
x
b
{\displaystyle x^{b}}
is given by:[ 9]
a
C
D
x
α
[
x
b
]
=
a
RL
I
x
⌈
α
⌉
−
α
[
D
x
⌈
α
⌉
[
x
b
]
]
=
Γ
(
b
+
1
)
Γ
(
b
−
⌈
α
⌉
+
1
)
⋅
a
RL
I
x
⌈
α
⌉
−
α
[
x
b
−
⌈
α
⌉
]
a
C
D
x
α
[
x
b
]
=
{
Γ
(
b
+
1
)
Γ
(
b
−
α
+
1
)
(
x
b
−
α
−
a
b
−
α
)
,
for
⌈
α
⌉
−
1
<
b
∧
b
∈
R
0
,
for
⌈
α
⌉
−
1
≥
b
∧
b
∈
N
{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[x^{b}\right]\right]={\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\left\lceil \alpha \right\rceil +1\right)}}\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[x^{b-\left\lceil \alpha \right\rceil }\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={\begin{cases}{\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\alpha +1\right)}}\left(x^{b-\alpha }-a^{b-\alpha }\right),\,&{\text{for }}\left\lceil \alpha \right\rceil -1<b\wedge b\in \mathbb {R} \\0,\,&{\text{for }}\left\lceil \alpha \right\rceil -1\geq b\wedge b\in \mathbb {N} \\\end{cases}}\end{aligned}}}
The Caputo fractional derivative of a exponential function
e
a
⋅
x
{\displaystyle e^{a\cdot x}}
is given by:
a
C
D
x
α
[
e
b
⋅
x
]
=
a
RL
I
x
⌈
α
⌉
−
α
[
D
x
⌈
α
⌉
[
e
b
⋅
x
]
]
=
b
⌈
α
⌉
⋅
a
RL
I
x
⌈
α
⌉
−
α
[
e
b
⋅
x
]
a
C
D
x
α
[
e
b
⋅
x
]
=
b
α
⋅
(
E
x
(
⌈
α
⌉
−
α
,
b
)
−
E
a
(
⌈
α
⌉
−
α
,
b
)
)
{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[e^{b\cdot x}\right]\right]=b^{\left\lceil \alpha \right\rceil }\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[e^{b\cdot x}\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&=b^{\alpha }\cdot \left(E_{x}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)-E_{a}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)\right)\\\end{aligned}}}
where
E
x
(
ν
,
a
)
=
a
−
ν
⋅
e
a
⋅
x
⋅
γ
(
ν
,
a
⋅
x
)
Γ
(
ν
)
{\textstyle E_{x}\left(\nu ,\,a\right)={\frac {a^{-\nu }\cdot e^{a\cdot x}\cdot \gamma \left(\nu ,\,a\cdot x\right)}{\Gamma \left(\nu \right)}}}
is the
E
t
{\textstyle \operatorname {E} _{t}}
-function and
γ
(
a
,
b
)
{\textstyle \gamma \left(a,\,b\right)}
is the lower incomplete gamma function .[ 10]
^ Diethelm, Kai (2019). "General theory of Caputo-type fractional differential equations" . Fractional Differential Equations . pp. 1–20. doi :10.1515/9783110571660-001 . ISBN 978-3-11-057166-0 . Retrieved 2023-08-10 .
^ Caputo, Michele (1967). "Linear Models of Dissipation whose Q is almost Frequency Independent-II" . ResearchGate . 13 (5): 530. Bibcode :1967GeoJ...13..529C . doi :10.1111/j.1365-246X.1967.tb02303.x .
^ Lazarević, Mihailo; Rapaić, Milan Rade; Šekara, Tomislav (2014). "Introduction to Fractional Calculus with Brief Historical Background" . ResearchGate : 8.
^ Dimitrov, Yuri; Georgiev, Slavi; Todorov, Venelin (2023). "Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations" . Fractal and Fractional . 7 (10): 750. doi :10.3390/fractalfract7100750 .
^ Sikora, Beata (2023). "Remarks on the Caputo fractional derivative" (PDF) . Matematyka I Informatyka Na Uczelniach Technicznych (5): 78–79.
^ Huseynov, Ismail; Ahmadova, Arzu; Mahmudov, Nazim (2020). "Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications" . ResearchGate : 1. arXiv :2012.11360 .
^ Weisstein, Eric W. (2024). "Binomial Coefficient" . mathworld.wolfram.com . Retrieved 2024-05-20 .
^ Sontakke, Bhausaheb Rajba; Shaikh, Amjad (2015). "Properties of Caputo Operator and Its Applications to Linear Fractional Differential Equations" (PDF) . Journal of Engineering Research and Applications . 5 (5): 23–24. ISSN 2248-9622 .
^ Weisstein, Eric W. "Fractional Derivative" . mathworld.wolfram.com . Retrieved 2024-05-20 .
^ Weisstein, Eric W. (2024). "E_t-Function" . mathworld.wolfram.com . Retrieved 2024-05-20 .