Applications of Fractional Calculus I've seen recently for the first time in Special Functions (by G. Andrews, R. Askey and R. Roy) the definitions of fractional integral 
$$(I_{\alpha }f)(x)=\frac{1}{\Gamma (\alpha )}\int_{a}^{x}(x-t)^{\alpha -1}f(t)dt\qquad \text{Re}\alpha >0$$
and fractional derivative 
$$\frac{d^{\nu }w^{\mu }}{dw^{\nu }}=\frac{\Gamma (\mu +1)}{\Gamma (\mu -\nu +1)}w^{\mu -\nu },$$
in The Hypergeometric Functions Chapter.
I would like to know some applications for Fractional Calculus and/or which results can only be obtained by it, if any.

A short paper for general public was published on Scribd :
"The fractionnal derivation"
http://www.scribd.com/JJacquelin/documents
