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 $f$ be continuous on $(0,\mathrm{\infty })$, then the Riemann–Liouville fractional integral ${}^{\text{RL}}\mathrm{I}$ states that

$${}_0^{\text{RL}}{\mathrm{I}}_x^{\alpha }\left[f\left(x\right)\right]=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\cdot \underset{0}{\overset{x}{\int }}\frac{f\left(t\right)}{{(x-t)}^{1-\alpha }}\mathrm{d}t$$

where $\mathrm{\Gamma }(\cdot )$ is the Gamma function.

Let's define ${\mathrm{D}}_x^{\alpha }:=\frac{{\mathrm{d}}^{\alpha }}{\mathrm{d}{x}^{\alpha }}$, say that ${\mathrm{D}}_x^{\alpha }{\mathrm{D}}_x^{\beta }={\mathrm{D}}_x^{\alpha +\beta }$ and that ${\mathrm{D}}_x^{\alpha }={}^{\text{RL}}{\mathrm{I}}_x^{-\alpha }$ applies. If $\alpha =m+z\in ℝ\wedge m\in {\mathbb{N}}_0\wedge 0<z<1$ then we could say ${\mathrm{D}}_x^{\alpha }={\mathrm{D}}_x^{m+z}={\mathrm{D}}_x^{z+m}={\mathrm{D}}_x^{z-1+1+m}={\mathrm{D}}_x^{z-1}{\mathrm{D}}_x^{1+m}={{}^{\text{RL}}\mathrm{I}}_x^{1-z}{\mathrm{D}}_x^{1+m}$. So if $f$ is also $C^m(0,\mathrm{\infty })$, then

$${\mathrm{D}}_x^{m+z}\left[f\left(x\right)\right]=\frac{1}{\mathrm{\Gamma }(1-z)}\cdot \underset{0}{\overset{x}{\int }}\frac{f^{(1+m)}\left(t\right)}{{(x-t)}^z}\mathrm{d}t.$$

This is known as the Caputo-type fractional derivative, often written as ${{}^{\text{C}}\mathrm{D}}_x^{\alpha }$.

Definition

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

$${}^{\text{C}}{\mathrm{D}}_x^{m+z}\left[f\left(x\right)\right]=\frac{1}{\mathrm{\Gamma }(1-z)}\cdot \underset{0}{\overset{x}{\int }}\frac{f^{(m+1)}\left(t\right)}{{(x-t)}^z}\mathrm{d}t$$

where $C^m(0,\mathrm{\infty })$ and $m\in {\mathbb{N}}_0\wedge 0<z<1$.

A popular equivalent definition is:

$${}^{\text{C}}{\mathrm{D}}_x^{\alpha }\left[f\left(x\right)\right]=\frac{1}{\mathrm{\Gamma }(\lceil \alpha \rceil -\alpha )}\cdot \underset{0}{\overset{x}{\int }}\frac{f^{\left(\lceil \alpha \rceil \right)}\left(t\right)}{{(x-t)}^{\alpha +1-\lceil \alpha \rceil }}\mathrm{d}t$$

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

Another popular equivalent definition is given by:

$${}^{\text{C}}{\mathrm{D}}_x^{\alpha }\left[f\left(x\right)\right]=\frac{1}{\mathrm{\Gamma }(n-\alpha )}\cdot \underset{0}{\overset{x}{\int }}\frac{f^{\left(n\right)}\left(t\right)}{{(x-t)}^{\alpha +1-n}}\mathrm{d}t$$

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

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

Properties and theorems

Basic properties and theorems

A few basic properties are:

A table of basic properties and theorems

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

Non-commutation

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

$${{}_{\mathrm{}}^{\mathrm{a}}\mathrm{D}}_x^{\alpha }{{}_{\mathrm{}}^{\mathrm{a}}\mathrm{D}}_x^{\beta }={{}_{\mathrm{}}^{\mathrm{a}}\mathrm{D}}_x^{\alpha +\beta }\ne {{}_{\mathrm{}}^{\mathrm{a}}\mathrm{D}}_x^{\beta }{{}_{\mathrm{}}^{\mathrm{a}}\mathrm{D}}_x^{\alpha }$$

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

Fractional Leibniz rule

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

$${{}_{\mathrm{}}^{\mathrm{a}}\mathrm{D}}_x^{\alpha }[g\left(x\right)\cdot h\left(x\right)]=\sum _{k=0}^{\mathrm{\infty }}[(\genfrac{}{}{0ex}{}{a}{k})\cdot g^{\left(k\right)}\left(x\right)\cdot {{}_{\mathrm{}}^{\mathrm{a}}\mathrm{D}}_x^{\alpha -k}\left[h\left(x\right)\right]]-\frac{{(x-a)}^{-\alpha }}{\mathrm{\Gamma }(1-\alpha )}\cdot g\left(a\right)\cdot h\left(a\right)$$

where $(\genfrac{}{}{0ex}{}{a}{b})=\frac{\mathrm{\Gamma }(a+1)}{\mathrm{\Gamma }(b+1)\cdot \mathrm{\Gamma }(a-b+1)}$ 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:

$${}_a^{\text{C}}{\mathrm{D}}_x^{\alpha }\left[f\left(x\right)\right]={}_a^{\text{RL}}{\mathrm{I}}_x^{\lceil \alpha \rceil -\alpha }\left[{\mathrm{D}}_x^{\lceil \alpha \rceil }\left[f\left(x\right)\right]\right]$$

