|
|
|
|
|
|
|
|
|
 Posted: Wed Jul 18, 2007 7:36 pm
Now, don't get me wrong, as the mathematician in me greatly appreciates the elegance of being able to take the concepts of differentiation and integration to any degree, but what are any practical uses for Fractional Calculus? Where might one encounter a real life fractional differential equation, and how often does one need to take a half derivative? Sorry for the shortness of my question, but I just haven't seen any need for it so far.
|
 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Posted: Thu Jul 19, 2007 12:47 am
I think it's used in thermodynamics, but I avoid that subject so I wouldn't know for sure.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Posted: Wed Aug 22, 2007 6:49 am
Offhand, i would point to Spectral Theorem and evolution equations (where it is used to replace the time derivative). The uses of fractional calculus are somewhat limited (though not just to the two i mentioned), that is in part due to the relative newness of it. I suspect as it becomes more accepted it will find more uses, as is generally the case.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Posted: Wed Sep 10, 2008 3:12 am
I can't really think of many physical applications. In fact, all I can think of is:OMG!!! That is so cool! I can do:
$$ dfrac{d^{q}}{dx^q} left( f(x) right) $$ !!!!!! Sweet!
Oh wait, is q have to be in Q or can we be in R. oooooooooo can we do it in C?
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Posted: Mon Sep 15, 2008 10:43 pm
The fractional integral is given by:  so it should be defined for all complex alpha, using certain continuity assumptions to calculate for alpha = 0, -1, -2...
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Posted: Sat Oct 04, 2008 11:43 pm
A Fourier transform is essentially its own inverse, up to a sign reversion of the argument argument and possibly a constant multiplier (this of course depends on the convention used for the transform). Say
F(ξ) = Φ(f) = Φ[f(x)](ξ) = Int[ f(x) exp(-2πiξx) dx ]; then
f(x) = Φ'(F) = Φ[F(ξ)](-x) = Int[ F(ξ) exp(2πiξx) dξ ],
where both integrals are over the real line and ' denotes inverse transform. Then we can define [(d/dx)^α]f(x) = Φ'[ (2πiξ)^α Φ(f) ]. Other conventions for the Fourier transform have formulae for the derivative. This is conceptually easier with a Laplace transform, since a Laplace transform can be viewed as a limited Fourier transform, although the inverse Laplace transform is a bit more hairy.
If there is a Markovian continuous random walk, the probability density of the particle satisfies a certain general partial differential equation. If the process in non-Markovian, the proper differential equation to describe it turns out to be fractional, at least under the assumptions of a Gaussian random walk with a non-linear mean-squared displacement satisfying a power-law, = Dt^α, α ≠ 1 (I'm not aware of a more general case, but I'm sure it's being studied somewhere). More physically, this can approximate certain kinds of diffusion phenomena, e.g., one with active transport in a cell. I've heard of some other applications, but I'm only vaguely aware of them.
There are also noncommutative PDEs. The noncommutative differential operators are elements of the polynomial ring of derivations of some algebra, and it's something I haven't wrapped my head around yet.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Reply
