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 $\left( 0,\, \infty \right)$, then the Riemann–Liouville fractional integral $$ states that

\left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac \, \operatornamet

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

Let's define $\operatorname_^ := \frac$, say that $\operatorname_^ \operatorname_^ = \operatorname_^$ and that $\operatorname_^ =$ applies. If $\alpha = m + z \in \mathbb \wedge m \in \mathbb_ \wedge 0 < z < 1$ then we could say $\operatorname_^ = \operatorname_^ = \operatorname_^ = \operatorname_^ = \operatorname_^\operatorname_^ = _^\operatorname_^$. So if $f$ is also $C^\left( 0,\, \infty \right)$, then

\left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac \, \operatornamet.

This is known as the Caputo-type fractional derivative, often written as $_^$.

Definition

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

\left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac \, \operatornamet

where $C^\left( 0,\, \infty \right)$ and $m \in \mathbb_ \wedge 0 < z < 1$.

A popular equivalent definition is:

\left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac\, \operatornamet

where $\alpha \in \mathbb_ \setminus \mathbb$ and $\left\lceil \cdot \right\rceil$ is the ceiling function. This can be derived by substituting $\alpha = m + z$ so that $\left\lceil \alpha \right\rceil = m + 1$ would apply and $\left\lceil \alpha \right\rceil + z = \alpha + 1$ follows.

Another popular equivalent definition is given by:

\left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac\, \operatornamet

where $n - 1 < \alpha < n \in \mathbb.$.

The problem with these definitions is that they only allow arguments in $\left( 0,\, \infty \right)$. This can be fixed by replacing the lower integral limit with $a$: \left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac\, \operatornamet. The new domain is $\left( a,\, \infty \right)$.

Properties and theorems

Basic properties and theorems

A few basic properties are:

Non-commutation

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

$\operatorname_^\operatorname_^ = \operatorname_^ \ne \operatorname_^\operatorname_^$

where $\alpha \in \mathbb_ \setminus \mathbb \wedge \beta \in \mathbb$.

Fractional Leibniz rule

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

\operatorname_^\left g\left( x \right) \cdot h\left( x \right) \right= \sum\limits_^\left \binom \cdot g^\left( x \right) \cdot \operatorname_^\left[ h\left( x \right) \right\right] - \frac \cdot g\left( a \right) \cdot h\left( a \right)

where $\binom = \frac$ 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:

\left f\left( x \right) \right= \left f\left( x \right) \right\right">\operatorname_^\left f\left( x \right) \right\right/math>

Furthermore, the following relation applies:

\left f\left( x \right) \right= \left f\left( x \right) \right- \sum\limits_^\left \frac \cdot f^\left( 0 \right) \right/math>

where $$ is the Riemann–Liouville fractional derivative.

Laplace transform

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

\mathcal_\left\\left( s \right) = s^ \cdot F\left( s \right) - \sum\limits_^\left s^ \cdot f^\left( 0 \right) \right/math>

where $\mathcal_\left\\left( s \right) = F\left( s \right)$.

Caputo fractional derivative of some functions

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

\begin
\left c \right&= \frac \cdot \int\limits_^ \frac\, \operatornamet = \frac \cdot \int\limits_^ \frac\, \operatornamet\\
\left c \right&= 0
\end

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

\begin
\left x^ \right&= \left x^ \right\right">\operatorname_^\left x^ \right\right= \frac \cdot \left x^ \right\ \left x^ \right&= \begin \frac \left( x^ - a^ \right),\, &\text \left\lceil \alpha \right\rceil - 1 < b \wedge b \in \mathbb\\ 0,\, &\text \left\lceil \alpha \right\rceil - 1 \geq b \wedge b \in \mathbb\\ \end
\end

The Caputo fractional derivative of an exponential function $e^$ is given by:

\begin
\left e^ \right&= \left e^ \right\right">\operatorname_^\left e^ \right\right= b^ \cdot \left e^ \right\
\left e^ \right &= b^ \cdot \left( E_\left( \left\lceil \alpha \right\rceil - \alpha,\, b \right) - E_\left( \left\lceil \alpha \right\rceil - \alpha,\, b \right) \right)\\
\end

where $E_\left( \nu,\, a \right) = \frac$ is the $\operatorname_$-function and $\gamma \left( a,\, b \right)$ is the lower incomplete gamma function.

References

{{reflist

Further reading

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

Fractional calculus