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.^[1]

The Caputo fractional derivative is motivated from the Riemann–Liouville fractional integral. Let ${\textstyle f}$ be continuous on ${\displaystyle \left(0,\,\infty \right)}$, then the Riemann–Liouville fractional integral ${\textstyle {^{\text{RL}}\operatorname {I} }}$ states that

$${\displaystyle {_{0}^{\text{RL}}\operatorname {I} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f\left(t\right)}{\left(x-t\right)^{1-\alpha }}}\,\operatorname {d} t}$$

where ${\textstyle \Gamma \left(\cdot \right)}$ is the Gamma function.

Let's define ${\textstyle \operatorname {D} _{x}^{\alpha }:={\frac {\operatorname {d} ^{\alpha }}{\operatorname {d} x^{\alpha }}}}$, say that ${\textstyle \operatorname {D} _{x}^{\alpha }\operatorname {D} _{x}^{\beta }=\operatorname {D} _{x}^{\alpha +\beta }}$ and that ${\textstyle \operatorname {D} _{x}^{\alpha }={^{\text{RL}}\operatorname {I} _{x}^{-\alpha }}}$ applies. If ${\textstyle \alpha =m+z\in \mathbb {R} \wedge m\in \mathbb {N} _{0}\wedge 0<z<1}$ then we could say ${\textstyle \operatorname {D} _{x}^{\alpha }=\operatorname {D} _{x}^{m+z}=\operatorname {D} _{x}^{z+m}=\operatorname {D} _{x}^{z-1+1+m}=\operatorname {D} _{x}^{z-1}\operatorname {D} _{x}^{1+m}={^{\text{RL}}\operatorname {I} }_{x}^{1-z}\operatorname {D} _{x}^{1+m}}$. So if ${\displaystyle f}$ is also ${\displaystyle C^{m}\left(0,\,\infty \right)}$, then

$${\displaystyle {\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(1+m\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t.}$$

This is known as the Caputo-type fractional derivative, often written as ${\textstyle {^{\text{C}}\operatorname {D} }_{x}^{\alpha }}$.

The first definition of the Caputo-type fractional derivative was given by Caputo as:

$${\displaystyle {^{\text{C}}\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(m+1\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t}$$

where ${\displaystyle C^{m}\left(0,\,\infty \right)}$ and ${\textstyle m\in \mathbb {N} _{0}\wedge 0<z<1}$.^[2]

A popular equivalent definition is:

$${\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}$$

where ${\textstyle \alpha \in \mathbb {R} _{>0}\setminus \mathbb {N} }$ and ${\textstyle \left\lceil \cdot \right\rceil }$ is the ceiling function. This can be derived by substituting ${\textstyle \alpha =m+z}$ so that ${\textstyle \left\lceil \alpha \right\rceil =m+1}$ would apply and ${\textstyle \left\lceil \alpha \right\rceil +z=\alpha +1}$ follows.^[3]

Another popular equivalent definition is given by:

$${\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(n-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(n\right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-n}}}\,\operatorname {d} t}$$

where ${\textstyle n-1<\alpha <n\in \mathbb {N} .}$.

The problem with these definitions is that they only allow arguments in ${\textstyle \left(0,\,\infty \right)}$. This can be fixed by replacing the lower integral limit with ${\textstyle a}$: ${\textstyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}$. The new domain is ${\textstyle \left(a,\,\infty \right)}$.^[4]

A few basic properties are:^[5]

A table of basic properties and theorems

Properties	${\displaystyle f\left(x\right)}$	${\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]}$	Condition
Definition	${\displaystyle f\left(x\right)}$	${\displaystyle f^{\left(\alpha \right)}\left(x\right)-f^{\left(\alpha \right)}\left(a\right)}$
Linearity	${\displaystyle b\cdot g\left(x\right)+c\cdot h\left(x\right)}$	${\displaystyle b\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[g\left(x\right)\right]+c\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[h\left(x\right)\right]}$
Index law	${\displaystyle \operatorname {D} _{x}^{\beta }}$	${\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha +\beta }}}$	${\displaystyle \beta \in \mathbb {Z} }$
Semigroup property	${\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\beta }}}$	${\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha +\beta }}}$	${\displaystyle \left\lceil \alpha \right\rceil =\left\lceil \beta \right\rceil }$

The index law does not always fulfill the property of commutation:

$${\displaystyle \operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }\operatorname {_{a}^{\text{C}}D} _{x}^{\beta }=\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha +\beta }\neq \operatorname {_{a}^{\text{C}}D} _{x}^{\beta }\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }}$$

where ${\displaystyle \alpha \in \mathbb {R} _{>0}\setminus \mathbb {N} \wedge \beta \in \mathbb {N} }$.

