Fractional Derivative Implications/Meaning?

Question

I've recently been studying the concept of taking fractional derivatives and antiderivatives, and this question has come to mind: If a first derivative, in Cartesian coordinates, is representative of the function's slope, and the second derivative is representative of its concavity, is there any qualitative relationship between a 1/2 derivative and its original function? Or a 3/2 derivative with its respective function?

Answer 1

(Non integer) fractional derivatives are non-local so the 1/2 derivative can not have a local meaning like tangent or curvature but would have to take into account the properties of the curve over a large extent (boundary conditions). See http://en.wikipedia.org/wiki/Fractional_calculus#Nature_of_the_fractional_derivative

In addition some of the applications of fractional derivations (http://en.wikipedia.org/wiki/Fractional_calculus#Applications) show the physical meaning is non-local.

Why is the fractional derivative non-local? Integrating a function is not unique or local, because it depends the values of the function over the entire range of integration. The generalization to fractional derivatives unifies differential and integral operators into one Differintegral operator. Whole derivatives are both unique and local. The apparent paradox of fractional derivatives being non-local is actually the natural case, just as integration is non-local.

There is a neat demonstration of calculating the fraction derivative from first principles as a limit in this article http://mathpages.com/home/kmath616/kmath616.htm and it shows how in the whole derivative case the extra terms zero out, but in the fractional case they do not.

Answer 2

There several approaches to fractional derivatives. I use the Grunwald-Letnikov derivative and its generalizations to complex plane and the two-sided derivatives. However, most papers use what I call "walking dead" derivatives: the Riemann-Liouville and Caputo. If you want to start, don't loose time with them. There are some attempts to give interpretations to the FD: Prof. Tenreiro Machado and also Prof. Podlubny. The best interpretation in my opinion is the system interpretation: there is a system (linear) called differintegrator with transfer function H(s)=s^a, for Re(s)>0 (forward, causal case) or Re(s) < 0 (backward, anti-causal case). The impulse response of the causal system is t^(a-1)/gamma(-a).u(t) where u(t) is the Heaviside function. Send me a mail and i'll send you some papers [email protected]