The derivation of fractional equations

Recently I saw some physical problems that can be modeled by equations with fractional derivatives, and I had some doubts: is it possible to write an action that results in an equation with fractional derivatives? For example, consider a hypothetical physical system with the principle of least action. Is there a "wave equation" with the time-derivative $3/2$? Does such a question make sense?

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Out of curiosity, what sort of physical problems work this way? –  spencer nelson Jan 31 '11 at 20:56
@Spencer I saw the first time in anomalous diffusion. You can see here, for example: pfi.uem.br/mfi/disserta_teses/teses_pdf/… (in portuguese). This is a investigation of a system governed by a non-Markovian Fokker-Planck equation that are related with the comb model. –  gsAllan Feb 1 '11 at 1:26
Forces proportional to velocity, such as friction for example, can be described by putting fractional derivatives in the Lagrangian. –  SMeznaric Jul 5 '13 at 23:11
Here is the tome our group did a lot of work out of: Metzler, R. (2000). The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339(1), 1–77. doi:10.1016/S0370-1573(00)00070-3. It is a great introduction to the subject. There should be an arxiv preprint available but I don't have the number handy. –  Michael Brown Jul 6 '13 at 0:14

When I've seen fractional derivatives I've assumed that one place where they would naturally arise is in physical situations where there's a fractional dependency on time.

For example, random walks typically result in movement proportional to $\sqrt{t}$. Googling for "fractional+derivative+random+walk" on arxiv.org finds some papers that explore this:

Dear @space_cadet, no, what you write is not correct. QGR is right that whenever fractional derivatives appear, the action is nonlocal. It's enough for some terms to appear, whether they're leading or not, and the action has to be nonlocal. At the end of your answer, you admit that you don't really know why they're nonlocal. Well, it's because the you may write $\partial_t^k$ as $E^k$ in the Fourier transformed energy representation, and the transformation of the operator $E^k$ back to the $t$ basis is nonlocal. –  Luboš Motl Jan 27 '11 at 17:19
@space_cadet, a fraction derivative is a simple example of "pseudo-differential operator". It's an operator that can be easily defined for example with fourier transform (like Lubos Motl sayed). You can define operator that in Fourier Transform is equivalent to $\sin{k}$...Fraction derivative is simpler, in Fourier Transform is a multiplication for your transform variable $k$ with fractional exponent ;-) –  Boy Simone Jan 28 '11 at 12:36