5 ms·

Fractional calculus

45 points by ertug 16y ago

arethuza 16y ago
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 http://people.tuke.sk/igor.podlubny/pspdf/pifcaa_r.pdf Which might help.
Avshalom 16y ago
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.
DougBTX 16y ago
Nice: Also notice that setting negative values for a yields integrals.
drbaskin 16y ago
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.
- deleted 16y ago
  [deleted]
- mnemonicsloth 16y ago
  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."
moolave 16y ago
Maybe this can explain the fractals and reverse-engineer them.
est 16y ago
you can drag drop play a fractional calculus in Mathematica http://mathworld.wolfram.com/FractionalDerivative.html http://mathworld.wolfram.com/FractionalDerivative.html http://mathworld.wolfram.com/FractionalIntegral.html http://mathworld.wolfram.com/FractionalIntegral.html

45 points by ertug 16y ago

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

Avshalom 16y ago

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.

DougBTX 16y ago

Nice: Also notice that setting negative values for a yields integrals.

drbaskin 16y ago

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.

deleted 16y ago
[deleted]
mnemonicsloth 16y ago
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."

moolave 16y ago

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

est 16y ago

you can drag drop play a fractional calculus in Mathematica http://mathworld.wolfram.com/FractionalDerivative.html http://mathworld.wolfram.com/FractionalDerivative.html http://mathworld.wolfram.com/FractionalIntegral.html http://mathworld.wolfram.com/FractionalIntegral.html