In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
D f(x) = \frac f(x)\,,
and of the integration ...
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
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
.
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
In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...
. 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:
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: