# Plot fractional trigonometric functions with the Mittag-Leffler function

Can anyone help please? Im trying to plot the solution $$X$$ of the system as in the paper attached - about fractional calculus which is $$X= [E_{\nu}(2t^{\nu})][2 \cos_{\nu}(3t^{\nu})+4 \sin_{\nu}(3t^{\nu})]$$ where $$E$$ is the Mittag-Leffler function and the defintions of $$\cos_{\nu}$$ and $$\sin_{\nu}$$ are in the paper also.

• Where are you stuck? – Kuba Apr 14 at 11:45
• What is the commands in mathematica to get the graphs of the solution \$ x = [ mittageleffler (2t^v)] [ 2cos _v(3t^v)+4Sin _v(3t^ )] as in the photo ; Thanks – Jojo Apr 14 at 11:58
• Maybe the downvoters can chill a bit, working with the Mittag-Leffler function is not easy, and Jojo is a beginner. – Roman Apr 14 at 18:47
• For future reference, I think it is best to add an opening paragraph before embedding images. I had to scroll through carefully to see that it was not an image-only post. – Carl Lange Apr 14 at 20:30

I'm not sure I'm understanding the notation right, but think that you can express all of these functions in terms of the builtin MittagLefflerE.

For the fractional cosine and sine definitions I'm using $$z=x^{\alpha}$$:

fcos[α_, z_] = Sum[(-z^2)^k/Gamma[1+2*α*k], {k, 0, ∞}]

MittagLefflerE[2 α, -z^2]

fsin[α_, z_] = Sum[(z*(-z^2)^k)/Gamma[(1+α)+2*k*α], {k, 0, ∞}]

z MittagLefflerE[2 α, 1 + α, -z^2]

For the fractional exponential I'm using $$z=i x^{\alpha}$$:

fexp[α_, z_] = fcos[α, z/I] + I*fsin[α, z/I]

MittagLefflerE[2 α, z^2] + z MittagLefflerE[2 α, 1 + α, z^2]

With these we can define your $$X$$-function:

X[ν_, t_] = fexp[ν, 2*t^ν]*(2*fcos[ν, 3*t^ν] + 4*fsin[ν, 3*t^ν])

(2 MittagLefflerE[2 ν, -9 t^(2 ν)] + 12 t^ν MittagLefflerE[2 ν, 1 + ν, -9 t^(2 ν)]) (MittagLefflerE[2 ν, 4 t^(2 ν)] + 2 t^ν MittagLefflerE[2 ν, 1 + ν, 4 t^(2 ν)])

Plot for $$\nu=1$$: using Evaluate to speed up the plotting dramatically,

Plot[Evaluate@X[1, t], {t, 0, 3}]

Plot for $$\nu=0.8$$: much slower as we cannot use the Evaluate trick,

I cannot read the labels on your other plots and so cannot reproduce them. What are their parameters $$\nu$$ and their axis labels?