I find it hard to believe that the article on fractional differentiation only briefly mentions the Fourier transform. For functions on R^n with vanishing integral (or periodic functions on R^n with vanishing integral), the Fourier transform allows you to define arbitrary powers of the (positive) Laplacian by taking the Fourier transform, multiplying by |\xi|^\alpha, and then taking the inverse Fourier transform. If n=1, this process yields the fractional derivatives in the linked article.

Something else that's great is that this works on (compact or asymptotically Euclidean) manifolds, too! You can make sense of the Laplacian on these spaces, and then spectral theory lets you define its fractional powers. The theory of pseudodifferential operators lets you realize these powers fairly explicitly as oscillatory integrals.

Fascinating stuff - I found this paper that tries to give geometrical/physical interpretations for fractional differentiation and integration:

http://people.tuke.sk/igor.podlubny/pspdf/pifcaa_r.pdf

Which might help.

Nice:

Also notice that setting negative values for a yields integrals.

If I remember the lecture I went to half a dozen years ago fractional calculus allows for some elegant solutions to cycloid curves and half infinite sheets of charge.

you can drag drop play a fractional calculus in Mathematica

http://mathworld.wolfram.com/FractionalDerivative.html

http://mathworld.wolfram.com/FractionalIntegral.html

Maybe this can explain the fractals and reverse-engineer them.