Editing Initialized fractional calculus

{{calculus|expanded=Specialized calculi}}
{{Multiple issues|
{{context|date=May 2024}}
{{refimprove|date=May 2024}}
{{too technical|date=May 2024}}
}}
In [[mathematical analysis]], '''initialization of the differintegrals''' is a topic in [[fractional calculus]], a branch of mathematics dealing with derivatives of non-integer order.

== Composition rule of Differintegrals ==

The [[Differintegral#Basic formal properties|composition law]] of the [[differintegral]] operator states that although: 

<math>\mathbb{D}^q\mathbb{D}^{-q} = \mathbb{I}</math>

wherein ''D''<sup>&minus;''q''</sup> is the left [[Inverse function|inverse]] of ''D<sup>q</sup>'', the converse is not necessarily true:

:<math>\mathbb{D}^{-q}\mathbb{D}^q \neq \mathbb{I}</math>

===Example===

Consider elementary integer-order [[calculus]]. Below is an integration and differentiation using the example function <math>3x^2+1</math>:

:<math>\frac{d}{dx}\left[\int (3x^2+1)dx\right] = \frac{d}{dx}[x^3+x+C] = 3x^2+1\,,</math>

Now, on exchanging the order of composition:

:<math>\int \left[\frac{d}{dx}(3x^2+1)\right] = \int 6x \,dx = 3x^2+C\,,</math>

Where ''C is'' the [[constant of integration]]. Even if it was not obvious, the initialized condition ''&fnof;''<nowiki>'</nowiki>(0)&nbsp;=&nbsp;''C'', ''ƒ''<nowiki>''</nowiki>(0)&nbsp;=&nbsp;''D'', etc. could be used. If we neglected those initialization terms, the last equation would show the composition of integration, and differentiation (and vice versa) would not hold.

==Description of initialization==

Working with a properly initialized differ integral is the subject of initialized fractional calculus. If the differ integral is initialized properly, then the hoped-for composition law holds.  The problem is that in differentiation, information is lost, as with ''C'' in the first equation.

However, in fractional calculus, given that the operator has been fractionalized and is thus continuous, an entire '''complementary function''' is needed.  This is called complementary function <math>\Psi</math>.

:<math>\mathbb{D}^q_t f(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)</math>

==See also==

*[[Initial conditions]]
*[[Dynamical systems]]

== References ==
* {{citation|title=Initialized Fractional Calculus|last1=Lorenzo|first1=Carl F.|last2=Hartley|first2=Tom T. |year=2000 |publisher=[[NASA]]|url=https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20000031631.pdf|ref=none}} (technical report).

[[Category:Fractional calculus]]