We know the physical meaning of the first and second derivative of the displacement with respect to time. But we do not have physical meanings for higher derivatives, like 10, 20, 27 etc. and the fractional derivative. By the way, we know how to calculate it.
Dear Pavel
To know about the History, definition, and calculation of fractional derivative, please have a look to the following links:
http://perrylocal.org/meinkea/files/2010/10/FRACTIONAL-DERIVATIVES.pdf
http://www3.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf
http://en.wikipedia.org/wiki/Fractional_calculus
Physical significance should be given high importance. This question is about physical significance of fractional calculus. Somehow, I too have the same kind of doubt about this field. Let us see what the others have to say!
Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems.
http://www.nonlinearbiomedphys.com/content/4/1/3
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being non-local is a useful property of fractional derivatives when it comes to modelling of complex phenomena ; see the attached document for a careful introduction in the domain of propagation through inhomogeneous media
on the other hand, such derivatives (at least the versions i know) must be taken with extra care ; for instance, the "leibniz rule" ( D(fg) = D(f) g + f D(g) ) does not hold ...
see http://theory.sinp.msu.ru/~tarasov/PDF/CNSNS2013.pdf for the proof
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I am Aware of some distinct application of fractional derivatives for physical Systems (modelling viscoelasticity of materials) and some first ideas for applying it to financial mathematics (Option pricing): From my Point of view, it is first of all an obvious generalisation of the mathematical concept of (classical) derivatives and, in the next step, maybe a useful tool (!) for modelling certain phenomena: As long as these approaches turn out to be useful to describe the reality and any Kind of Problems I do not see the Need for giving them a "physical meaning". - And if I consider classical number theory which was for a very long time at the core of "pure mathematics" and is nowadays an indispensible part of any cryptographic system (or non-Euclidean geometry and so on) there are a lot of examples which started at a rather theoretical level (or as pure mathematics) and turned out to be rather important and useful centuries later ...
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