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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.

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up vote 5 down vote accepted

Fractional derivatives can be used to establish connections between various special functions. The book An Atlas of Functions makes heavy use of this, especially derivatives of order 1/2 and -1/2.

Also, the existence of fractional derivatives is related to the convergence of Fourier transforms. For example, if a function has a 1/2 a derivative that means you can multiply its Fourier transform by $x^{1/2}$ and it is still in $L^2$. But I haven't seen much use in actually computing fractional derivatives, only knowing that they exist.

On a somewhat related note, see my answer to a question on Math Overflow related to Sobolev spaces.

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I wouldn't say there are results that can only be obtained through differintegration. It only happens that there are problems whose solutions look neater when we bring in the machinery of differintegrals.

Spanier and Oldham and Miller and Ross remain useful references on the applications of differintegration. The first reference has a chapter on how certain diffusion problems have a neater formulation when differintegrals are used. For the second reference, the application that jumped out at me was Abel's solution to the so-called tautochrone problem: finding the curve such that the time needed for a particle to descend from a given position to the bottom of the curve (assuming there is no friction) is independent of position.

Though Huygens and other mathematicians have already obtained this solution long before Abel, he decided to use an integral equation formulation that can then be solved with the help of differintegration. In particular, he arrived at the equation


which when reformulated as a differintegral is


I won't spoil the rest of the solution; I'd suggest that you read Miller and Ross if you're interested.

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Miller and Ross looks very nice indeed.

As far as applications are concerned: Applications of Fractional Differential Equations. For some strange reason, the original link is not available. Instead, you can look at it by using Google Docs viewer.

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Additional applications have arisen recently in fractional diffusion processes, mathematial biology (random eye movements follow a fractional process), solar physics, and many other places. In these cases the fractional derivatives are used (as noted above) primarily in writing models as fractional differential equations. A more detailed discussion of the fractional diffusion equation and its applications can be found here.

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A short paper for general public was published on Scribd : "The fractionnal derivation"

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Concerning "which results can be obtained", there are many ways of doing this, most of which involving complicated formulas or some other means.

The easiest method in my opinion is the simplistic induction. Take $D^n_x:=\frac{d^n}{dx^n}$ and it will be fairly obvious that $D^n_xe^{ax}=a^ne^{ax}$. One most easily proves this through induction, using the fact that $D^n_xD^k_x=D^{n+k}_x$ for $n\in\mathbb Q$, then $\mathbb R$ by assuming continuity.

Other formulas can be made, like generalizing the $n$th derivative through the limit definition of a derivative.

Some commonly known fractional derivatives include $D^n_xe^{ax}=a^ne^{ax}$, $D^n_x\sin(x)=\sin(x+\frac{n\pi}2)$, and $D^n_xx^\mu=\frac{\Gamma(\mu+1)}{\Gamma(\mu-n+1)}x^{\mu-n}$ if $-\mu\notin\mathbb N$.

Less known, for example, would be if $-\mu\in\mathbb N$, in which case, $D^n_x\frac1x=\frac{\ln(x)-\gamma-\psi^{(0)}(-n)}{x^{n+1}\Gamma(-n)}$. Obviously, the less commonly known fractional derivatives are much more complicated.

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