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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?

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Please read the FAQ. Regarding your question, this is standard undergraduate material, for example see: en.wikipedia.org/wiki/Fourier_transform and look up the equation for the Fourier transform of an iterated derivative. – Ryan Budney Apr 20 '10 at 3:57
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I understand that it must be frustrating to see a question that seems too low-level posted. Before posting this question, I tried to do due diligence by researching it and asking several math grad students and a (in industry) PHD (who hadn't heard of it before!). Perhaps you could expand on what qualifies as a `research level math question'? Additionally, thinking about a fractional derivative in the indirect manner you describe seems suboptimal, further defending the validity of asking for a more meaningful definition. (I hadn't heard of it this way before hand, but..) – Christopher Olah Apr 20 '10 at 4:54
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There is a lovely little book on this subject whose entire thesis is to answer the question you've just asked. It's called "An Introduction to the Fractional Calculus and Fractional Differential Equations" by Miller and Ross. I think it's fairly cheap on amazon – Dylan Wilson Aug 6 '10 at 7:33
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Back when I was studying these, I treated the leap from integer-order derivatives/integrals to arbitrary-order differintegrals (I really have no love for the term "fractional") in the same way that I had treated how the gamma functions extend the factorial, and how general exponents extend the normal integer powers even before that. This is more of finding out how far you can stretch the rules that used to apply only to integer values. As for books, I always read Miller/Ross, Spanier/Oldham, and Podlubny side-by-side. (We really still are far off from notation everybody can be happy with!) – J. M. Aug 6 '10 at 9:39
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Subsequently, there has been an illuminating answer to a related question, "Geometric interpretation of the half-derivative?". In particular, there is a beautiful "mechanical interpretation of the half-derivative." – Joseph O'Rourke Jan 16 at 1:28
up vote 17 down vote accepted

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

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Thank you. This looks good and I've started reading it. – Christopher Olah Apr 20 '10 at 5:26

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.

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The probably "physically convincing" part of differintegrals for me was when they were applied to simplify the PDEs that frequently crop up in diffusion problems. "Fractional Differential Equations" by Podlubny (the same guy who wrote the paper cited above) shows how it's done. – J. M. Aug 6 '10 at 9:41
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Misleading or not, there's a long history of 'physical interpretations' of purely mathematical ideas giving insight into pure mathematics. – Dan Piponi Jan 16 at 0:40

If one solves diffusion problems, magnetic or thermal, by the use of the LaPlace transform there results s raised to fractional powers. Usually s denotes the first derivative with respect to time and I interpret s raised to a fractional power as a fractional derivative with respect to time. This occurs in all skin effect calculations and is not trouble if you have a program that inverts the LaPlace transform. I think the formation of ice on water is a direct physical example of the ice thickness being proportional to the 1/2 derivative of time.

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I don't understand the sentence "of the ice thickness being proportional to the 1/2 derivative of time": the the 1/2-derivative of time with respect to what? – André Henriques Jan 16 at 0:06
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That claim about ice seemed interesting so I had to investigate. I found a paper igsoc.org:8080/journal/52/179/j05j055.pdf It solves the heat equation by taking the "square root" of the differential operators on each side. I've no idea what conditions are required to make this a valid operation. – Dan Piponi Jan 16 at 0:39

Personally, if I was entering this subject blind I would feel cheated if not shown the extensive pure mathematical power of the fractional derivative. Being that it is more useful than just being used to solve differential equations or physical problems.

The first thing is to look at Cauchy's integral formula which is most aptly

$$\int_a^x \int_a^{x_{n-1}}...\int_a^{x_1}f(x_0)\,dx_0dx_1dx_2...dx_{n-1} = \frac{1}{n-1!}\int_a^x f(y)y^{n-1}\,dy$$

which is a strikingly powerful equation. The natural generalisation arises by considering the operator $I_a f = \int_a^x f(y)\,dy$ and simply writing $$I_a^n = I_a ... (n\,times)...I_a f = \frac{1}{n-1!}\int_a^x f(y)y^{n-1}\,dy$$

where a natural conclusion is to define

$$I_{a}^z f = \frac{1}{\Gamma(z)}\int_a^x f(y)y^{z-1}\,dy$$

which through no obvious or simple method

$$I_a^{z_0}I_a^{z_1} = I_a^{z_0 + z_1}$$

This gives not only one iterated "fractional" integral but infinitely many for each $a$. The perspective result, or canonical fact, is that each fractional integral satisfies

$$I_a^z (x-a)^r = \frac{\Gamma(r+1)}{\Gamma(r+z+1)}(x-a)^{r+z}$$

and $I_a (x-b)^r$ when $b \neq a$ is defined using a binomial expansion.

