What is the actual meaning of a fractional derivative? We're all use to seeing differential operators of the form $\frac{d}{dx}^n$ where $n\in\mathbb{Z}$. But it has come to my attention that this generalises to all complex numbers, forming a field called fractional calculus which apparently even has applications in physics!
These derivatives are defined as fractional iterates. For example, $(\frac{d}{dx}^\frac{1}{2})^2 = \frac{d}{dx}$ or $(\frac{d}{dx}^i)^i = \frac{d}{dx}^{-1}$
But I can't seem to find a more meaningful definition or description. The derivative means something to me; these just have very abstract definitions. Any help?
 A: If one solves diffusion problems, magnetic or thermal, by the use of the LaPlace transform there results s raised to fractional powers. Usually s denotes the first derivative with respect to time and I interpret s raised to a fractional power as a fractional derivative with respect to time. This occurs in all skin effect calculations and is not trouble if you have a program that inverts the LaPlace transform. I think the formation of ice on water is a direct physical example of the ice thickness being proportional to the 1/2 derivative of time. 
A: Probabilistically, you can give a perfectly clear meaning to many fractional derivatives.
I will look at definitions of fractional\nonlocal derivatives that are Markovian generators of stochastic processes with jumps. I hope to convince the reader that

*

*Different definitions arise naturally,

*there is a clear interpretation of many properties (like nonlocality or killing/not-killing constants), and

*generalizations are natural and meaningful for applications.

