Basic Examples (6)

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

({{"[◼]", "ConformableD" }})[x,{x,1/2}]

In general, for 0<α<1, we have:

Assuming[
{α∈Reals,0<α<1},
Simplify@({{"[◼]", "ConformableD" }})[x,{x,α}]
]

Arbitrary-order fractional derivative of a function in x:

({{"[◼]", "ConformableD" }})[f[x],{x,α}]

Plot these for f(x)=(x^2):

Plot[Evaluate@Table[({{"[◼]", "ConformableD" }})[(x^2),{x,α}],{α,{0,1/2,9/10,1}}],
{x,0,1},PlotLegends->Thread[α=={0,1/2,9/10,1}]
]

Conformable fractional derivative of Sin:

fr=Table[
({{"[◼]", "ConformableD" }})[Sin[x],{x,α}],
{α,{1/2,9/10,2.5,3.5}}
]

Plot[fr,{x,0,10π}]

Conformable fractional derivative of MittagLefflerE:

({{"[◼]", "ConformableD" }})[MittagLefflerE[a,x],{x,0.8}]

Conformable fractional derivative of JacobiSN at various values of m:

Table[
({{"[◼]", "ConformableD" }})[JacobiSN[u,m],{u,α}],
{α,0,1,1/3}
]

Conformable fractional derivative of some transcedental functions:

({{"[◼]", "ConformableD" }})[{Exp[x],Log[x],Sinh[x]},{x,α}]

Scope (5)

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

({{"[◼]", "ConformableD" }})[f[x]g[x],{x,1/2}]

({{"[◼]", "ConformableD" }})[f[x]/g[x],{x,1/2}]

({{"[◼]", "ConformableD" }})[f[g[x]],{x,1/2}]

Conformable derivative of the power function with respect to x:

({{"[◼]", "ConformableD" }})[(x^n),{x,α}]

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

({{"[◼]", "ConformableD" }})[(x^n),{x,2}]==D[(x^n),{x,2}]

Conformable derivatives of BesselJ function:

({{"[◼]", "ConformableD" }})[BesselJ[ν,x],{x,α}]

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

({{"[◼]", "ConformableD" }})[MeijerG[{{a1},{a2}},{{b1},{b2}},x],{x,α}]

FreeQ[%,MeijerG]

Options (2)

Assumptions (2)

By default, no assumptions are considered:

Simplify@({{"[◼]", "ConformableD" }})[(x^5),{x,α}]

Assuming 2<α<3:

({{"[◼]", "ConformableD" }})[(x^5),{x,α},Assumptions->{α∈Reals,2<α<3}]

Applications (1)

Solve the basic conformable fractional differential equation (Subscript[T, α])(y)(x)=y(x):

DSolve[
({{"[◼]", "ConformableD" }})[f[x],{x,α},Assumptions->{α∈Reals,0<α<1}]==f[x],
f[x],x
]

Properties and Relations (2)

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

({{"[◼]", "ConformableD" }})[f,{x,-1}]

At the same time, α has to be real:

({{"[◼]", "ConformableD" }})[f,{x,I}]

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

({{"[◼]", "ConformableD" }})[Sin[x],{x,0}]