Wolfram Research

Function Repository Resource:

FractionalOrderD

Source Notebook

Compute the fractional derivative of an expression

Contributed by: Oleg Marichev & Paco Jain (Wolfram Research)

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

gives the α-order fractional integro-derivative of f with respect to x.

Details and Options

ResourceFunction["FractionalOrderD"] automatically maps over lists in its first argument.
All expressions that do not explicitly depend on the given variable are taken to have zero partial derivative.

Examples

Basic Examples (5) 

Compute a fractional derivative of a power function:

In[1]:=
ResourceFunction["FractionalOrderD"][x^2, {x, 3/2}]
Out[1]=

The first-order fractional derivative is equivalent to differentiation:

In[2]:=
ResourceFunction["FractionalOrderD"][x^2, {x, 1}]
Out[2]=
In[3]:=
D[x^2, x]
Out[3]=

The negative-first-order fractional derivative is equivalent to indefinite integration:

In[4]:=
ResourceFunction["FractionalOrderD"][x^2, {x, -1}]
Out[4]=
In[5]:=
Integrate[x^2, x]
Out[5]=

Compute the fractional derivative of symbolic order α of an exponential function:

In[6]:=
ResourceFunction["FractionalOrderD"][E^x, {x, \[Alpha]}]
Out[6]=

Compute the fractional derivative of symbolic order α of a logarithmic function:

In[7]:=
ResourceFunction["FractionalOrderD"][Log[x], {x, -1}]
Out[7]=

Scope (1) 

FractionalOrderD works with symbolic orders α:

In[8]:=
ResourceFunction["FractionalOrderD"][x^n, {x, \[Alpha]}]
Out[8]=

Possible Issues (1) 

FractionalOrderD may return results in terms of Unevaluated sums:

In[9]:=
ResourceFunction["FractionalOrderD"][ChebyshevT[n, x], {x, \[Alpha]}]
Out[9]=

Interactive Examples (1) 

Varying α from 0 to -1 smoothly relates a function and its derivative:

In[10]:=
f[x] := x^3 Sin[x];
Manipulate[
 Plot[Evaluate@{f[x], D[f[x], x], ResourceFunction["FractionalOrderD"][f[x], {x, m}]}, {x, 0, 5}], {m, 0, -1}]
Out[10]=

Neat Examples (1) 

Create a table of fractional derivatives:

In[11]:=
flist = {E^(x), x^2, Log[x], Sin[x], ArcTan[x], Cosh[x], ArcCosh[x], EllipticE[x], BesselJ[0, x]};
In[12]:=
Grid[Join[{{f[x], Inactive[
ResourceFunction["FractionalOrderD"]][f[x], {x, n}]}}, Transpose[{flist, ResourceFunction["FractionalOrderD"][flist, {x, n}]}]], Background -> {None, {{None, GrayLevel[.9]}}, {{1, 1} -> Hue[.6, .4, 1], {1, 2} -> Hue[.6, .4, 1]}}, BaseStyle -> {FontFamily -> Times, FontSize -> 12}, Dividers -> All,
   FrameStyle -> Hue[.6, .4, .8], Spacings -> {2, 1}] // TraditionalForm
Out[12]=

Publisher

Wolfram|Alpha Math Team

Version History

  • 3.1.4 – 07 November 2022
  • 3.1.3 – 11 October 2022
  • 3.1.2 – 11 October 2022
  • 3.1.1 – 23 August 2022
  • 3.1.0 – 23 August 2022
  • 3.0.0 – 23 August 2022
  • 2.1.0 – 11 September 2021
  • 2.0.0 – 05 October 2020
  • 1.0.0 – 23 September 2020

Related Resources

Author Notes

There are different ways of defining the fractional integro-derivative. The current implementation (v 1.0) implements only the Riemann-Liouville formulation. Another frequently used definition, the Grunwald-Letnikov formulation should perhaps be implemented in a future version.

Known issues:

JSS Examples

License Information