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 be continuous on, then the Riemann–Liouville fractional integral states that
where is the Gamma function.
Let's define, say that and that applies. If then we could say. So if is also, then
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:
where and.
A popular equivalent definition is:
where and is the ceiling function. This can be derived by substituting so that would apply and follows.
Another popular equivalent definition is given by:
where.
The problem with these definitions is that they only allow arguments in. This can be fixed by replacing the lower integral limit with :. The new domain is.

Properties and theorems

Basic properties and theorems

A few basic properties are:

Properties			Condition
Definition
Linearity
Index law
Semigroup property

Non-commutation

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

Fractional Leibniz rule

The Leibniz rule for the Caputo fractional derivative is given by:
where 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:
Furthermore, the following relation applies:
where is the Riemann–Liouville fractional derivative.

Laplace transform

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

Caputo fractional derivative of some functions

The Caputo fractional derivative of a constant is given by:
The Caputo fractional derivative of a power function is given by:
The Caputo fractional derivative of an exponential function is given by:
where is the -function and is the lower incomplete gamma function.