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NFractionalD
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NFractionalD

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NFractionalD[f,{x,α},x0]

gives a numerical approximation to the Riemann–Liouville fractional derivative of order α of the function f at the point x0.

Details and Options

Examples

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Basic Examples  (5)Summary of the most common use cases

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

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-9ijqxw
Direct link to example
Out[1]=1

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

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-y2w9t3
Direct link to example
Out[1]=1

Plot the -order fractional derivative of an exponential function:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-8dwjik
Direct link to example
Out[1]=1

Numerical fractional derivative of MittagLefflerE at a point:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-gl9r59
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Out[1]=1

Generate a list of numerical values of the fractional derivative of the MittagLefflerE function:

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In[2]:=2
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https://wolfram.com/xid/0dqw9zwuyov-p4s7hs
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Plot this fractional derivative:

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In[3]:=3
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https://wolfram.com/xid/0dqw9zwuyov-dcm5gq
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Out[3]=3

Numerical fractional integral:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-ge8q19
Direct link to example
Out[1]=1

Scope  (9)Survey of the scope of standard use cases

Plot the -order fractional derivative of the square function with respect to x:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-q6t8ng
Direct link to example
Out[1]=1

Calculate the 0.23-order fractional derivative of the Exp function numerically and symbolically:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-joioqz
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0dqw9zwuyov-gvkdq9
Direct link to example
Out[2]=2

Calculate the value of the half-order derivative of the ArcTan function at some point:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-b0uzfd
Direct link to example
Out[1]=1

Plot this function:

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In[2]:=2
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https://wolfram.com/xid/0dqw9zwuyov-6h0lup
Direct link to example
Out[2]=2

The -order fractional derivative of the Sin function:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-2gpsww
Direct link to example
Out[1]=1

The -order fractional integral of the Sin function:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-nf4fb4
Direct link to example
Out[1]=1

The half-order fractional derivative of the AiryAi function:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-xjal6t
Direct link to example
Out[1]=1

Numerically calculate the fractional derivative of the BesselJ function:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-e3yxgu
Direct link to example
Out[1]=1

Numerically calculate the fractional derivative of the MeijerG function:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-4k5kwb
Direct link to example
Out[1]=1

Plot the numerically calculated fractional derivatives and integrals of a trigonometric product:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-v8dpud
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0dqw9zwuyov-hkl3c3
Direct link to example
Out[2]=2

Options  (2)Common values & functionality for each option

Method  (2)

NFractionalD has two built-in methods, the Riemann–Liouville and the Grünwald–Letnikov methods:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-mr32es
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0dqw9zwuyov-2efjy6
Direct link to example
Out[2]=2

If the Method is not specified, NFractionalD automatically uses the Riemann–Liouville approach:

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In[3]:=3
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https://wolfram.com/xid/0dqw9zwuyov-79lsi5
Direct link to example
Out[3]=3

In some cases, the Grünwald–Letnikov method is able to calculate the numerical fractional derivative:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-1z2pdx
Direct link to example
Out[1]=1

It is not possible to calculate this fractional derivative using the Riemann–Liouville method:

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In[2]:=2
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https://wolfram.com/xid/0dqw9zwuyov-cc3wod
Direct link to example
Out[2]=2

Applications  (3)Sample problems that can be solved with this function

NFractionalD is able to numerically calculate fractional derivatives when FractionalD fails:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-fgc8rk
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0dqw9zwuyov-fztxfs
Direct link to example
Out[2]=2

FractionalD outputs may contain DifferenceRoot sequences:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-qbhxtp
Direct link to example
Out[1]=1

For a specific order of fractional differentiation, this is a cumbersome sum of HypergeometricPFQ functions:

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In[2]:=2
✖
https://wolfram.com/xid/0dqw9zwuyov-0lugr2
Direct link to example
Out[2]=2

However, the numerically calculated fractional derivative plot might be insightful:

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In[3]:=3
✖
https://wolfram.com/xid/0dqw9zwuyov-rpoch9
Direct link to example
Out[3]=3

