Question

We're all use to seeing differential operators of the form $\frac{d}{dx}^n$ where $n\in\mathbb{Z}$. But it has come to my attention that this generalises to all complex numbers, forming a field called fractional calculus which apparently even has applications in physics!

These derivatives are defined as fractional iterates. For example, $(\frac{d}{dx}^\frac{1}{2})^2 = \frac{d}{dx}$ or $(\frac{d}{dx}^i)^i = \frac{d}{dx}^{-1}$

But I can't seem to find a more meaningful definition or description. The derivative means something to me; these just have very abstract definitions. Any help?

Answer 1

I understand where Ryan's coming from, though I think the question of how to interpret fractional calculus is still a reasonable one. I found this paper to be pretty neat, though I have no idea if there are any better interpretations out there.

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

Answer 2

If the original poster is satisfied, that everything should be ok. However, I find this approach of giving a 'physical interpretation' of a purely mathematical idea slightly misleading. You can give a physical meaning to complex numbers, sure, but their mathematical meaning is far more interesting and compelling; I would rather speak of an application to physics.

As to fractional derivatives, they become quite easy to understand if you think that the Fourier transform takes the derivative of a function into multiplication by the variable: $\widehat f'=i\xi\cdot \hat f$. So higher order derivatives can be defined as multiplication of $\hat f$ by powers of $\xi$, and it is no wonder that you can use this idea to define fractional derivatives, or actually generic 'functions of $d/dx$'. This leads to pseudodifferential operators etc.etc.

The main reason why this idea is not just a game but on the contrary is enormously useful, also in physics, is that using this kind of calculus you can give explicit (well, almost) expressions to fundamental things such as solutions to differential equations, and manipulate or estimate them in a very effective way.

Answer 3

"The definition of fractional derivative is very important. You know, consideringt the case of advanced calculus, we should define the measure firstly, and then we can define the fractional derivative. While the measure is related to fractal sets, it is not so easy to define a new reasonable and applicable one since fractal objects in our real world, On the other hand, are not really fractals. "

I mean, we should define or use a fractional derivatie from measure of the sets logically, which is more reasonable. And it's useful in real applications.