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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!

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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. –  Qmechanic Dec 11 '13 at 15:40
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I vaguely remember one of these in a PDE class. For some examples, see Fractional diffusion equations and processes with randomly varying time. –  Emilio Pisanty Dec 11 '13 at 15:42
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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. –  Carl Witthoft Dec 11 '13 at 15:54
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If anyone has an argument why no new physics has been introduce that would be appreciated as well. –  Anode Dec 11 '13 at 16:59
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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. –  Abhinav Dec 11 '13 at 17:39
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1 Answer

up vote 3 down vote accepted

All our sophisticated mathematical tools - Derivatives and Integrals, Fourier Transforms, Groups and Representations, Riemann Tensors, Kähler manifolds, etc. are merely descriptive techniques. What exists is what exists, independent of how we try to describe it.

New mathematical ideas often help us see known phenomena more clearly, or deal with the mathematical analysis more easily. Imagine studying the gravitational field of a planet or star in rectangular coordinates instead of spherical. Ugh! (A good exercise for undergrad students...hee hee!)

A well-aligned mathematical view will let us see phenomena otherwise hidden in messy data or a blizzard of algebraic terms. I think of the moons of Saturn and waves induced in the rings - a very complex system requiring wise choices of coordinate systems. In high energy physics, prediction of the Omega hadron as the final piece of a puzzle was allowed by use of group representation theory.

So, application of innovative mathematics to physics can certainly facilitate discovery of physical phenomena.

Specifically for fractional derivatives, though, it's unlikely to help. Any interesting peaks, wiggles, jumps in the semi-derivative of a function or in numerical analysis of raw data, is unlikely to show anything not already clearly visible in the data or its plain first-order derivative.

If you can think of some type of feature in data or an expression that's easily visible the way it is, but hard to notice in the semi-derivative or semi-integral, you'd be onto something. Good luck finding such a thing.

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