I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator.
I started with the book The Fractional Calculus by Oldham and Spanier, and it comes as no surprise that I favor the Grünwald-Leitnikov derivative. It seems to me a great definition, because it directly generalizes the basic definition of the derivative $\frac {df} {dx}=\lim_{h \rightarrow 0} \frac {f(x)-f(x-h)} {h}$. And it also produces the integral when $\alpha$ is set to be a negative number.
Another (which I think is the Liouville definition, but I'm not sure) generalizes the property of differentiating an exponential $\frac {d^k} {dx^k} e^{rx} = r^ke^{rx}$ and thus if $\frac {d^\alpha} {dx^\alpha}f(x)=\sum A_ne^{nx}$ then $f(x)=\sum A_n n^\alpha e^{nx}$.
A definition, which is used really often for some reason, is the Caputo derivative. Lot of people find it natural that $\frac {d^{\frac 1 2}} {dx^{\frac 1 2}} [1]=0$, but I think it is "evident" that it should be proportional to $x^{-\frac 1 2}$.
Now comes the actual question. Why are there so many definitions of the fractional derivative? Are some of them "better" than the others in some sense? And lastly, is there a general framework, wherein "functions" of differential operators, maybe more general than (fractional) powers, can be given an explicit meaning?
Best Answer
The reason is that the fractional derivative is not a local operator. The usual derivative is a local derivative in the sense that the value of the derivative at one point only depends on the value of the function in a neighborhood of that point. This is not the case for the fractional derivative and that cannot be due to some general theoretical result due to Peetre.
So the definition depends on the domain of definition of the functions under scrutiny. This is not the same definition if we are looking at functions defined on ${\bf R}$ or on $[0,1]$ or on $[0,\infty)$ and of course the derivative of say $\sin$ is not the same in these three cases. Same for the derivative of the constant function.
Fractional derivatives are a particular example of operators obtained using the functional calculus on some operator space. The result of such operation of course depends on the functional space under consideration, which itself is dictated by the context and the problems at hand.
tl;dr: there is not a best definition and the fractional derivatives do not share the nice local properties of the usual derivative, so beware.