Talk:Fractional calculus/Archive1

message: http://www.xuru.org/fc/toc.asp
I didn't see it already in the article, but didn't want to add it if it's already there.
JWhiteheadcc (talk) 15:36, 16 January 2008 (UTC)
thread_title:  A good introduction to Fractional Calculus: 

message: pardon the pun.  absolve the alliterations.  I think "heuristics" and "Half derivative of a simple function" cover pretty much the same stuff.  But I think half-derivative is written more clearly, and with less narrative ("A fairly natural question to ask...",  "It turns out...").  I say heuristics should be merged into half-derivative. Kevin Baastalk 17:19, 14 January 2009 (UTC)
thread_title:  self-similarity of this article 

message: There are some interesting issues that show up w/fractional calculus concerning boundary conditions, composition, and the like.
When you integrate a function like 5x+3, you get 5x^2/2+3x + C.  This is because when you differentiate 5x^2/2+3x + 2 or 5x^2/2+3x - 7 or anything like that you get the same answer: 5x+3 .  For composition to hold, integration must be the inverse of differentiation so that many-to-one mapping in differentation (infinity-to-one) must map back from one-to-infinity, and we do this by adding an undefined constant, 'C'.  Now when we integrate twice or differentiate twice, this infinity-to-one stuff happens twice, so we get two undefined constants. (This is all assuming an "indefinite" integral right now.)  But when we integrate/differentiate one and a half times, do we get one and a half undefined constants?  Ofcourse not, so what happens?
It is best to visualize it - picture differentiation/integration as an exponentially-shaped funnel - differentiation is pushing into the funnel, such that the result is smaller and that constant is popped out, while integration is pulling out of the funnel - such that we have to pop a constant in.  Notice in typical calculus we are only pushing in discrete steps - for this, picture planes - or slices, if you will, cutting through the funnel, infinitely thin spaced one unit apart. when we integrate or differentiate by 1, we push all of these planes up or down by exactly one unit.  (thus, in the end, the picture looks the same as before)  We see that when we push down, we lose information because a finite becomes an infinitesimal, and when we pull up, we gain a constant because an infinitesimal becomes finite.
When we do fractional integration/differentation, we have to picture the whole volume of the funnel filled and the up/down motion being continuous rather than discrete.  In our discrete example, each slice represented a polynominal order, the first was a*x^0, the next one higher, b*x^1, etc. In the continuous case, where we fill all the space in between, we have a continuum of polynominals - a(t)*x^t, or a "spectrum", if you will.  As we move it up and down we have a continuum of those a(t) constants being pushed out or in.
Think of a function, instead of being a short string of symbols generating a curve, as a spectrum, such as: f(x) = a(t)*sin(x*t)+b(t)*cos(x*t) .  Integrating or differentiating by a real number, 'q', will shift the entire spectrum - D_x^q ( a(t)*sin(x*t)+b(t)*cos(x*t) ) = a(t)*D_x^q( sin(x*t) ) + b(t)*D_x^q( cos(x*t) ) .   Notice that in addition to adding a pi/2 phase shift per degree of integration/differentation, we have to apply a fractionalized version of the chain rule to the cos(x*t) and sin(x*t), bringing 't' out to the front.  something like " a(t) * t^q * sin(x*t+q*pi/2) ", i believe.
In one differentiation, some of those t's brought out to the front will be zero, hence our loss of information.  If we leave them as t's and integrate back, we will get back our original function.  But if we evaluate the expression - replacing all t's with their calculated values, the zeros remain and there are no t's to be pushed back, except that instead of multiplying by t when t=0 we might be dividing by t when it equals zero, resulting in an undefined expression.  This undefined expression is our "+ C".  This dividing by zero or multiplying by zero is the infinitesimal-to-finite or finite-to-infinitesimal conversion; the infinity-to-one.
And this is how we lose composition; by not leaving everything as t's.  As we pull from the funnel we generate a whole continuum of undefined expressions (+ C)'s, just as when we push through it we continuosly drive parts of the spectrum to zero.
Kevin Baastalk 16:14, 9 June 2009 (UTC)
thread_title:  composition / initialization / boundary conditions - explanation. 

