Talk:Fractional calculus

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This article has been tagged since January 2007.

This is a large and multi-faceted topic. This will be the mother-page for a large section. Here's a rough outline:

Introduction
History
Semiotic base
1. Differintegrals
  1. Riemann-Liouville
  2. Grunwald-Lietnikov
  3. Weyl
  4. Interpretation
2. Relation to Standard Transformations
  1. Laplace transform
  2. Fourier transform
3. Properties and Techniques
  1. General Properties
  2. Differintegration of some special functions
Geometric structure of
1. Relation to Diffusion
  1. anomalous(non-fickian) diffusion
  2. fractional brownian motion
2. Relation to Fractals & Chaos Theory
Advanced topics
1. Multiple-order differintegration
  1. extraordinary differential equations
  2. partial fractional derivatives
2. Special Forms of Fractional Calculus
  1. Initialized fractional calculus
  2. Local fractional derivative(LFD)
3. Morphological(Synthesis of Structure and Change) aspects
  1. fractional reaction-diffusion equations
  2. fractional calculus in continuum mechanics
  3. fractal operators
Applications of Fractional Calculus
1. Mathematics
2. Physics
3. Engineering
Contemporary Trends in Fractional Calculus

And, ofcourse, I am open to suggestions. I will, however, be stubborn on there being a 'geometric structure of' section, in whatever form. I hope this helps get this moving.

-User:Kevin_baas 2003.05.06

---

I think some of that may be hard to swallow for an undergraduate math student. (Minor note: fractional calculus deals with complex numbered orders of differintegration as well.) Charles, thank you very much for your contributions to this page! I've been waiting for someone besides me to work in this area. :) Kevin Baas 19:33, 16 Apr 2004 (UTC)

OK - let me explain that I was working today on the basis of the half-page article in the big Soviet mathematical encyclopedia. So it's not going to look like a tutorial, at this point.

Charles Matthews 19:42, 16 Apr 2004 (UTC)

1 Alternative version
2 Hmmm…
3 complicated?
4 and now for the Amateur Hour (me)
- 4.1 Soapbox: Need for a history section // Why not help newbies more?

[edit] Alternative version

This page used to be quite different. The current and the older version both have their advantages. I invite contributors to look at the older version here, and combine the best of both worlds, while making the article more in line with the protocols agreed to on the WikiProject Mathematics pages. Kevin Baas | talk 19:56, 2004 Sep 24 (UTC)

I think it would also be helpful to point out that we now have pages on functional calculus and pseudo-differential operator, that contribute significantly to the context; and, less obviously, there is material on the Sobolev space page that also uses fractional differentiation, defined via Fourier transform.

Charles Matthews 20:56, 24 Sep 2004 (UTC)

Kevin, are you done with the page or is this work in progress? Gadykozma 23:25, 24 Sep 2004 (UTC)

This page, or the alternative? Every page is a work in progress. The version here is primarily Charles Matthews', the alternative version is primarily mine, before charles radically altered the page. Why do you ask? Kevin Baas | talk 02:19, 2004 Sep 25 (UTC)

I'm afraid I cannot add much mathematical intuition beyond what I anyway wrote under Sobolev space. Editorially, I only think that it's better to start with modest goals (i.e. the current article) and expand the article step by step. Here it is also important to keep in sync with Differintegral so that there won't be any unnecessary duplication of material. Gadykozma 02:49, 25 Sep 2004 (UTC)

[edit] Hmmm…

Interesting article. I certainly haven't explored the subject in depth—this article is all that I've read on it—, but I'm already beginning to wonder about the uniqueness of $D p / q$ (to deal only with the rational case for now; it would be easy to extend that to real- and complex-valued exponents). For a given function $f$ , might there be two distinct operators $D 1$ and $D 2$ such that $\big(D_1^{p/q}\big)^q = f' = \big(D_2^{p/q}\big)^q$ ?

Take the polynomial case:

$f(x) = \sum_{i=0}^n a_ix^i.$

If we make the desirable (I suppose) assumptions that $D p / q (f + g) = D p / q (f) + D p / q (g)$ and that $D p / q (k f) = k D p / q (f)$ (for constant $k$ ), then one way to define $D 1 / q (f)$ for a non-zero integer $q$ is

$D^{1/q}(f) = \sum_{i=0}^n {\Gamma(i+1)\over \Gamma(i+1-1/q)} a_ix^{i-1/q}.$

And that implies that

$D^{p/q}(f) = \sum_{i=0}^n {\Gamma(i+1)\over \Gamma(i+1-p/q)} a_ix^{i-p/q}$

for integers $p$ and $q$ (again, $q\neq0$ ). But is that definition unique? And is it easy to extend to general functions? How about function composition: do we get $D^{p/q}\big(f(g)\big) = D^{p/q}\big(f(g)\big) D^{p/q}(g)$ for functions $f$ and $g$ ? Do we even want to define $D p / q$ that way?

Just some random musings. Forgive me if this is just a lot of ignorant babbling. Shorne 05:37, 16 Oct 2004 (UTC)

(PS: Why do \bigl and the like not work? Shorne 05:37, 16 Oct 2004 (UTC))

Shorne hi. You forgot one important assumption, and that's translation invariance: you want that

(D p / q)(f (x + a)) = (D p / q f)(x + a)

. However, even with this assumption there is more than one solution. The easiest way to see this is in the Fourier domain. There, differentiation is just mutiplication by n. So it's root must be multiplication by $\sqrt{n}$ . However, any choice of signs would also give you a square root. In other words, for any choice of a series

ε n

of $\pm 1$ , you can construct a "root of differential" operator by taking Fourier transform, multiplying by $\epsilon_n \sqrt{n}$ , and taking inverse Fourier transform.

