It is easier to understand if you do a Fourier transform first . Then a fractional derivative becomes just a fractional power of the wavenumber (in fact fractional derivatives are defined this way). More generally, we can take a not necessarily linear or polynomial function of the wave number. Non linear functions of the wavenumber occur naturally as dispersion relations. For example in solid state physics the momentum of a phonon is almost always a non linear function of its wavenumber.

To my knowledge, fractional derivatives appear when we consider anomalous diffusion (http://en.wikipedia.org/wiki/Anomalous_diffusion),

The most commonly referenced physical/biological system with anomalous diffusion is transport of proteins inside the cells.

The fractional calculus is even recently used in quantum physics.

I think that the fractional calculus rising the dimension of the physical system.

Dear Jan: That's exactly where I heard this concept. What I am looking forward to is a physical meaning to it.

Dear Petar: This read is very interesting. Thanks.

Dear Hamdi: What kind of dimension?

Read Igor Podlubny notes for physical meaning of fractional derivative.

If you have access to content at physicstoday.com, there was an excellent article published in the November 2002 issue ("Fractional Kinetics" by Sokolov, Klafter and Blumen) on the topic of fractional derivatives and some physical applications: http://www.physicstoday.org/resource/1/phtoad/v55/i11/p48_s1

It is easier to understand if you do a Fourier transform first . Then a fractional derivative becomes just a fractional power of the wavenumber (in fact fractional derivatives are defined this way). More generally, we can take a not necessarily linear or polynomial function of the wave number. Non linear functions of the wavenumber occur naturally as dispersion relations. For example in solid state physics the momentum of a phonon is almost always a non linear function of its wavenumber.