NFractionalD

NFractionalD[f,{x,α},x0]

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

Details and Options

Examples

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Basic Examples  (5)

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

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

Plot the -order fractional derivative of an exponential function:

Numerical fractional derivative of MittagLefflerE at a point:

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

Plot this fractional derivative:

Numerical fractional integral:

Scope  (9)

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

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

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

Plot this function:

The -order fractional derivative of the Sin function:

The -order fractional integral of the Sin function:

The half-order fractional derivative of the AiryAi function:

Numerically calculate the fractional derivative of the BesselJ function:

Numerically calculate the fractional derivative of the MeijerG function:

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

Options  (2)

Method  (2)

NFractionalD has two built-in methods, the RiemannLiouville and the GrünwaldLetnikov methods:

If the Method is not specified, NFractionalD automatically uses the RiemannLiouville approach:

In some cases, the GrünwaldLetnikov method is able to calculate the numerical fractional derivative:

It is not possible to calculate this fractional derivative using the RiemannLiouville method:

Applications  (3)

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

FractionalD outputs may contain DifferenceRoot sequences:

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

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

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

The timing for the calculated 151 values:

Plot this fractional derivative:

Properties & Relations  (6)

NFractionalD is defined for all real :

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

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

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

Plot the fractional derivative using the numerical NFractionalD approach:

Compare with the symbolic FractionalD approach:

For negative orders , NCaputoD coincides with NFractionalD:

Possible Issues  (3)

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

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

NFractionalD takes only numeric points of evaluation:

Neat Examples  (1)

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

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.

CMS

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

BibTeX

@misc{reference.wolfram_2022_nfractionald, author="Wolfram Research", title="{NFractionalD}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/NFractionalD.html}", note=[Accessed: 04-February-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_nfractionald, organization={Wolfram Research}, title={NFractionalD}, year={2022}, url={https://reference.wolfram.com/language/ref/NFractionalD.html}, note=[Accessed: 04-February-2023 ]}