Talk:Fractional calculus

ambiguous "Fractional derivative of a basic power function" section hiding the most important formula

AFAIK (and understand from the text), the last part of this section starting with "For a general function f(x)" is really about any kind of f() and not only for f() being a basic power function, as expected by the section title.

Also, this general formula is probably the most important part of this page since it explain how to practically compute it (when you don't want to do it in Fourier or Laplace space ).

So I guess it should deserve its own paragraph. — Preceding unsigned comment added by Fabrice.Neyret (talk • contribs) 17:40, 9 May 2022 (UTC)

I take issue with this section, but not for the same reasons you do. This wording implies that this is the way of computing the fractional derivative of a power function, which it is not. There are many different fractional derivatives as detailed in the later in the page under "Fractional integrals" and "Fractional derivative" and they do not follow this form. As well as saying that was the general formula for fractional derivatives is also misleading if not false. The formula in this section is pretty much the same as the Riemann-Liouville fractional integral.

I do however think that it would make more sense if it were tweaked and moved as a "Special case of basic power functions" section in the Riemann–Liouville_integral article.

Same with the Laplace transform section. I think it can be moved the Riemann-Liouvile integral article as motivation for it's definition alongside the Cauchy repeated integral rule. Coffeevector (talk) 06:52, 2 September 2022 (UTC)

Which character does the article use for the index of differentiation?

The nice illustration with caption "The animation shows the derivative operator oscillating ..." appears to use a Greek lowercase alpha for the index, whereas the article appears to use a Roman lowercase A. (Unless my eyes are deceiving me.)

It's probably best if they both use the same character, especially because the caption does not define the meaning of that character but assumes it is understood. 2601:200:C000:1A0:9D6A:3426:156B:13FB (talk) 23:36, 17 June 2022 (UTC)

Two equivalent definitions of Caputo fractional derivative

There seem to be two equivalent definitions given of the Caputo fractional derivative, one using α and one using ν, which is confusing. One of them should be removed. Since α seems to be used consistently throughout the article, I would suggest removing the definition that uses ν. Some editing of the surrounding text will also be required. Benjamin Rich (talk) 15:10, 22 December 2023 (UTC)

Proposal for the Addition of Information on Fractional Operators

Considering the following references:

Acceleration of the order of convergence of a family of fractional fixed-point methods and its implementation in the solution of a nonlinear algebraic system related to hybrid solar receivers

Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods

Would it be possible to add the following information on fractional operators?

References

Set $O_{x,\alpha }^{n}(h)$ of Fractional Operators

Fractional calculus, a branch of mathematics dealing with derivatives of non-integer order, emerged nearly simultaneously with traditional calculus. This emergence was partly due to Leibniz's notation for derivatives of integer order: ${\frac {d^{n}}{dx^{n}}}$. Thanks to this notation, L'Hopital was able to inquire in a letter to Leibniz about the interpretation of taking $n={\frac {1}{2}}$ in a derivative. At that moment, Leibniz couldn't provide a physical or geometric interpretation for this question, so he simply replied to L'Hopital in a letter that "... is an apparent paradox from which, one day, useful consequences will be drawn". The name "fractional calculus" originates from a historical question, as this branch of mathematical analysis studies derivatives and integrals of a certain order $\alpha \in \mathbb {R}$. Currently, fractional calculus lacks a unified definition of what constitutes a fractional derivative. Consequently, when the explicit form of a fractional derivative is unnecessary, it is typically denoted as follows:

${\frac {d^{\alpha }}{dx^{\alpha }}}.$

Fractional operators have various representations, but one of their fundamental properties is that they recover the results of traditional calculus as $\alpha \to n$. Considering a scalar function $h:\mathbb {R} ^{m}\to \mathbb {R}$ and the canonical basis of $\mathbb {R} ^{m}$ denoted by $\{{\hat {e}}_{k}\}_{k\geq 1}$, the following fractional operator of order $\alpha$ is defined using Einstein notation :

$o_{x}^{\alpha }h(x):={\hat {e}}_{k}o_{k}^{\alpha }h(x).$

Denoting $\partial _{k}^{n}$ as the partial derivative of order $n$ with respect to the $k$-th component of the vector $x$, the following set of fractional operators is defined:

$O_{x,\alpha }^{n}(h):=\left\{o_{x}^{\alpha }:\exists o_{k}^{\alpha }h(x){\text{ and }}\lim _{\alpha \to n}o_{k}^{\alpha }h(x)=\partial _{k}^{n}h(x)\ \forall k\geq 1\right\},$

with its complement:

$O_{x,\alpha }^{n,c}(h):=\left\{o_{x}^{\alpha }:\exists o_{k}^{\alpha }h(x)\ \forall k\geq 1{\text{ and }}\lim _{\alpha \to n}o_{k}^{\alpha }h(x)\neq \partial _{k}^{n}h(x){\text{ for at least one }}k\geq 1\right\}.$

Consequently, the following set is defined:

$O_{x,\alpha }^{n,u}(h):=O_{x,\alpha }^{n}(h)\cup O_{x,\alpha }^{n,c}(h).$

Extension to Vectorial Functions

For a function $h:\Omega \subset \mathbb {R} ^{m}\to \mathbb {R} ^{m}$, the set is defined as:

${}_{m}O_{x,\alpha }^{n,u}(h):=\left\{o_{x}^{\alpha }:o_{x}^{\alpha }\in O_{x,\alpha }^{n,u}([h]_{k})\ \forall k\leq m\right\},$

where $[h]_{k}:\Omega \subset \mathbb {R} ^{m}\to \mathbb {R}$ denotes the $k$-th component of the function $h$. Calfracsets (talk) 06:27, 12 August 2024 (UTC)