# Caputo fractional derivative

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.

## Motivation

The Caputo fractional derivative is motivated from the Riemann–Liouville fractional integral. Let f{\textstyle f} be continuous on $\left(0,\,\infty \right)$, then the Riemann–Liouville fractional integral $^{RL}$I{\textstyle {^{\text{RL}}\operatorname {I} }} states that
${_{0}^{\text{RL}}\operatorname {I} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f\left(t\right)}{\left(x-t\right)^{1-\alpha }}}\,\operatorname {d} t$
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 $f$ is also $C^{m}\left(0,\,\infty \right)$, then
${\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(1+m\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t.$
This is known as the Caputo-type fractional derivative, often written as $^C$D$_x$$^α${\textstyle {^{\text{C}}\operatorname {D} }_{x}^{\alpha }}.

## Definition

The first definition of the Caputo-type fractional derivative was given by Caputo as:
${^{\text{C}}\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(m+1\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t$
where $C^{m}\left(0,\,\infty \right)$ and m∈N$_0$∧0<z<1{\textstyle m\in \mathbb {N} _{0}\wedge 0<z<1}.
A popular equivalent definition is:
${^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t$
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.
Another popular equivalent definition is given by:
${^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(n-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(n\right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-n}}}\,\operatorname {d} t$
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)}.

## Properties and theorems
### Basic properties and theorems

A few basic properties are:
A table of basic properties and theorems :
| Properties | $f\left(x\right)$ | ${_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]$ | Condition |
| Definition | $f\left(x\right)$ | $f^{\left(\alpha \right)}\left(x\right)-f^{\left(\alpha \right)}\left(a\right)$ | |
| Linearity | $b\cdot g\left(x\right)+c\cdot h\left(x\right)$ | $b\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[g\left(x\right)\right]+c\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[h\left(x\right)\right]$ | |
| Index law | $\operatorname {D} _{x}^{\beta }$ | ${_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha +\beta }}$ | $\beta \in \mathbb {Z}$ |
| Semigroup property | ${_{a}^{\text{C}}\operatorname {D} _{x}^{\beta }}$ | ${_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha +\beta }}$ | $\left\lceil \alpha \right\rceil =\left\lceil \beta \right\rceil$ |

#### Non-commutation

The index law does not always fulfill the property of commutation:
$\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }\operatorname {_{a}^{\text{C}}D} _{x}^{\beta }=\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha +\beta }\neq \operatorname {_{a}^{\text{C}}D} _{x}^{\beta }\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }$
where $\alpha \in \mathbb {R} _{>0}\setminus \mathbb {N} \wedge \beta \in \mathbb {N}$.

#### Fractional Leibniz rule

The Leibniz rule for the Caputo fractional derivative is given by:
$\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }\left[g\left(x\right)\cdot h\left(x\right)\right]=\sum \limits _{k=0}^{\infty }\left[{\binom {a}{k}}\cdot g^{\left(k\right)}\left(x\right)\cdot \operatorname {_{a}^{\text{RL}}D} _{x}^{\alpha -k}\left[h\left(x\right)\right]\right]-{\frac {\left(x-a\right)^{-\alpha }}{\Gamma \left(1-\alpha \right)}}\cdot g\left(a\right)\cdot h\left(a\right)$
where (a/b)=(Γ(a+1))/(Γ(b+1)⋅Γ(a−b+1)) {\textstyle {\binom {a}{b}}={\frac {\Gamma \left(a+1\right)}{\Gamma \left(b+1\right)\cdot \Gamma \left(a-b+1\right)}}} is the binomial coefficient.

### Relation to other fractional differential operators

Caputo-type fractional derivative is closely related to the Riemann–Liouville fractional integral via its definition:
${_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[f\left(x\right)\right]\right]$
Furthermore, the following relation applies:
${_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[{\frac {x^{k-\alpha }}{\Gamma \left(k-\alpha +1\right)}}\cdot f^{\left(k\right)}\left(0\right)\right]$
where ${_{a}^{\text{RL}}\operatorname {D} _{x}^{\alpha }}$ is the Riemann–Liouville fractional derivative.

### Laplace transform

The Laplace transform of the Caputo-type fractional derivative is given by:
${\mathcal {L}}_{x}\left\{{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]\right\}\left(s\right)=s^{\alpha }\cdot F\left(s\right)-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[s^{\alpha -k-1}\cdot f^{\left(k\right)}\left(0\right)\right]$
where L$_x${f(x)}(s)=F(s){\textstyle {\mathcal {L}}_{x}\left\{f\left(x\right)\right\}\left(s\right)=F\left(s\right)}.

## Caputo fractional derivative of some functions

The Caputo fractional derivative of a constant $c$ is given by:
${\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {\operatorname {D} _{t}^{\left\lceil \alpha \right\rceil }\left[c\right]}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {0}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&=0\end{aligned}}$
The Caputo fractional derivative of a power function $x^{b}$ is given by:
${\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[x^{b}\right]\right]={\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\left\lceil \alpha \right\rceil +1\right)}}\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[x^{b-\left\lceil \alpha \right\rceil }\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={\begin{cases}{\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\alpha +1\right)}}\left(x^{b-\alpha }-a^{b-\alpha }\right),\,&{\text{for }}\left\lceil \alpha \right\rceil -1<b\wedge b\in \mathbb {R} \\0,\,&{\text{for }}\left\lceil \alpha \right\rceil -1\geq b\wedge b\in \mathbb {N} \\\end{cases}}\end{aligned}}$
The Caputo fractional derivative of an exponential function $e^{a\cdot x}$ is given by:
${\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[e^{b\cdot x}\right]\right]=b^{\left\lceil \alpha \right\rceil }\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[e^{b\cdot x}\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&=b^{\alpha }\cdot \left(E_{x}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)-E_{a}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)\right)\\\end{aligned}}$
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.