FractionalD[f,{x,α}]
gives the Riemann–Liouville fractional derivative 
of order α of the function f.
FractionalD
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])](Files/FractionalD.en/5.png "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)](Files/FractionalD.en/8.png "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 all close allBasic Examples (4)
Calculate the half-order fractional derivative of a quadratic function with respect to x:
FractionalD[x^2, {x, 1 / 2}]Arbitrary-order fractional derivative of a quadratic function with respect to x:
FractionalD[x^2, {x, α}]Plot these fractional derivatives for different 
's:
Plot[Evaluate[Table[(2 x^2 - α/Gamma[3 - α]), {α, {0, 1 / 2, 9 / 10, 1}}]], {x, 0, 1}]Calculate the 
-order fractional derivative of a constant with respect to x:
FractionalD[c, {x, α}]Fractional derivative of MittagLefflerE:
FractionalD[MittagLefflerE[a, x], {x, 3 / 2}]Scope (8)
Fractional derivative of the power function with respect to x:
FractionalD[x ^ n, {x, 1 / 3}]0.23-order fractional derivative of the Exp function with respect to x:
FractionalD[Exp[x], {x, 0.23}]For positive integer 
, the fractional Riemann–Liouville derivative coincides with the ordinary derivative:
FractionalD[ArcTan[x], {x, 2}]D[ArcTan[x], {x, 2}]For negative integer 
, FractionalD differs from the ordinary indefinite integral by a constant:
FractionalD[Sin[a x], {x, -1}]Integrate[Sin[a x], x]Together[%% - %]Fractional derivatives of Sin function are written in terms of HypergeometricPFQ:
FractionalD[Sin[x], {x, α}]Fractional derivatives of BesselJ function:
FractionalD[BesselJ[ν, x], {x, α}]Fractional derivatives of MeijerG function are given in terms of another MeijerG function:
FractionalD[MeijerG[{{a1}, {a2}}, {{b1}, {b2}}, x], {x, α}]Laplace transform of a fractional integral in general form:
LaplaceTransform[FractionalD[f[t], {t, -2 / 3}], t, s]Substitute the exponential function:
% /. {f[t_] -> Exp[t]}//SimplifyGet the same result by applying LaplaceTransform to the FractionalD of Exp:
LaplaceTransform[FractionalD[Exp[t], {t, -2 / 3}], t, s]//FullSimplifyOptions (1)
Assumptions (1)
FractionalD may return a ConditionalExpression:
FractionalD[MittagLefflerE[a, x], {x, α}]Restricting parameters using Assumptions will simplify the output:
FractionalD[MittagLefflerE[a, x], {x, α}, Assumptions -> {a > 0, α∈ℤ, α ≥ 0}]Applications (2)
Calculate the half-order fractional derivative of the cubic function:
FractionalD[x ^ 3, {x, 1 / 2}]Get the ordinary derivative of the cubic function repeating the half-order fractional differentiation:
FractionalD[%, {x, 1 / 2}]Recover the initial function using fractional integration:
FractionalD[FractionalD[FractionalD[%, {x, -1 / 3}], {x, -1 / 3}], {x, -1 / 3}]Consider the following fractional order integral equation:
eqn = FractionalD[y[t], {t, -2 / 3}] - y[t] == 1;Solve it for the Laplace transform:
Solve[LaplaceTransform[eqn, t, s], LaplaceTransform[y[t], t, s]]sol = InverseLaplaceTransform[%, s, t]eqn /. sol /. t -> 0.3Properties & Relations (6)
FractionalD is defined for all real 
:
FractionalD[x ^ 2, {x, 4 / 5}]FractionalD[x ^ 2, {x, -4 / 5}]The 0-order fractional derivative of a function is the function itself:
FractionalD[x ^ 2, {x, 0}]FractionalD is not defined for complex order 
:
FractionalD[x ^ 2, {x, -4 / 5 + I}]
In general, the fractional derivative of a constant is not 0:
FractionalD[c, {x, α}]FractionalD results may contain DifferenceRoot sequences:
FractionalD[Sin[x] Cos[x], {x, α}, Assumptions -> {α∉ℤ, α > 0}]This general expression is simplified to a finite sum of HypergeometricPFQ instances if 
is a given real number:
% /. α -> 0.2Calculate the fractional derivative of a function at some point:
N[FractionalD[Sin[x]Cos[x - 1], {x, 1 / 2}] /. x -> 1 / 3]//AbsoluteTimingUse the NFractionalD function for faster numerical calculations:
NFractionalD[Sin[x]Cos[x - 1], {x, 1 / 2}, 1 / 3]//AbsoluteTimingNeat Examples (1)
Create a table of α
fractional and n
ordinary derivatives of a few special functions:
flist = {E ^ (x), x ^ 2, Log[x], Sin[x], ArcTan[x], BesselJ[0, x], Hypergeometric1F1[a, b, x]};Grid[Join[{{f[x], Subsuperscript[D, x, α][f[x]], D[f[x], {x, n}]}}, Transpose[{flist, Map[FractionalD[#, {x, α}]&, flist], Map[D[#, {x, n}]&, flist]}]], Background -> {None, {{None, GrayLevel[.7]}}, {{1, 1} -> Hue[.6, .4, 1], {1, 2} -> Hue[.6, .4, 1], {1, 3} -> Hue[.6, .4, 1]}}, BaseStyle -> {FontFamily -> Times, FontSize -> 12}, Dividers -> All, FrameStyle -> Hue[.6, .4, .8], Spacings -> {2, 1}]//TraditionalFormText
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.
APA
Wolfram Language. (2022). FractionalD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FractionalD.html