I am sorry - but I am not able to help with this problem.

Geoff Joyce

Dear Jihad Asad,

The paper 'Leibniz rule for fractional derivatives generalized and an application to infinite series' seems to be useful for you. Follow the link below to check out

http://epubs.siam.org/doi/abs/10.1137/0118059

Best wishes,

Arlúcio

http://epubs.siam.org/doi/abs/10.1137/0118059

Dear Jihad,

I would say that the answer for your question is very complicated in the general case. If I understood correctly, the unknown x in the file is a function of t. Then you not only need the Leibniz formula, but also the chain rule for fractional derivatives. You can find these rules e.g. from Chapter 3 of the book "The Analysis of Fractional Differential Equations - An Application-Oriented Exposition Using Differential Operators of Caputo Type" by K. Diethelm.

Even in the simplest cases the answer cannot be expressed in terms of elementary functions. For example, if b=infinity and x(t)=t, you will end up to the Beta function provided 0

Dear Jihad,

 Follow the link below to check out

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwiMko6F6fTLAhUEXg8KHQGlA9gQFggpMAE&url=http%3A%2F%2Fwww.springer.com%2Fcda%2Fcontent%2Fdocument%2Fcda_downloaddocument%2F9783642145735-c1.pdf%3FSGWID%3D0-0-45-993489-p174025563&usg=AFQjCNF1q-ttS7FTbYCzt6O3uBg-FFroYw

Dear Jihad Asad,

I think it may be useful to your mentioned question. I would like to give you the following attachment. Please find it.

Thanks and regards,

Krushna