Fractional derivatives extend ordinary integer-order derivatives to noninteger orders. Unlike integer-order differentiation, fractional differentiation does not have a single canonical definition. Two principal definitions are the Riemann-Liouville fractional derivative and the Caputo fractional derivative.
For noninteger order 
and positive integer 
with 
, both can be defined using the Riemann-Liouville
operator of fractional integration. The
Riemann-Liouville fractional
derivative differentiates after fractional
integration,
 |
whereas the Caputo fractional derivative applies fractional integration after ordinary differentiation,
 |
The two definitions generally differ by terms determined by the behavior of 
at the lower terminal 
. For example, the Riemann-Liouville
fractional derivative of a nonzero constant
function is generally nonzero, while the Caputo
fractional derivative of a constant is zero. The semiderivative
is a fractional derivative of order 
.
The Riemann-Liouville fractional derivative with lower terminal 
is implemented in the Wolfram
Language as FractionalD[f,
x,
alpha
].
A fractional integral can be defined similarly. The study of fractional derivatives and integrals is called fractional calculus.
