The fractional derivative of a function to order a is often defined implicitly by the fourier transform. The fractional derivative in a point x is a local property only when a is an integer.
Applications of the fractional calculus includes partial differential equations, especially parabolic ones where it is sometimes useful to split a timederivative into fractional time[?].
There are many well known fields of application where we can use the fractional calculus. Just a few of them are:

History (fill this in (it started about 300 years ago.))
Differintegrals The combined differentation/integral operator used in fractional calculus is called the differintegral, and it has a couple of different forms which are all equavalent. (provided that they are initialized(used) properly.)
By far, the most common form is the RiemannLouiville form:
Closely related topics anomalous diffusion[?]  fractional brownian motion[?]  fractals and fractional calculus[?] 
extraordinary differential equations[?]  partial fractional derivatives[?]  fractional reactiondiffusion equations[?]  fractional calculus in continuum mechanics[?]
http://mathworld.wolfram.com/FractionalCalculus[?]
http://www.diogenes.bg/fcaa/[?]
http://www.nasatech.com/Briefs/Oct02/LEW17139[?]
http://unr.edu/homepage/mcubed/FRG[?]
"An Introduction to the Fractional Calculus and Fractional Differential Equations"
"The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order (Mathematics in Science and Engineering, V)"
"Fractals and Fractional Calculus in Continuum Mechanics"
"Physics of Fractal Operators"
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