Dear colleague

Actually the question is very general, it depends if you are talking about global or local derivatives. In both there are some advances, for example with Francisco Martínez and Pshtiwan Mohammed we have worked for Laplace Transformation in the local non conformable case. In the local case the definition (and of course its practical use) depends on the kernel.

Thanks

Dear Juan Napoles

Thank you for your time and reply. You are right and I know that. I think there is a misunderstanding here because the question is exactly mentioned. I can't obtain L^-1 {F(s)} . Please look at it again and share your knowledge we us.

Thanks

Dear Dr.

Mohammad M. Amirian

Greating!

It is my pleasure to help you, you can read the following links

https://www.ijedr.org/papers/IJEDR1801106.pdf

http://neutrino.aquaphoenix.com/ReactionDiffusion/SERC4chap3.pdf

https://www.researchgate.net/publication/312257467_Fractional_double_Laplace_transform_and_its_properties

https://www.researchgate.net/publication/322148582_Fractional_Laplace_transform_and_fractional_calculus

https://www.sciencedirect.com/science/article/pii/S1007570418302934

http://ijrast.com/index.php/IJRAST/article/download/45/41/

https://link.springer.com/chapter/10.1007/978-0-387-21746-8_5

http://www.m-hikari.com/imf/imf-2017/17-20-2017/p/romeroIMF17-20-2017.pdf

With best wishes.

Dear A. M. Abdallah

Thank you so much for sharing such great materials. Regards,

Mohammad

Dear Mohammad M. Amirian ,

The following is the Laplace transform of the stable pd of order \alpha on R_+

G(s) = exp( - s^(\alpha) ), Re(s) \ge 0.

Assume that Y and X_{\alpha} are independent rs-s, where X_{\alpha} is stable of order \alpha, and Y is exponential. Then we obtain for the Laplace transform of Y^{1\alpha}*X_{\alpha} with the use of the expectation functional E{ } the following:

E{ exp( - s * Y^{1\alpha} * X_{\alpha} ) } = E{ exp ( - (s * Y^{1/\alpha} )^{\alpha} } =

=E{ exp ( - s^{\alpha} * Y ) = 1/( 1+ s^{\alpha} )

Perhaps this helps :)

Best, Joachim Domsta

Without the explicit formula for the Laplace transform, one can answer some related questions. In particular to determine the limits of the L.T. at the origin and behaviour at infinity. They depends on zeros of the denominator, especially on the parameters $k$ and $alpha$.

Let me add to Victor's important suggestion, that some standard ways to fulfill his useful advice is to use Tauberian type theorems based on Karamata's regular variability, as it is given e.g. in William Feller's book "An Introduction to Probability Theory and Its Applications". Some exemplary usage of the rv-property a little bit aside the problem can be found in Janos Galambos book "The Asymptotic Theory of Extreme Order Statistics". Further references can be found through the key words implied by this note.

Seemingly, for this reduced problem (to asymptotics) the problem with zeros of the denominator is substantial only in case when at least one of them is positive, under assumption that we are talking about onesided LT. Then the choice of appropriate branches of analyticity might be leading even to non-existence of the preimage of the LT at the given function.