It is useful to look at the most simple stochastic  jump process and its corresponding generator.  Take a Markov chain $P=\{p_{i,j}\}_{i,j\in \text{State space}}$ (which is intrinsically jumpy)  and write out its generator
$$
\mathcal G f(x):=(P-I)f(x)=\sum_{y\in\text{ State space}}(f(y)-f(x))p_{x,y},\quad x\in\text{ State space}.
$$
Here the intuition is clear: the infinitesimal jump (working with unit time in this case) from $x$ to $y$ is assigned intensity/probability $p_{x,y}$. The operator $\mathcal G$ is non-local. If we modify the process (impose boundary conditions), say by forcing the process to be absorbed at $a\in\text{ State space}$ once it tries to jump to a state $y\notin \Omega\subset \text{State space},$ we obtain a new generator
$$
\mathcal G^{\text{abs}} f(x):=(P^{\text{abs}}-I)f(x)=\sum_{y\in\Omega}(f(y)-f(x))p_{x,y}+(f(a)-f(x))\sum_{y\notin\Omega}p_{x,y},\quad x\in\Omega.
$$
If we instead decide to kill it (by testing against functions with $f(a)=0$, for example), the new generator will be
$$
\mathcal G^{\text{kill}} f(x):=(P^{\text{kill}}-I)f(x)=\sum_{y\in\Omega}(f(y)-f(x))p_{x,y}-f(x)\sum_{y\notin\Omega}p_{x,y},\quad x\in\Omega.
$$
So from one single process we can obtain many different generators/fractional derivative (as mentioned in a comment above, the boundary conditions are reflected in the representation of the operator away from the boundary due to the non-locality of $\mathcal G$).
Let us now move to the Riemann-Liouville and Caputo derivatives of order $\beta\in(0,1)$. Consider the  three fractional derivatives for $x<a$
\begin{align}
D^{\beta}_{\infty}f(x)&:= \int_0^{\infty}(f(x+y)-f(x))\nu(y)dy, \\
^{C}D^{\beta}_a f(x):&= \int_0^{a-x}(f(x+y)-f(x))\nu(y)dy &+(f(a)-f(x))\int_{a-x}^\infty\nu(y)dy,\\
^{RL}D^{\beta}_af(x)&:= \int_0^{a-x}(f(x+y)-f(x))\nu(y)dy &-f(x)\int_{a-x}^\infty\nu(y)dy,
\end{align}
where $\nu(y):=\frac{-\Gamma(-\beta)^{-1}}{y^{1+\beta}}$. Similarly as for the  Markov chain above: the operator $D^{\beta}_{\infty}$ is the generator of a $\beta$-stable subordinator $X^\beta(s)$, the operator $^{C}D^{\beta}_a$ is the generator of a $\beta$-stable subordinator $X^\beta(s)$ absorbed at $\{a\}$ on the first attempt to jump outside $\Omega:=(-\infty,a)$, and the operator $^{RL}D^{\beta}_a$ is the generator of a $\beta$-stable subordinator $X^\beta(s)$ killed on the first attempt to jump outside $\Omega:=(-\infty,a)$.
Integrating by parts we can rewrite the three operators above in their Riemann-Liouville integral representation, namely
\begin{align}
D^{\beta}_{\infty}f(x)&= \int_x^{\infty}f'(y)\frac{(y-x)^{-\beta}}{\Gamma(1-\beta)}dy \\
^{C}D^{\beta}_a f(x)&= \int_x^{a}f'(y)\frac{(y-x)^{-\beta}}{\Gamma(1-\beta)}dy,\\
^{RL}D^{\beta}_af(x)&= \frac{d}{dx}\int_x^{a}f(y)\frac{(y-x)^{-\beta}}{\Gamma(1-\beta)}dy,
\end{align}
where the last two operators are your standard definitions of Caputo and Riemann-Liouvile derivatives (right and left versions will correspond to the processes $X^\beta(s)$ and $-X^{\beta}(s)$ respectively).  We can now say that the Caputo derivative $^{C}D^{\beta}_a$ (Riemann-Liouville derivative  $^{RL}D^{\beta}_a$) kills (does not kill) constants  as it is the generator of a process (killed process). Again you can see that (naturally) $^{C}D^{\beta}_a$ and $^{RL}D^{\beta}_a$ contain boundary information in their representation away from the boundary (in sharp difference with local differential operators).
Some references: Caputo, Riemann-Liouville, and Grünwald-Leitnikov derivatives from a stochastic point of view in this book. Reflecting boundary conditions and other options for Caputo derivatives of order $\beta\in(1,2)$ here and here.
By substituting a general Lévy measure $\nu(x,dy)$ in the formulas above (generalizing fractional derivatives), many meaningful stochastic processes and their versions on a bounded domain can be studied through their generators (see book, article ). Similar arguments can be carried over for some fractional Laplacians (see this book for example).
A: I understand where Ryan's coming from, though I think the question of how to interpret fractional calculus is still a reasonable one. I found this paper to be pretty neat, though I have no idea if there are any better interpretations out there. 
http://people.tuke.sk/igor.podlubny/pspdf/pifcaa_r.pdf
