Can You Obtain New Physics from the use of Fractional Derivatives?

I was curious if anyone could give me an example of the use of fractional derivatives in physics and explain what they offer that "conventional" mathematics does not (in terms of new physics and not just another method to work the problem)?

If you look at the conservation of mass example on the wikipedia page they cite the paper http://www.stt.msu.edu/~mcubed/fcom.pdf they derive a general equation for conservation of mass in equation 13. It seems like the fractional derivative is then used as a method to approach their problem, but I dont think it includes new physics (correct me if I'm wrong on this). Is there anything gained by stating that fractional conservation of mass holds in this case (apart from gaining another way to approach the problem)?

Any thoughts on this example or others would be appreciated!

• Related: physics.stackexchange.com/q/4005/2451 and links therein. More on fractional derivatives: physics.stackexchange.com/search?q=%22fractional+derivatives%22 . If you like this question you may also enjoy reading this and this Phys.SE posts. Dec 11 '13 at 15:40
• I vaguely remember one of these in a PDE class. For some examples, see Fractional diffusion equations and processes with randomly varying time. Dec 11 '13 at 15:42
• Dang -- I've got a whole book on the subject but it's packed away at home. Anyway, the wiki page, en.wikipedia.org/wiki/Fractional_calculus , lists several examples of practical usage. Dec 11 '13 at 15:54
• If anyone has an argument why no new physics has been introduce that would be appreciated as well. Dec 11 '13 at 16:59
• In general, the use of any new mathematical techniques shouldn't lead to new phenomena being discovered. All 'physics' in a system is a result of the inherent laws of physics that govern the system. The methods you use might help uncover the new physics, but other methods, like, say simulations, would have uncovered it in the first place. Dec 11 '13 at 17:39