Talk:Differintegral

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[edit] Oddity

Moved here from initialized fractional calculus;

A certain oddity about the differintegral should be pointed out. If the differintegral is "uninitialized", then although:

\mathbb{D}^q\mathbb{D}^{-q} = \mathbb{I}

(That is, Dq is the left inverse of D-q.), the converse is not necessarily true.

\mathbb{D}^{-q}\mathbb{D}^q \neq \mathbb{I}

But hold on, the ship isn't sunk yet! Let's take a look at integral calculus, to get a better idea of what's happening. First, let's integrate, then differentiate, using the arbitrary function 3x2+1:

d\left[\int (3x^2+1)dx\right]/dx = d[x^3+x+c]/dx = 3x^2+1

Well, that was pretty straightforward, and it worked. Now, what happens when we exchange the order of composition?

\int [d(3x^2+1)/dx]dx = \int 6xdx = 3x^2+c

Hmmm... I think it's fairly obvious what that integration constant is. Even if it wasn't obvious, we would simply use the initialization terms such as f'(0) = c, f' '(0)= d, etc. If we neglected those initilization terms, the last equation would fail our test. This is exactly the problem that we encountered with the differintegral. If the differintegral is initialized properly, then the composition holds. The problem is that in differentation, we lose state information, as we lost the c in the first equation. (see dynamical systems).

In fractional calculus, however, since the operator has been fractionalized and is thus continuous, an entire complimentary function is needed, not just a constant or set of constants.

{}_a\mathbb{D}^q_tf(x)=\frac{1}{\Gamma(n-q)}\frac{d^n}{dx^n}\int_{a}^{t}(t-\tau)^{n-q-1}f(\tau)d\tau + \Psi(x)

Charles Matthews 06:41, 22 Apr 2004 (UTC)

[edit] oddity.. almost duplicate..

Moved here from initialization of the differintegrals (NB this duplicates the above, but not exactly):

Before discussing initialization of the differintegrals in fractional calculus, a certain oddity about the differintegral should be pointed out. Although:

\mathbb{D}^q\mathbb{D}^{-q} = \mathbb{I}

That is, Dq is the left inverse of D-q. On first glance, the converse is not necessarily true.

\mathbb{D}^{-q}\mathbb{D}^q \neq \mathbb{I}

However, let's take a look at integral calculus, to get a better idea of what's happening. First, let's integrate, then differentiate, using the arbitrary function 3x2+1:

d\left[\int (3x^2+1)dx\right]/dx = d[x^3+x+c]/dx = 3x^2+1

The process did work successfully. On exchanging the order of composition:

\int [d(3x^2+1)/dx]dx = \int 6xdx = 3x^2+c

The integration constant here is clear. Even if it wasn't obvious, we would simply use the initialization terms such as f'(0) = c, f' '(0)= d, ect. If we neglected those initialization terms, the last equation would fail our test.

This is exactly the problem that we encountered with the differintegral. If the differintegral is initialized properly, then the hoped-for composition law holds. The problem is that in differentiation, we lose information, as we lost the c in the first equation. (see dynamical systems).

In fractional calculus, however, since the operator has been fractionalized and is thus continuous, an entire complementary function is needed, not just a constant or set of constants. We call this complementary function "Ψ".

\mathbb{D}^q_tf(t)=\frac{1}{\Gamma(n-q)}\frac{d^n}{dt^n}\int_{0}^{t}(t-\tau)^{n-q-1}f(\tau)d\tau + \Psi(x)

(Working with a properly initialized differintegral is the subject of initialized fractional calculus).

Charles Matthews 06:44, 22 Apr 2004 (UTC)

Re the above - I have not managed to understand any exact statement on 'initialization', here. I think both of the above are too tentative to stand on a page for ever. The initialization issue, I think, is to do with what I wrote on fractional calculus about needing to take into account boundary conditions, because the fractional operators are not in general locally defined. I have had serious doubts about whether the Weyl differintegral (periodic boundary conditions) is really 'the same as' the others.

Charles Matthews 07:29, 22 Apr 2004 (UTC)