Effectively generate a list of numerical values of fractional derivatives of a complex function using NFractionalD:

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-ibqmwt
Direct link to example

The timing for the calculated 151 values:

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In[2]:=2
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https://wolfram.com/xid/0dqw9zwuyov-b1o5f5
Direct link to example
Out[2]=2

Plot this fractional derivative:

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In[3]:=3
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https://wolfram.com/xid/0dqw9zwuyov-7ph0ls
Direct link to example
Out[3]=3

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

NFractionalD is defined for all real :

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-z2ojjm
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0dqw9zwuyov-n1trsx
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0dqw9zwuyov-by8bld
Direct link to example
Out[3]=3

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

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-heuuox
Direct link to example
Out[1]=1

NFractionalD gives the output in WorkingPrecision (if not specified, it is MachinePrecision):

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In[1]:=1
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https://wolfram.com/xid/0dqw9zwuyov-u2ihnv
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0dqw9zwuyov-fe5t59
Direct link to example
Out[2]=2

If the Method option is not specified, NFractionalD uses the "RiemannLiouville" method:

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In[1]:=1
✖
https://wolfram.com/xid/0dqw9zwuyov-neou5t
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0dqw9zwuyov-6cv5to
Direct link to example
Out[2]=2

Plot the fractional derivative using the numerical NFractionalD approach:

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In[1]:=1
✖
https://wolfram.com/xid/0dqw9zwuyov-um8zu6
Direct link to example
Out[1]=1

Compare with the symbolic FractionalD approach:

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In[2]:=2
✖
https://wolfram.com/xid/0dqw9zwuyov-508p2e
Direct link to example
Out[2]=2

For negative orders , NCaputoD coincides with NFractionalD:

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In[1]:=1
✖
https://wolfram.com/xid/0dqw9zwuyov-8dz1if
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0dqw9zwuyov-t2n3hc
Direct link to example
Out[2]=2

Possible Issues  (3)Common pitfalls and unexpected behavior

NFractionalD will generate an error message if the Method option is not correct:

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In[1]:=1
✖
https://wolfram.com/xid/0dqw9zwuyov-dxj3jy
Direct link to example
Out[1]=1

NFractionalD will generate an error message if the precision of input is less than the WorkingPrecision:

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In[1]:=1
✖
https://wolfram.com/xid/0dqw9zwuyov-ibopf7
Direct link to example
Out[1]=1

NFractionalD takes only numeric points of evaluation:

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In[1]:=1
✖
https://wolfram.com/xid/0dqw9zwuyov-h5ph01
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0dqw9zwuyov-6iqsnn
Direct link to example
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

Plot the Sin function and its half, first and -order derivatives:

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In[1]:=1
✖
https://wolfram.com/xid/0dqw9zwuyov-wuw4q5
Direct link to example
Out[1]=1
Wolfram Research (2022), NFractionalD, Wolfram Language function, https://reference.wolfram.com/language/ref/NFractionalD.html.
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Wolfram Research (2022), NFractionalD, Wolfram Language function, https://reference.wolfram.com/language/ref/NFractionalD.html.

Text

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

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

CMS

Wolfram Language. 2022. "NFractionalD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NFractionalD.html.

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Wolfram Language. 2022. "NFractionalD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NFractionalD.html.

APA

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

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

BibTeX

@misc{reference.wolfram_2025_nfractionald, author="Wolfram Research", title="{NFractionalD}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/NFractionalD.html}", note=[Accessed: 04-April-2025 ]}

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@misc{reference.wolfram_2025_nfractionald, author="Wolfram Research", title="{NFractionalD}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/NFractionalD.html}", note=[Accessed: 04-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_nfractionald, organization={Wolfram Research}, title={NFractionalD}, year={2022}, url={https://reference.wolfram.com/language/ref/NFractionalD.html}, note=[Accessed: 04-April-2025 ]}

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@online{reference.wolfram_2025_nfractionald, organization={Wolfram Research}, title={NFractionalD}, year={2022}, url={https://reference.wolfram.com/language/ref/NFractionalD.html}, note=[Accessed: 04-April-2025 ]}