Defining $\frac{d}{dx}_a^z = I_a^{-z}$ for $\Re(z) < 0$ and $\frac{d}{dx}_a^z = \frac{d}{dx}^n I_a^{n-z}$ for $\Re(z) < n$ we arrive at a fractional derivative.

This seemingly convenient and beautiful expression gives us something rather ugly though. Since $\frac{d}{dx} e^x = e^x$ we would like $\frac{d}{dx}^z e^x = e^x$, but this is not so. By uniform convergence and all that jazz

$$\frac{d}{dx}_a^z e^x = \sum_{n=0}^\infty \frac{x^{n-z}}{\Gamma(n+1-z)}$$

which is not $e^x$.

Therefore another fractional derivative is required. Taking $a = -\infty$ then we arrive at the commonly called "exponential differintegral" which can be written

$$\frac{d}{dx}^{-z} f(x) = \frac{1}{\Gamma(z)}\int_0^\infty f(x-y)y^{z-1}\,dx$$ defined for $f$ satisfying specific decay conditions at negative infinity. As one can see this fractional derivative fixes $e^x$ but diverges for any polynomial.

Now we can generalize this even further!

Consider $f(w)$ entire on $\mathbb{C}$, and for convenience assume $f(w)w \to 0$ as $w \to \infty$ when $|\arg(w)| < \kappa$ and call this space of function $D_\kappa$

Then we have the disastrously large formula

$$\frac{d^z}{dw^z} f(w) = \frac{e^{i\theta z}}{\Gamma(-z)}\Big{(}\sum_{n=0}^\infty f^{(n)}(w)\frac{(-e^{i\theta})^n}{n!(n-z)} + \int_1^\infty f(w-e^{i\theta}y)y^{-z-1}\,dy\Big{)}$$

which holds for all $|\theta| < \kappa$ and $\Re(z) > -1$.

Now some people would rashly think what is the point of this? Some interesting things happen in this scenario, firstly the differintegral can be thought of as a modified Mellin transform. Giving us things like Ramanujan's master theorem in a slicker notation. It further emphasizes that this operator arises in a very natural sense (the Mellin transform being prominent in many areas of mathematics). It says $\frac{d^z}{dw^z}$ for $\Re(z) > 0$ takes $D_\kappa$ to itself. So we have a semigroup $\{\frac{d^z}{dw^z} | \Re(z) > 0\}$ acting on $D_\kappa$.

Furthermore, when looking at the fourier transform definition of a fractional derivative, it is in fact this clunky looking exponential derivative that's really pulling the strings. Where it may seem cleaner in Fourier transforms, it is much more general in its Mellin form.

All in all it is quite a mysterious object, and is underused in my opinion.

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Fractional derivative arise is diffusion problems as the previous poser noticed. The Abel equation of the tauthochrone is another classical example. The physical interpretation is still debatable but it is often attributed to memory effects or underlying fractal behaviors giving rise to power laws. Classical references are Oldham and Spanier 1974 (https://www.amazon.com/Fractional-Calculus-Mathematics-Science-Engineering/dp/0125255500/ref=sr_1_1?s=books&ie=UTF8&qid=1469461451&sr=1-1&keywords=Oldham+and+Spanier+1974), Podlubny (https://www.amazon.com/Fractional-Differential-Equations-198-Introduction/dp/0125588402/ref=sr_1_2?s=books&ie=UTF8&qid=1469461515&sr=1-2&keywords=Podlubny), Kilbas and Marichev (https://www.amazon.com/Fractional-Integrals-Derivatives-Theory-Applications/dp/2881248640) etc.

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