FractionalD
FractionalD[f,{x,α}]
gives the Riemann–Liouville fractional derivative 
of order α of the function f.
Details and Options
- FractionalD is also known as the Riemann–Liouville differintegral of f.
- FractionalD generalizes D to fractional order and unifies the notions of derivatives and integrals from calculus.
- FractionalD plays a foundational role in fractional calculus since other types of fractional derivatives such as CaputoD can be defined in terms of it.
- The Riemann–Liouville fractional derivative of
of order 
is defined as 
, where ![n=max(0,TemplateBox[{alpha}, Ceiling])](Files/FractionalD.en/5.png "n=max(0,TemplateBox[{alpha}, Ceiling])")
. - The derivatives of fractional order "interpolate" between the derivatives of integer orders, as shown below for the function
and its fractional derivatives of order 
given by ![2/TemplateBox[{{3, -, alpha}}, Gamma] x^(2-alpha)](Files/FractionalD.en/8.png "2/TemplateBox[{{3, -, alpha}}, Gamma] x^(2-alpha)")
for 
: - The order α of a fractional derivative can be symbolic or an arbitrary real number.
- FractionalD[array,{x,α}] threads FractionalD over each element of array.
- FractionalD takes different Assumptions on the parameters of input functions.
- All expressions that do not explicitly depend on the given variable
are interpreted as constants.
Examples
open allclose allBasic Examples (4)
Calculate the half-order fractional derivative of a quadratic function with respect to x:
Arbitrary-order fractional derivative of a quadratic function with respect to x:
Plot these fractional derivatives for different 
's:
Calculate the 
-order fractional derivative of a constant with respect to x:
Fractional derivative of MittagLefflerE:
Scope (8)
Fractional derivative of the power function with respect to x:
0.23-order fractional derivative of the Exp function with respect to x:
For positive integer 
, the fractional Riemann–Liouville derivative coincides with the ordinary derivative:
For negative integer 
, FractionalD differs from the ordinary indefinite integral by a constant:
Fractional derivatives of Sin function are written in terms of HypergeometricPFQ:
Fractional derivatives of BesselJ function:
Fractional derivatives of MeijerG function are given in terms of another MeijerG function:
Laplace transform of a fractional integral in general form:
Substitute the exponential function:
Get the same result by applying LaplaceTransform to the FractionalD of Exp:
Options (1)
Assumptions (1)
FractionalD may return a ConditionalExpression:
Restricting parameters using Assumptions will simplify the output:
Applications (2)
Calculate the half-order fractional derivative of the cubic function:
Get the ordinary derivative of the cubic function repeating the half-order fractional differentiation:
Recover the initial function using fractional integration:
Consider the following fractional order integral equation:
Properties & Relations (6)
FractionalD is defined for all real 
:
The 0-order fractional derivative of a function is the function itself:
FractionalD is not defined for complex order 
:
In general, the fractional derivative of a constant is not 0:
FractionalD results may contain DifferenceRoot sequences:
This general expression is simplified to a finite sum of HypergeometricPFQ instances if 
is a given real number:
Calculate the fractional derivative of a function at some point:
Use the NFractionalD function for faster numerical calculations:
fractional and 
ordinary derivatives of a few special functions:Text
Wolfram Research (2022), FractionalD, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalD.html.
CMS
Wolfram Language. 2022. "FractionalD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FractionalD.html.
APA
Wolfram Language. (2022). FractionalD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FractionalD.html