A: Fractional derivative arise is diffusion problems as the previous poser noticed. 
The Abel equation of the tauthochrone is another classical example.
The physical interpretation is still debatable but it is often attributed to memory effects or underlying fractal behaviors giving rise to power laws.
Classical references are Oldham and Spanier 1974 (https://www.amazon.com/Fractional-Calculus-Mathematics-Science-Engineering/dp/0125255500/ref=sr_1_1?s=books&ie=UTF8&qid=1469461451&sr=1-1&keywords=Oldham+and+Spanier+1974), Podlubny (https://www.amazon.com/Fractional-Differential-Equations-198-Introduction/dp/0125588402/ref=sr_1_2?s=books&ie=UTF8&qid=1469461515&sr=1-2&keywords=Podlubny), Kilbas and Marichev 
(https://www.amazon.com/Fractional-Integrals-Derivatives-Theory-Applications/dp/2881248640) etc. 
A: Personally, if I was entering this subject blind I would feel cheated if not shown the extensive pure mathematical power of the fractional derivative. Being that it is more useful than just being used to solve differential equations or physical problems.
The first thing is to look at Cauchy's integral formula which is most aptly
$$\int_a^x \int_a^{x_{n-1}}...\int_a^{x_1}f(x_0)\,dx_0dx_1dx_2...dx_{n-1} = \frac{1}{n-1!}\int_a^x f(y)y^{n-1}\,dy$$
which is a strikingly powerful equation. The natural generalisation arises by considering the operator $I_a f = \int_a^x f(y)\,dy$ and simply writing
$$I_a^n =\underbrace{ I_a ... I_a}_{n \text{ times}} f = \frac{1}{(n-1) !}\int_a^x f(y)y^{n-1}\,dy$$
where a natural conclusion is to define
$$I_{a}^z f = \frac{1}{\Gamma(z)}\int_a^x f(y)y^{z-1}\,dy$$
which through no obvious or simple method
$$I_a^{z_0}I_a^{z_1} = I_a^{z_0 + z_1}$$
This gives not only one iterated "fractional" integral but infinitely many for each $a$. The perspective result, or canonical fact, is that each fractional integral satisfies
$$I_a^z (x-a)^r = \frac{\Gamma(r+1)}{\Gamma(r+z+1)}(x-a)^{r+z}$$
and $I_a (x-b)^r$ when $b \neq a$ is defined using a binomial expansion.
Defining $\frac{d}{dx}_a^z = I_a^{-z}$ for $\Re(z) < 0$ and $\frac{d}{dx}_a^z = \frac{d}{dx}^n I_a^{n-z}$ for $\Re(z) < n$ we arrive at a fractional derivative.
This seemingly convenient and beautiful expression gives us something rather ugly though. Since $\frac{d}{dx} e^x = e^x$ we would like $\frac{d}{dx}^z e^x = e^x$, but this is not so. By uniform convergence and all that jazz
$$\frac{d}{dx}_a^z e^x = \sum_{n=0}^\infty \frac{x^{n-z}}{\Gamma(n+1-z)}$$
which is not $e^x$.
Therefore another fractional derivative is required. Taking $a = -\infty$ then we arrive at the commonly called "exponential differintegral" which can be written
$$\frac{d}{dx}^{-z} f(x) = \frac{1}{\Gamma(z)}\int_0^\infty f(x-y)y^{z-1}\,dx$$ defined for $f$ satisfying specific decay conditions at negative infinity. As one can see this fractional derivative fixes $e^x$ but diverges for any polynomial.
Now we can generalize this even further!
Consider $f(w)$ entire on $\mathbb{C}$, and for convenience assume $f(w)w \to 0$ as $w \to \infty$ when $|\arg(w)| < \kappa$ and call this space of function $D_\kappa$
Then we have the disastrously large formula
$$\frac{d^z}{dw^z} f(w) = \frac{e^{i\theta z}}{\Gamma(-z)}\Big{(}\sum_{n=0}^\infty f^{(n)}(w)\frac{(-e^{i\theta})^n}{n!(n-z)} + \int_1^\infty f(w-e^{i\theta}y)y^{-z-1}\,dy\Big{)}$$
which holds for all $|\theta| < \kappa$ and $\Re(z) > -1$.
Now some people would rashly think what is the point of this? Some interesting things happen in this scenario, firstly the differintegral can be thought of as a modified Mellin transform. Giving us things like Ramanujan's master theorem in a slicker notation. It further emphasizes that this operator arises in a very natural sense (the Mellin transform being prominent in many areas of mathematics). It says $\frac{d^z}{dw^z}$ for $\Re(z) > 0$ takes $D_\kappa$ to itself. So we have a semigroup $\{\frac{d^z}{dw^z} | \Re(z) > 0\}$ acting on $D_\kappa$.
Furthermore, when looking at the fourier transform definition of a fractional derivative, it is in fact this clunky looking exponential derivative that's really pulling the strings. Where it may seem cleaner in Fourier transforms, it is much more general in its Mellin form.
All in all it is quite a mysterious object, and is underused in my opinion.