Yes, the boundary conditions and initialization amount to the same thing. Initialized fractional calculus refers to a paper published by the NASA John Glenn Research Center, cited on the main page. The problem with merely specifying the 'boundary conditions', in the sense of a function integrated from 'a' to 'b', or even specifying f'=w, f' '=x, f' ' '=y, etc, is that this does not fully specify the region of integration. One needs an infinite set of such constants: an entire complimentary function.
Regarding the Weyl differeintegral: perhaps it has a set of applications unique to it, but this does not at all preclude it's equivalency in regards to fractional calculus. In all the literature I've seen on fractional calculus, the Weyl differintegral has been treated without prejudice, although it is not used as often as, say, the Riemann-Louiville differintegral. If not ininitalized properly, the composition rule does not hold on either of them, and they produce different results. But when initialization is taken into account, they become indistinguishable from each other in effect.
However, I am not well-versed on the Weyl differintegral, and I'm not familiar with the application that you have studied it in. Even so, a difference in application does not imply a difference in fundamental formal properties, and as long as your argument rests on that leap alone, I consider it unsubstantiated.
Honestly, I could care less about the Weyl differintegral. I'm just telling you my experience and logic. It would be prudent for you to check out the literature before making any criticisms or exclusions. Kevin Baas 23:39, 22 Apr 2004 (UTC)

[edit] NASA citation

Please tell me if the NASA paper you cite has been published in a refereed journal.

The Weyl version, as I have said before, is something referred to in a major treatise, Zygmund's Trigonometric Series. It should therefore be given a proper write-up. The whole point, according to G. H. Hardy, of doing mathematical analysis, is to get beyond fundamental formal properties.

Charles Matthews 08:17, 23 Apr 2004 (UTC)

NASA:I do not know where it has been published. I do know, however, that it is mathematically sound, and that it is applied. But I was not complete: The complementary function goes back to the early beginnings of fractional calculus: it goes back to Riemann. Some of the books cited on the mother page give reference to this.
Weyl: I don't dispute the fact that it should be given a proper write-up. Whatever gave you this impression?
The point: the point is not "what is the point of mathematics?" - that is off-track. The point is that a square and a triangle are both geometric figures. Likewise, a Weyl and a Riemann-Louiville differintegral are both differintegrals. Everything discussed in the fractional calculus section applies equally well to the Weyl differintegral as any other differintegral.
It is certainly possible to have unique content on the Weyl differintegral page, and to have links to and/or from topics regarding trigonometric series. This is not precluded by the weyl differintegral being linked from the fractional calculus discussion, as well as discussed therein. Wikipedia is not heriarchial. Kevin Baas 17:04, 23 Apr 2004 (UTC)

[edit] ImportantLabeledEquation

I don't quite like the dotted boxes, be they made with blockquote or with the new ImportantLabeledEquation template. I will post this as discussion on Wikipedia talk:WikiProject Mathematics. Oleg Alexandrov 3 July 2005 01:06 (UTC)

[edit] Definitions and explanation

This page needs a definition of the differintegral at the top of the page. All the equations need to define what all the variables inside it mean. The article on fractional calculus doesn't give decent enough perspective for someone to understand this page. Like.. what do the superscripts mean? Fresheneesz 17:22, 26 November 2005 (UTC)

[edit] relationship among definitions

This article should explain the relationship between the various definitions it gives. Are they all equivalent? If not, which ones differ, and how? Do they all have all the properties listed further down?

Also, what's the basis for the distinction between "standard" definitions and definitions via transform? The definition via the Fourier transform appears the most obvious to me; why is it less standard? Joriki 23:43, 13 February 2006 (UTC)

[edit] Composition rule

Does the composition rule hold always without exception? Consider this:

\mathbb{D}^{\frac{5}{6}}\mathbb{D}^{2}(t)=\mathbb{D}^{\frac{17}{6}}(t)=\frac{1}{\Gamma(-\frac{5}{6})t^{\frac{11}{6}}}

However, it is clear that

\mathbb{D}^{2}(t)=0--

Therefore

\mathbb{D}^{\frac{5}{6}}\mathbb{D}^{2}(t)=0

--88.101.219.136 08:26, 4 July 2007 (UTC)

[edit] Error in the definition via Fourier Transform?

The definition via Fourier transform gives the Fourier Transform as a derivative as a product of i*t times the original function but the multiplication takes place in the frequency domain and not the time domain, so it should be i*omega. I am making the appropriate change. —Preceding unsigned comment added by 152.2.176.132 (talk) 15:06, 17 October 2007 (UTC)

I think that the definition via Laplace transform is incorrect as well. The correct rule for differentiation is
\mathcal{L}\left\{\frac{df}{dt}\right\} = s\cdot\mathcal{L} \left\{ f(t) \right\}-f(0)
--195.113.191.162 (talk) 07:33, 14 December 2007 (UTC)

[edit] Merger? and applications

First, this is at least 50% a duplicate of the more fundamental article on fractional calculus. Since it is also a "low-priority" article, perhaps it should be merged?

Second, on neither of these pages is there any discussion of application. I understand that it has application at least in analyzing so-called "meta-materials" (e.g. optical media with negative refractive index) - it would be awesome if I could read about this here; I've only heard hints of it from those in the know. 208.120.110.46 (talk) 03:14, 7 April 2008 (UTC)