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FractionalD
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FractionalD

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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]).
  • 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) 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 all

Basic Examples  (4)Summary of the most common use cases

Calculate the half-order fractional derivative of a quadratic function with respect to x:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-9ijqxw
Out[1]=1

Arbitrary-order fractional derivative of a quadratic function with respect to x:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-06lozd
Out[1]=1

Plot these fractional derivatives for different 's:

In[2]:=2
✖
https://wolfram.com/xid/0cg37llh7-pggxbv
Out[2]=2

Calculate the -order fractional derivative of a constant with respect to x:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-y2w9t3
Out[1]=1

Fractional derivative of MittagLefflerE:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-gl9r59
Out[1]=1

Scope  (8)Survey of the scope of standard use cases

Fractional derivative of the power function with respect to x:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-zamx21
Out[1]=1

0.23-order fractional derivative of the Exp function with respect to x:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-gvkdq9
Out[1]=1

For positive integer , the fractional Riemann–Liouville derivative coincides with the ordinary derivative:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-b0uzfd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg37llh7-lfctso
Out[2]=2

For negative integer , FractionalD differs from the ordinary indefinite integral by a constant:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-mdyb9p
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg37llh7-l7c047
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg37llh7-dv7zz3
Out[3]=3

Fractional derivatives of Sin function are written in terms of HypergeometricPFQ:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-yyvdki
Out[1]=1

Fractional derivatives of BesselJ function:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-e3yxgu
Out[1]=1

Fractional derivatives of MeijerG function are given in terms of another MeijerG function:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-4k5kwb
Out[1]=1

Laplace transform of a fractional integral in general form:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-o0d9hj
Out[1]=1

Substitute the exponential function:

In[2]:=2
✖
https://wolfram.com/xid/0cg37llh7-en3a4l
Out[2]=2

Get the same result by applying LaplaceTransform to the FractionalD of Exp:

In[3]:=3
✖
https://wolfram.com/xid/0cg37llh7-7l8v0j
Out[3]=3

Options  (1)Common values & functionality for each option

Assumptions  (1)

FractionalD may return a ConditionalExpression:

In[2]:=2
✖
https://wolfram.com/xid/0cg37llh7-bais0b
Out[2]=2

Restricting parameters using Assumptions will simplify the output:

In[3]:=3
✖
https://wolfram.com/xid/0cg37llh7-mu7kqu
Out[3]=3

Applications  (2)Sample problems that can be solved with this function

Calculate the half-order fractional derivative of the cubic function:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-il7axk
Out[1]=1

Get the ordinary derivative of the cubic function repeating the half-order fractional differentiation:

In[2]:=2
✖
https://wolfram.com/xid/0cg37llh7-2aq2lc
Out[2]=2

Recover the initial function using fractional integration:

In[3]:=3
✖
https://wolfram.com/xid/0cg37llh7-t34o6x
Out[3]=3

Consider the following fractional order integral equation:

In[1]:=1
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https://wolfram.com/xid/0cg37llh7-xlagoz

Solve it for the Laplace transform:

In[2]:=2
✖
https://wolfram.com/xid/0cg37llh7-i1d1r4
Out[2]=2

Find the inverse transform:

In[3]:=3
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https://wolfram.com/xid/0cg37llh7-km5asc
Out[3]=3

Verify this solution:

In[4]:=4
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https://wolfram.com/xid/0cg37llh7-1p7fu3
Out[4]=4

Properties & Relations  (6)Properties of the function, and connections to other functions

FractionalD is defined for all real :

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-z2ojjm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg37llh7-n1trsx
Out[2]=2

The 0-order fractional derivative of a function is the function itself:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-vm2in3
Out[1]=1

FractionalD is not defined for complex order :

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-ri9o7g
Out[1]=1

In general, the fractional derivative of a constant is not 0:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-heuuox
Out[1]=1

FractionalD results may contain DifferenceRoot sequences:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-fgc8rk
Out[1]=1

This general expression is simplified to a finite sum of HypergeometricPFQ instances if is a given real number:

In[2]:=2
✖
https://wolfram.com/xid/0cg37llh7-0lugr2
Out[2]=2

Calculate the fractional derivative of a function at some point:

In[1]:=1
✖
https://wolfram.com/xid/0cg37llh7-rpoch9
Out[1]=1

Use the NFractionalD function for faster numerical calculations:

In[2]:=2
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https://wolfram.com/xid/0cg37llh7-s892kc
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

Create a table of α^(th) fractional and n^(th) ordinary derivatives of a few special functions:

In[1]:=1
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https://wolfram.com/xid/0cg37llh7-q4y35d
In[2]:=2
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https://wolfram.com/xid/0cg37llh7-70w5xz
Wolfram Research (2022), FractionalD, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalD.html.
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Wolfram Research (2022), FractionalD, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalD.html.

Text

Wolfram Research (2022), FractionalD, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalD.html.

✖
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.

✖
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

✖
Wolfram Language. (2022). FractionalD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FractionalD.html

BibTeX

@misc{reference.wolfram_2025_fractionald, author="Wolfram Research", title="{FractionalD}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/FractionalD.html}", note=[Accessed: 10-July-2025 ]}

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@misc{reference.wolfram_2025_fractionald, author="Wolfram Research", title="{FractionalD}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/FractionalD.html}", note=[Accessed: 10-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_fractionald, organization={Wolfram Research}, title={FractionalD}, year={2022}, url={https://reference.wolfram.com/language/ref/FractionalD.html}, note=[Accessed: 10-July-2025 ]}

✖
@online{reference.wolfram_2025_fractionald, organization={Wolfram Research}, title={FractionalD}, year={2022}, url={https://reference.wolfram.com/language/ref/FractionalD.html}, note=[Accessed: 10-July-2025 ]}