Function Repository Resource:

FractionalD

Source Notebook

Calculate a fractional derivative

Contributed by: Sami Yrjänheikki

ResourceFunction["FractionalD"][f,{x,α}]

gives the fractional derivative .

Details and Options

If α is an integer, ResourceFunction["FractionalD"][f,{x,α}] is equivalent to D[f,{x,α}].

The Cauchy method uses the formula

.

The Caputo method uses the formula

.

Examples

Basic Examples (2)

Half-derivative of x²:

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$ResourceFunction["FractionalD"][x^2, {x, 1/2}]$

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Taking the half-derivative twice yields the usual first derivative:

In[2]:=

$ResourceFunction["FractionalD"][ ResourceFunction["FractionalD"][x^2, {x, 1/2}], {x, 1/2}]$

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Scope (5)

If α is an integer, the result is the normal derivative:

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Half-derivative of elementary functions:

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$ResourceFunction["FractionalD"][x^a, {x, 1/2}]$

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Half-derivative of trig functions:

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Half-derivative of inverse trig functions:

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A more exotic fractional derivative:

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Options (1)

Method (1)

Choosing between Cauchy and Caputo may result in performance differences:

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Possible Issues (1)

FractionalD uses integration under the hood. Sometimes the integral cannot be computed:

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Neat Examples (1)

The fractional derivative smoothly interpolates between the function and its derivative:

In[15]:=

$Manipulate[ Plot[{x^2, 2 x, Evaluate@ResourceFunction["FractionalD"][x^2, {x, m}]}, {x, 0, 1}], {m, 0.001, 0.999}]$

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Publisher

Sami Yrjänheikki

Version History

1.0.0 – 15 April 2020

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License