Furthermore, the following relation applies:

$${}_a^{\text{C}}{\mathrm{D}}_x^{\alpha }\left[f\left(x\right)\right]={}_a^{\text{RL}}{\mathrm{D}}_x^{\alpha }\left[f\left(x\right)\right]-\sum _{k=0}^{\lfloor \alpha \rfloor }[\frac{x^{k-\alpha }}{\mathrm{\Gamma }(k-\alpha +1)}\cdot f^{\left(k\right)}\left(0\right)]$$

where ${}_a^{\text{RL}}{\mathrm{D}}_x^{\alpha }$ is the Riemann–Liouville fractional derivative.

Laplace transform

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

$${ℒ}_x\left\{{}_a^{\text{C}}{\mathrm{D}}_x^{\alpha }\left[f\left(x\right)\right]\right\}\left(s\right)=s^{\alpha }\cdot F\left(s\right)-\sum _{k=0}^{\lceil \alpha \rceil -1}[s^{\alpha -k-1}\cdot f^{\left(k\right)}\left(0\right)]$$

where ${ℒ}_x\left\{f\left(x\right)\right\}\left(s\right)=F\left(s\right)$.

Caputo fractional derivative of some functions

The Caputo fractional derivative of a constant $c$ is given by:

$$\begin{array}{cc}{}_a^{\text{C}}{\mathrm{D}}_x^{\alpha }\left[c\right]& =\frac{1}{\mathrm{\Gamma }(\lceil \alpha \rceil -\alpha )}\cdot \underset{a}{\overset{x}{\int }}\frac{{\mathrm{D}}_t^{\lceil \alpha \rceil }\left[c\right]}{{(x-t)}^{\alpha +1-\lceil \alpha \rceil }}\mathrm{d}t=\frac{1}{\mathrm{\Gamma }(\lceil \alpha \rceil -\alpha )}\cdot \underset{a}{\overset{x}{\int }}\frac{0}{{(x-t)}^{\alpha +1-\lceil \alpha \rceil }}\mathrm{d}t\\ {}_a^{\text{C}}{\mathrm{D}}_x^{\alpha }\left[c\right]& =0\end{array}$$

The Caputo fractional derivative of a power function $x^b$ is given by:

$$\begin{array}{cc}{}_a^{\text{C}}{\mathrm{D}}_x^{\alpha }\left[x^b\right]& ={}_a^{\text{RL}}{\mathrm{I}}_x^{\lceil \alpha \rceil -\alpha }\left[{\mathrm{D}}_x^{\lceil \alpha \rceil }\left[x^b\right]\right]=\frac{\mathrm{\Gamma }(b+1)}{\mathrm{\Gamma }(b-\lceil \alpha \rceil +1)}\cdot {}_a^{\text{RL}}{\mathrm{I}}_x^{\lceil \alpha \rceil -\alpha }\left[x^{b-\lceil \alpha \rceil }\right]\\ {}_a^{\text{C}}{\mathrm{D}}_x^{\alpha }\left[x^b\right]& =\{\begin{array}{cc}\frac{\mathrm{\Gamma }(b+1)}{\mathrm{\Gamma }(b-\alpha +1)}(x^{b-\alpha }-a^{b-\alpha }),& \text{for }\lceil \alpha \rceil -1<b\wedge b\in ℝ\\ 0,& \text{for }\lceil \alpha \rceil -1\ge b\wedge b\in \mathbb{N}\end{array}\end{array}$$

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

$$\begin{array}{cc}{}_a^{\text{C}}{\mathrm{D}}_x^{\alpha }\left[e^{b\cdot x}\right]& ={}_a^{\text{RL}}{\mathrm{I}}_x^{\lceil \alpha \rceil -\alpha }\left[{\mathrm{D}}_x^{\lceil \alpha \rceil }\left[e^{b\cdot x}\right]\right]=b^{\lceil \alpha \rceil }\cdot {}_a^{\text{RL}}{\mathrm{I}}_x^{\lceil \alpha \rceil -\alpha }\left[e^{b\cdot x}\right]\\ {}_a^{\text{C}}{\mathrm{D}}_x^{\alpha }\left[e^{b\cdot x}\right]& =b^{\alpha }\cdot (E_x(\lceil \alpha \rceil -\alpha ,b)-E_a(\lceil \alpha \rceil -\alpha ,b))\end{array}$$

where $E_x(\nu ,a)=\frac{a^{-\nu }\cdot e^{a\cdot x}\cdot \gamma (\nu ,a\cdot x)}{\mathrm{\Gamma }\left(\nu \right)}$ is the ${\mathrm{E}}_t$-function and $\gamma (a,b)$ is the lower incomplete gamma function.

Further reading

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