Function Repository Resource:

ConformableD

Calculate the conformable fractional derivative

Contributed by: Naman T.

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

gives the conformal fractional derivative T_α(f)(x) of order α of the function f.

Details and Options

The conformable fractional derivative is a natural generic derivative on fractional orders introduced by Khalil et al (2014), see https://doi.org/10.1016/j.cam.2014.01.002 for details.

The basic definition is made on α∈[0,1) as follows:

for x>0, which can then be easily generalised for any α∈[n,n+1), n natural.

ResourceFunction["ConformableD"][array,{x,α}] threads ResourceFunction["ConformableD"] over each element of array.

ResourceFunction["ConformableD"] takes different Assumptions on the parameters of the input functions.

All inputs free of the independent variable are treated as constants.

Examples

Basic Examples (6)

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

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In general, for 0<α<1, we have:

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$Assuming[ {\[Alpha] \[Element] Reals, 0 < \[Alpha] < 1}, Simplify@ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][x, {x, \[Alpha]}] ]$

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Arbitrary-order fractional derivative of a function in x:

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$ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][f[x], {x, \[Alpha]}]$

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Plot these for f(x)=x²:

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$Plot[Evaluate@Table[ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][ x^2, {x, \[Alpha]}], {\[Alpha], {0, 1/2, 9/10, 1}}], {x, 0, 1}, PlotLegends -> Thread[\[Alpha] == {0, 1/2, 9/10, 1}] ]$

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Conformable fractional derivative of Sin:

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$fr = Table[ ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][Sin[x], {x, \[Alpha]}], {\[Alpha], {1/2, 9/10, 2.5, 3.5}} ]$

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$Plot[fr, {x, 0, 10 \[Pi]}]$

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Conformable fractional derivative of MittagLefflerE:

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Conformable fractional derivative of JacobiSN at various values of m:

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$Table[ ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][JacobiSN[u, m], {u, \[Alpha]}], {\[Alpha], 0, 1, 1/3} ]$

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Conformable fractional derivative of some transcedental functions:

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$ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][{Exp[x], Log[x], Sinh[x]}, {x, \[Alpha]}]$

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Scope (5)

Conformable fractional derivative satisfies the product, quotient, and chain rules:

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Conformable derivative of the power function with respect to x:

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$ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][x^n, {x, \[Alpha]}]$

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For positive integral values of α, the conformable derivative coincides with the ordinary derivative:

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$ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][x^n, {x, 2}] == D[x^n, {x, 2}]$

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Conformable derivatives of BesselJ function:

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$ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][BesselJ[\[Nu], x], {x, \[Alpha]}]$

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Conformable fractional derivatives of MeijerG function are given in terms of multiple MeijerG functions:

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$ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][ MeijerG[{{a1}, {a2}}, {{b1}, {b2}}, x], {x, \[Alpha]}]$

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Options (2)

Assumptions (2)

By default, no assumptions are considered:

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$Simplify@ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][x^5, {x, \[Alpha]}]$

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Assuming 2<α<3:

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$ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][x^5, {x, \[Alpha]}, Assumptions -> {\[Alpha] \[Element] Reals, 2 < \[Alpha] < 3}]$

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Applications (1)

Solve the basic conformable fractional differential equation T_α(y)(x)=y(x):

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$DSolve[ ResourceFunction[ CloudObject[ "https://www.wolframcloud.com/obj/namantaggar11/DeployedResources/Function/ConformableD"]][f[x], {x, \[Alpha]}, Assumptions -> {\[Alpha] \[Element] Reals, 0 < \[Alpha] < 1}] == f[x], f[x], x ]$

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Properties and Relations (2)

ConformableD is defined for all positive and real values of α:

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At the same time, α has to be real:

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The 0-order conformable fractional derivative of a function is not itself:

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Wolfram Cloud Resource Object