Can you add some background information to the equation?  Maybe yours boundary conditions are wrong?

ALREADY I understood.Thanks

@Markiyan Hirnyk 

But for this example:

fracdiff(exp(x), x, 1/2)=erf(sqrt(x))*exp(x)

Maple give wong result,it should be: (sqrt(Pi)*sqrt(x)*exp(x)*erf(sqrt(x))+1)/(sqrt(Pi)*sqrt(x))

Formuła fractional integro-differentiation from :

http://functions.wolfram.com/ElementaryFunctions/Exp/20/03/  

for exp(x).

@Markiyan Hirnyk 

4*(int(sin(x^2)*cos(y^2), [y = 0 .. sqrt(-x^2+1), x = 0 .. 1], numeric))

or

int(sin(r^2*cos(theta)^2)*cos(r^2*sin(theta)^2)*r, [r = 0 .. 1, theta = 0 .. 2*Pi], numeric)

but symbolic solution Maple can't find it.

MMA yes can find it:

Integrate[Sin[r^2*Cos[t]^2]*Cos[r^2*Sin[t]^2]*r, {r, 0, 1}, {t, 0, 2*Pi}]=\[Pi] Sin[1/2]^2

@Markiyan Hirnyk 

MMA code: Integrate[Sin[x^2]*Cos[y^2], {x, y} \[Element] Disk[]]

Pi*Sin(1/2)^2

Yes thank you for great answer,but why I got a strange result (infinity+infinity*Ik......)? .Mayby is a bug?

@Markiyan Hirnyk 

Thanks for more complicated answer.

@acer 

Thank You

@vv 

@kianooshkargar 

Not better way to solve it numerically?

PDE_series.mw

@torabi

Reference  paper from https://arxiv.org/pdf/1304.2505.pdf

@torabi 

1.This method is not perfect ,because it is a numerical method.

2.Maple can't inverse  Laplace Transform,because it is impossible to solve.That way I'm using numerical method.

3.If alpha=1/2  the second condition is falls: diff(T(z, 0), z) = Pi

4.F(s) is a function of s (dummy parameter) from dsolve for z=0.

With my head and with the help of Google. ->


"Fractional Calculus: Definitions and Applications"

http://digitalcommons.wku.edu/cgi/viewcontent.cgi?article=1115&context=theses

@Adam Ledger 

My worksheets they are FREE to download.You can use them for any purpose 

I,m read yours comments  and after an hour I wrote this worksheets.

A Talbot method of Inverse of Laplace Transfom I discovered a few months ago.

Magic numbers from: https://arxiv.org/pdf/1304.2505.pdf

@Carl Love 

Thanks.

@DSkoog