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
{\displaystyle D}
D
f
(
x
)
=
d
d
x
f
(
x
)
,
{\displaystyle Df(x)={\frac {d}{dx}}f(x)\,,}
and of the integration operator
J
{\displaystyle J}
J
f
(
x
)
=
∫
0
x
f
(
s
)
d
s
,
{\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,}
and developing a calculus for such operators generalizing the classical one.
In this context, the term powers refers to iterative application of a linear operator
D
{\displaystyle D}
to a function
f
{\displaystyle f}
, that is, repeatedly composing
D
{\displaystyle D}
with itself, as in
D
n
(
f
)
=
(
D
∘
D
∘
D
∘
⋯
∘
D
⏟
n
)
(
f
)
=
D
(
D
(
D
(
⋯
D
⏟
n
(
f
)
⋯
)
)
)
{\displaystyle D^{n}(f)=(\underbrace {D\circ D\circ D\circ \cdots \circ D} _{n})(f)=\underbrace {D(D(D(\cdots D} _{n}(f)\cdots )))}
.
For example, one may ask for a meaningful interpretation of
D
=
D
1
2
{\displaystyle {\sqrt {D}}=D^{\frac {1}{2}}}
as an analogue of the functional square root for the differentiation operator, that is, an expression for some linear operator that, when applied twice to any function, will have the same effect as differentiation. More generally, one can look at the question of defining a linear operator
D
a
{\displaystyle D^{a}}
for every real number
a
{\displaystyle a}
in such a way that, when
a
{\displaystyle a}
takes an integer value
n
∈
Z
{\displaystyle n\in \mathbb {Z} }
, it coincides with the usual
n
{\displaystyle n}
-fold differentiation
D
{\displaystyle D}
if
n
>
0
{\displaystyle n>0}
, and with the
n
{\displaystyle n}
-th power of
J
{\displaystyle J}
when
n
<
0
{\displaystyle n<0}
.
One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator
D
{\displaystyle D}
is that the sets of operator powers
{
D
a
∣
a
∈
R
}
{\displaystyle \{D^{a}\mid a\in \mathbb {R} \}}
defined in this way are continuous semigroups with parameter
a
{\displaystyle a}
, of which the original discrete semigroup of
{
D
n
∣
n
∈
Z
}
{\displaystyle \{D^{n}\mid n\in \mathbb {Z} \}}
for integer
n
{\displaystyle n}
is a denumerable subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.
Fractional differential equations, also known as extraordinary differential equations, are a generalization of differential equations through the application of fractional calculus.