message: This  may be useful at some point for fleshing out the article. It discusses (briefly!) an application of fractional derivatives.
CRGreathouse (t | c) 23:38, 15 March 2010 (UTC)
thread_title:  Applications 

message: Blue and purple are hard do differentiate for people, like myself, with red-green colorblindness. 24.18.246.152 (talk) 02:33, 26 August 2010 (UTC)
thread_title:  Example plot used is not color blind friendly 

message: Hi, I prepared animation using gnuplot: http://smp.if.uj.edu.pl/~baryluk/fractional_diff/fractional_diff.gif (9MB)
It shows derivatives of x^4. I shows D^p x^4, where p varies from across animation from 0 to 5. For negative argument I show real part of complex result. Hopes this will help somehow. In the same directory in files fractional_diff.gp and fractional_diff.gp one will find simple code for this animation (do whatever you want with them). Witold Baryluk --91.213.255.7 (talk) 22:02, 9 November 2010 (UTC)
replies: text: I, for one, uploaded it: . Kevin Baastalk 19:44, 10 November 2010 (UTC)
thread_title:  New plot, animation 

message: I removed the recently-added "Generalizations" section for the following reasons:
* The so-called "Udita fractional operator" is not referred to in any indepedent reliable secondary sources.  This is a non-negotiable requirement for inclusion in an encyclopedia.  Our mathematics articles aim to cover things that can be found in secondary sources (like textbooks), not cutting-edge research like recent PhD dissertations, or even recent journal articles that have not gained widespread acceptance.  In particular, the term itself "Udita fractional operator" does not seem to appear anywhere outside wikipedia, and this is also against our policies.
* The Erdélyi–Kober operator does not seem to be a generalization of any of the operators mentioned in the article, which each accepts as an input a function f.  But there is no f in that section.  If we are going to mention it here, then it only makes sense to say in what way this generalizes those operators.  
--Sławomir Biały (talk) 18:58, 29 July 2012 (UTC)
thread_title:  Generalizations section 

message: An anonymous user removed this reference and was reverted, for not having given a reason. Actually he did give a reason: "Dannon is an amateur mathematician who makes wild mistakes".  I don't know if one can express things in this way, but I had a look and I can add this: H. Vic Dannon is not known to Math. Reviews, and the journal "Gauge Institute Journal" is not referenced in Math. Reviews.  Apparently more than half of the papers there are written by Vic Dannon.  I have serious doubts about the notability of this reference.  Bdmy (talk) 22:52, 20 June 2013 (UTC)
thread_title:  H. Vic Dannon in "References" 

message: I'm a physicist and there are instances in physics where taking "the square root of a differential operator" comes in handy, for example the Klein-Gordon operator is second order in time while we want it to be first order to get unitary time evolution. 
Anyhow, I know the interpretation of a derivative of a function is the slope of the tangent at that point, but what is the natural interpretation of a fractional derivative? Is it a kind of deformation of the function? Is this deformation continuous or differentiable in itself? 94.211.47.103 (talk) 11:17, 11 November 2013 (UTC)
replies: text: There probably isn't in general >_<. First it depends on which definition is used, secondly interpretations seem to be open (the book by Herrmann has some geometric interpretations). The only concrete way I know is to just plot the function after fractional differentiation. M∧Ŝc2ħεИτlk 11:43, 29 May 2015 (UTC)
thread_title:  Interpretation of non-integer derivatives 

message: It would be useful to have table summarizing which properties of (the usual sort of) differentiation are obeyed by the various fractional derivatives.  (e.g. Is a fractional derivative a linear operator? , is the chain rule satisfied? , is the product rule satisfied? )  The current version of the article only gives the hint: " Unfortunately the comparable process for the derivative operator D is significantly more complex, but it can be shown that D is neither commutative nor additive in general."
In expositions of fractional derivatives that I've found on the web, authors expend so much effort just in defining a fractional derivative that they run out of time and energy before they can state its properties!
Tashiro~enwiki (talk) 16:06, 21 February 2017 (UTC)
thread_title: Table of properties for fractional derivatives ? 

