Talk:Fractional calculus

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

1. Introduction
2. History
3. 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
4. Geometric structure of
   1. Relation to Diffusion
      1. anomalous(non-fickian) diffusion
      2. fractional brownian motion
   2. Relation to Fractals & Chaos Theory
5. 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
6. Applications of Fractional Calculus
   1. Mathematics
   2. Physics
   3. Engineering
7. 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

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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)

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)

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:

$f(x)=\sum _{i=0}^{n}a_{i}x^{i}.$

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

$D^{1/q}(f)=\sum _{i=0}^{n}{\Gamma (i+1) \over \Gamma (i+1-1/q)}a_{i}x^{i-1/q}.$

And that implies that

$D^{p/q}(f)=\sum _{i=0}^{n}{\Gamma (i+1) \over \Gamma (i+1-p/q)}a_{i}x^{i-p/q}$

for integers $p$ and $q$ (again, $q\neq 0$). 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 $\epsilon _{n}$ 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)

Fractals?

IMHO, there should be a section on the relationship of fractional calculus to fractals. There are books that this is discussed in. Many of them bring up the Wierstraus function. I wrote a little paper (doc) on it myself elucidating the intuitive spatial relations, which I think makes the connection clear (if one can comprehend the rather idiosyncratic terminology - a lot of allusion to information theory) and answers the "Hmmm...". Kevin Baas^talk 19:32, 2004 Dec 30 (UTC)

Clarification. I wrote a little supplementary (doc) to clarify my little paper. Kevin Baas^talk 00:03, 2005 Jan 3 (UTC)

Point of the paper regarding the "Hmmm..." and D₁, D₂, is that for a unique x (statistically-uncorrelated/linearly-independant/non-redundant/orthogonal) parameter function, integrated over a unique y (bla bla bla) parameter region, there is a unique x-y (bla bla bla) parameter result.

Given two unique operands and a unique differential metric, the solution is completely specified, and thus the operator is unique.

The apparent confusion could result from not realizing that the integration operator requires a complete specification of not one, but two operands: the integrated function and the region of integration, and that the number of integrations (single, double, etc.) ("degree" of integration) is rigidly fixed to the dimensionality of the region of integration.

Or the confusion could result from equivocating multiple valid metrics with multiple valid operators. The chain-rule/differential-geometry shows that there is not a problem; in d(f(x)) = df(x)dx, a change in dx results in a change in df(x)dx, by no "fault" of the operator d. Kevin Baas^talk 21:56, 2005 Jan 3 (UTC)

Fractional derivative - nonlocalizable

The above little paper(doc) is a geometrically intuitive explanation for the fractional derivative "peripheral vision" (non-localizable) property. It shows how to make sense of it geometrically, and that it really amounts to nothing special - we've just been making a mistake in our conception of a derivative as a unary operator. it doesn't have any of the citations filled in in the esoteric page, but that page isn't important. one just needs to know basic calculus and maybe a little about fractals to be able to understand the main idea. I think it's important, and forgive me for endorsing it here, but i jsut want to get it out there, and frac calc is a pretty esoteric field, and i'm not in college, so it's hard to find an outlet. Kevin Baas^talk 01:04, 2005 Feb 18 (UTC)

Kevin - please see Wikipedia:What Wikipedia is not 1.3.5, i.e. under Wikipedia is not a soapbox and Primary research. Charles Matthews 06:25, 18 Feb 2005 (UTC)

I have read the policy. I am not suggesting putting this in the article. I'm just posting a link on the talk page for anyone who is interested and/or wants to try to get a clearer picture of the spatial meaning of fractional calculus. People having a clearer understanding may benefit the article, and in any case benefits the general knowledge. Kevin Baas^talk 20:13, 2005 Feb 20 (UTC)

If you say I just want to get it out there, you are using WP for a purpose which is not the intended purpose. Charles Matthews 20:33, 20 Feb 2005 (UTC)

I suggest you read the Wikipedia:What Wikipedia is not that you refered me to more carefully. Kevin Baas^talk 20:35, 2005 Feb 20 (UTC)

How about this: If you have done primary research on a topic, publish your results in normal peer-reviewed journals, or elsewhere on the web.

Clear enough? Charles Matthews 20:40, 20 Feb 2005 (UTC)

that refers to the article, not the talk page. how about this:

"Self-promotion. While you are free to write about yourself or projects you have a strong personal involvement in, remember that the standards for encyclopedic articles apply to such pages just like any other. A very few somewhat famous Wikipedians have significantly contributed to encyclopedia articles about themselves and their accomplishments, and this has mostly been accepted after some debate. Creating overly abundant links and references to autobiographical articles is not acceptable."

Clear enough? Kevin Baas^talk 20:42, 2005 Feb 20 (UTC)