As for your PS question, the mechanics are expalined in meta:Help:Formula so check there. Gadykozma 14:16, 16 Oct 2004 (UTC)

[edit] complicated?

"Unfortunately the comparable process for the derivative operator D is significantly more complex..." really? The one in Mathworld isn't especially complicated in concept; integrate up the fraction (integration being so tidy) then differentiate down:

D μ = D m I m - μ

with integer $m\ge\mu>0$

Y'don't even have to use the least m. Kwantus 2005 July 2 01:18 (UTC)

With m restricted to the least value, Loverro p11 calls that the Lefthand form and the reverse $D μ = I m - μ D m$ the Righthand or Caputo form. The latter is apparently more practical, producing 0 for constant functions and working better in DEs. Kwantus 2005 July 2 18:17 (UTC)

[edit] and now for the Amateur Hour (me)

Question: I am a non-mathematician who chanced upon fractional diff/integ quite by accident many years ago, specifically by noting that since fractional diff/integ is trivial for simple sinusoidal plane waves -- a proportional -/+ phase shift does the trick -- then I could define a meaningful continuum for diff/integ just by taking the Fourier transform of a function and shifting its components proportionally. It's rather fun, as you can see the results approximating the regular fencepost values of -2, -1, 0, 1, 2 as they approach those points. Sort of slinky-thinky, yes, but such visualizations might help some readers. Comments, anyone? --Terry Dactyl 04:53, 15 August 2005 (UTC)

And an amateur question. Any practical applications of this, as there are for ordinary caluclus? An example or two, of interest to non-mathematicians, would help this article shine. Derex 19:15, 7 June 2006 (UTC)

[edit] Soapbox: Need for a history section // Why not help newbies more?

Some further discussion and questions about this article:

There is a rich history to this topic that goes back to the foundations of calculus with Leibniz and Newton, as briefly mentioned on this page. None of that seems to show up in this page. I'd would try to edit the article myself, but frankly, the show of mathematical notational expertise on this page is a bit too intimidating. Still, I find it a bit bizarre that for such a historically rich topic there isn't even a header for the history of this topic. Giving two external references, one in French without any link (!) and one link at the end, just doesn't seem to cut it.
Why oh why do so many pages like this in Wikipedia seem absolutely dedicated to scaring the bejeebers out of anyone other than a pure mathematician who might want to come by and actually learn something they can understand about the topic? This article is so notation-rich and method-rich that the only people who would seem to have much chance of following it are the people who already do understand it. What is the point of that? I keep thinking that I'm reading a discussion by expert wine connoisseurs of the various delicate parameters that add richness to the topic. That's fine -- important even -- but wouldn't a more complete article help get the general drift over to less sophisticated tasters? Shouldn't the idea be writing articles that try to bring in new folks and get them interested in learning more, so that someday they, too, can become aware of the full depth of it?
I would again in this case point to the example of how fractional calculus could be introduced with minimal notations by going through a discussion of how how sinusoidals behave when differentiated or integrated (they shift left or right by wavelength-proptotional units), and how that can be used in an intuitive fashion to create continuous definitions of differentiation and integration as left and right shifts of wave forms. Does that over simplify things? Great gollumpus gumdrops (please pardon the profanity) yes, of course it does, insanely so! But isn't that the whole point when trying to convey a basic concept over to newbies -- simplifying things to the point where they can have that little click of intuition that tells them "Hey, I think I get the drift of this idea afterall, wow!"?
From sinusoidal-only you can then point out this odd idea that you can try adding just a couple of them with different wavelengths, then pointing out that you still get the right results at the fencepost (integer) differentiations and integrations. You then refer the readers to this wonderful concept of a basis set, and how sinusoidals form such a basis set. (Side Note: Error on my part! I first typed "base set", thought it look wrong, but for some reason went with it anyway. Spelled correctly, "Basis set" is in fact covered in the article Basis (linear_algebra). However, I remain disappointed that even in that case the article starts with the assumption that the reader must understand linear algebra before being able to grasp the concept of a set of elements from which all points of interest can be constructed or expressed.) From that you introduce Fourier transforms, and point out how extraordinarily useful this little concept is -- and, if you don't mind tossing in a bit of physics, point out that this simple concept underlies the entire mysterious-sounding uncertainty principle of quantum physics. Hey, it's a better intro and a lot more precise intro to such a concept than the wonky philosophy stuff leads the reader away from precision and mathematical formulation of the concept!)

In other words: Lead 'em in! Bring 'em along! Get their curiousity up! To me a topic like this one -- fractional calculus that is, since I've digressed a bit 8^) -- is a perfect example of the kind of fascinating concept-extension that makes math fun.

Another is the beautiful way that complex numbers almost magically encompass wave-related mathematics. (I am stealing Roger Penrose's terminology and perspective there, since I'm currently going through his delightfully exploratory book The Road to Reality. Penrose's take on the topic really is a lot of fun to read, even if you already know it well and were already impressed by it.)

All of the above takes far less writing than it might seem, especially with good use of diagrams and references to other (apparently non-existent in some cases!) Wikipedia articles. But more importantly, it might give a few readers enough interest and self-confidence to actually dig into the real meat of the article that is already here.

Enough... as in "wow, I think I just said waaaaaay too much, but what the heck, I believe what I said and don't particularly feel like taking it back... 8^)

Cheers, Terry Bollinger 03:23, 19 June 2006 (UTC)

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