

Fractional calculus - ertug
http://en.wikipedia.org/wiki/Fractional_calculus

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drbaskin
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.

~~~
mnemonicsloth
_I find it hard to believe that the article... only briefly mentions..._

Wikipedia's math content is really frustrating this way. The linked article
was pretty good, but many deserve a disclaimer:

"This document delivers utility bounded above by what you paid for it."

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arethuza
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.

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DougBTX
Nice:

 _Also notice that setting negative values for_ a _yields integrals._

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Avshalom
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.

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est
you can drag drop play a fractional calculus in Mathematica

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

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

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