The Leibniz rule for the Caputo fractional derivative is given by:

$${\displaystyle \operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }\left[g\left(x\right)\cdot h\left(x\right)\right]=\sum \limits _{k=0}^{\infty }\left[{\binom {a}{k}}\cdot g^{\left(k\right)}\left(x\right)\cdot \operatorname {_{a}^{\text{RL}}D} _{x}^{\alpha -k}\left[h\left(x\right)\right]\right]-{\frac {\left(x-a\right)^{-\alpha }}{\Gamma \left(1-\alpha \right)}}\cdot g\left(a\right)\cdot h\left(a\right)}$$

where ${\textstyle {\binom {a}{b}}={\frac {\Gamma \left(a+1\right)}{\Gamma \left(b+1\right)\cdot \Gamma \left(a-b+1\right)}}}$ is the binomial coefficient.^[6][7]

Caputo-type fractional derivative is closely related to the Riemann–Liouville fractional integral via its definition:

$${\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[f\left(x\right)\right]\right]}$$

Furthermore, the following relation applies:

$${\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[{\frac {x^{k-\alpha }}{\Gamma \left(k-\alpha +1\right)}}\cdot f^{\left(k\right)}\left(0\right)\right]}$$

where ${\displaystyle {_{a}^{\text{RL}}\operatorname {D} _{x}^{\alpha }}}$ is the Riemann–Liouville fractional derivative.

The Laplace transform of the Caputo-type fractional derivative is given by:

$${\displaystyle {\mathcal {L}}_{x}\left\{{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]\right\}\left(s\right)=s^{\alpha }\cdot F\left(s\right)-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[s^{\alpha -k-1}\cdot f^{\left(k\right)}\left(0\right)\right]}$$

where ${\textstyle {\mathcal {L}}_{x}\left\{f\left(x\right)\right\}\left(s\right)=F\left(s\right)}$.^[8]

The Caputo fractional derivative of a constant ${\displaystyle c}$ is given by:

$${\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {\operatorname {D} _{t}^{\left\lceil \alpha \right\rceil }\left[c\right]}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {0}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&=0\end{aligned}}}$$

The Caputo fractional derivative of a power function ${\displaystyle x^{b}}$ is given by:^[9]

$${\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[x^{b}\right]\right]={\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\left\lceil \alpha \right\rceil +1\right)}}\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[x^{b-\left\lceil \alpha \right\rceil }\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={\begin{cases}{\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\alpha +1\right)}}\left(x^{b-\alpha }-a^{b-\alpha }\right),\,&{\text{for }}\left\lceil \alpha \right\rceil -1<b\wedge b\in \mathbb {R} \\0,\,&{\text{for }}\left\lceil \alpha \right\rceil -1\geq b\wedge b\in \mathbb {N} \\\end{cases}}\end{aligned}}}$$

The Caputo fractional derivative of an exponential function ${\displaystyle e^{a\cdot x}}$ is given by:

$${\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[e^{b\cdot x}\right]\right]=b^{\left\lceil \alpha \right\rceil }\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[e^{b\cdot x}\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&=b^{\alpha }\cdot \left(E_{x}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)-E_{a}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)\right)\\\end{aligned}}}$$

where ${\textstyle E_{x}\left(\nu ,\,a\right)={\frac {a^{-\nu }\cdot e^{a\cdot x}\cdot \gamma \left(\nu ,\,a\cdot x\right)}{\Gamma \left(\nu \right)}}}$ is the ${\textstyle \operatorname {E} _{t}}$-function and ${\textstyle \gamma \left(a,\,b\right)}$ is the lower incomplete gamma function.^[10]

• Ricardo Almeida, A Caputo fractional derivative of a function with respect to another function