message: I have never heard of this derivative, and looking at the edit page it seems someone wants to put his name all over the Wiki article. According to Google, the IP 2001:4200:5000:fffd::d belongs to the University of the Free State, Orange Free State, South Africa. The first author of the cited paper (to be found at http://arxiv.org/pdf/1602.03408v1) is from this exact place. Seems original research or at least low relevance to me, similar to the Udita fractional derivative.  — Preceding unsigned comment added by 92.29.134.85 (talk) 23:14, 8 August 2016 (UTC)
replies: text: I think this section should be removed. This refers to a 2016 paper in a small journal. Until this paper has a large number of citations, the relevance to the field can be disputed. Further, there are many other versions of fractional derivatives in the literature that are not included in this article. 128.227.222.223 (talk) 17:51, 7 November 2016 (UTC)
replies: text: This 2016 article by Abdon Atangana has had over 100 citations in other articles. GeorgePock (talk) 16:01, 20 September 2017 (UTC)
thread_title:  Atangana-Baleanu derivative 

message: 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
##Differintegrals
###Riemann-Liouville
###Grunwald-Lietnikov
###Weyl
###Interpretation
##Relation to Standard Transformations
###Laplace transform
###Fourier transform
##Properties and Techniques
###General Properties
###Differintegration of some special functions
#Geometric structure of
##Relation to Diffusion
###anomalous(non-fickian) diffusion
###fractional brownian motion
##Relation to Fractals & Chaos Theory
#Advanced topics
##Multiple-order differintegration
###extraordinary differential equations
###partial fractional derivatives
##Special Forms of Fractional Calculus
###Initialized fractional calculus
###Local fractional derivative(LFD)
##Morphological(Synthesis of Structure and Change) aspects
###fractional reaction-diffusion equations
###fractional calculus in continuum mechanics
###fractal operators
#Applications of Fractional Calculus
##Mathematics
##Physics
##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 April 2004 (UTC)
thread_title: Untitled

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

text: 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 September 2004 (UTC)
thread_title:  Alternative version 

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thread_title:  External links modified 

message: I don't understand why we mention the Machado derivative and then assert, "(This derivative does not exist anywhere in the literature)." If the derivative doesn't exist, why is it in the article? And is it true that it doesn't exist? Machado is indeed an author of papers on fractional calculus. Centibyte(talk) 17:04, 3 October 2018 (UTC)
thread_title:  Machado derivative 

message: In the section Caputo-Fabrizio fractional derivative there is no reference to the mentioned paper.
I report here all details I know, but I don't know how to add them to the page, please someone can do it? Thank you.
The details of the paper are the following:
* Title: A new Definition of Fractional Derivative without Singular Kernel
* Authors: Michele Caputo and Mauro Fabrizio
* Journal: Progress in Fractional Differentiation and Applications
* Journal abbreviation: Progr. Fract. Differ. Appl.
* DOI: 10.12785/pfda/010201 (WARNING: DOI is wrong)
* Volume: 1
* Issue: 2
* Pages: 73-85
* Year: 2015
References for these information can be found on the journal website
http://www.naturalspublishing.com/ContIss.asp?IssID=255
This is the full text of the paper
http://www.naturalspublishing.com/files/published/0gb83k287mo759.pdf
The paper is indexed in Scopus:
https://www.scopus.com/record/display.uri?eid=2-s2.0-85051078999&origin=resultslist
This is a PDF print of the Scopus page, for whose that don't have access to Scopus
https://drive.google.com/file/d/1eadN5q_kcuRDl_cMnlU9idnQIKt_a1d2/view?usp=sharing  — Preceding unsigned comment added by Enzotib (talk • contribs) 10:54, 7 August 2020 (UTC)
replies: text: Done, and thank you for providing the information, that made adding the reference a lot easier! I went ahead and chose not to add the DOI since I checked and, you're right, it doesn't work. Integral Python click here to argue with me 21:53, 7 August 2020 (UTC)
thread_title:  Reference to paper of Caputo-Fabrizio 

message: Chaotic dynamics in a novel COVID-19 pandemic model described by commensurate and incommensurate fractional-order derivatives. 
I came to this article trying to make some sense of the above paper but didn't get anywhere with either ;).  Maybe someone can summarize it and add a cite if it makes any sense.  I'm unfamiliar with the topic but it seems mathematically suspicious.  Is it legit?  Fractional calculus I mean.  Can it interpret things that more direct methods (stochastic PDE, etc.) don't? 2601:648:8202:350:0:0:0:C115 (talk) 07:02, 3 February 2022 (UTC)
thread_title:  Epidemiology application