Exploring the Applications and Interpretations of Fractional Derivatives

[starzero]

I recently read a paper on fractional derivatives. That is how to take derivatives of fractional order rather than the usual integral order. The paper made perfect sense to me, however I wondered:

1) Are there geometric interpretations of fractional derivatives? Kind of like how first derivatives give us slopes of tangent lines and second derivatives tell us about a functions concavity.

2) Are there physicsal applications?

Thanks to anyone who has answers or can point me in the right direction.

 

[pwsnafu]

The only geometrical interpretation I have seen is Igor Podlubny's Geometric and physical interpretation of fractional integration and fractional differentiation which is on Arxiv. There is also Parvate & Gangal Calculus on fractal subsets of real line, again on Arxiv. The latter is very hard to read though.

As for applications? Oh God. Where to begin. Podlubny's book Fractional Differential Equations has a whole chapter on it. We not only have a dedicated journal Fractional Calculus and Applied Analysis but also the application only Fractional Differential Equations. Take a look at the References section of the Wikipedia page on the subject. Or Podlubny's http://people.tuke.sk/igor.podlubny/fc_resources.html" .

The million dollar question is what is the physical interpretation. Too many papers fall into "here is an experiment and we can model it with fractional derivatives" and give no reason as to why. Papers on this topic range from capital-b Bad to down right mathematical BS. The best paper I have seen is Neel, Abdennadher and Solofoniaina A continuous variant for Grunwald-Letnikov fractional derivatives. I highly recommend this as it derives everything from first principles.

NB: everything I wrote above is about real fractional calculus. I haven't seen any applications of complex orders, nor of fractional derivatives of complex functions.

 

[JJacquelin]

There are a lot of physical applications. Some are listed in :
K.B.Oldham, J.Spanier, "The fractionnal Calculus", Academic Press, N.-Y., 1974.
An interesting application in electrotechnology, in the field of impedances calculus, is a generalization of the resistance, inductance, capacitance to a more general notion : see section 7, p.4 in the paper "La dérivation fractionnaire"
http://www.scribd.com/JJacquelin/documents
[ written in French, but the table in p.4 showing the components generalization can be as well understood in English ]

 

[starzero]

Thank you both for the information.