Initialized fractional calculus

In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus.

Composition rule of Differintegrals

The composition law of the differintegral operator states that although:

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

wherein D^−q is the left inverse of D^q, the converse is not necessarily true:

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

Example

Consider elementary integer-order calculus. Below is an integration and differentiation using the example function $3x^{2}+1$:

${\frac {d}{dx}}\left[\int (3x^{2}+1)dx\right]={\frac {d}{dx}}[x^{3}+x+C]=3x^{2}+1\,,$

Now, on exchanging the order of composition:

$\int \left[{\frac {d}{dx}}(3x^{2}+1)\right]=\int 6x\,dx=3x^{2}+C\,,$

Where C is the constant of integration. Even if it was not obvious, the initialized condition ƒ'(0) = C, ƒ''(0) = 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 $\Psi$.

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

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

The Fractional Calculus of Sets (FCS), first mentioned in the article titled "Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods", is a methodology derived from fractional calculus. The main concept behind FCS is the characterization of the elements of fractional calculus using sets due to the large number of fractional operators available. This methodology originated from the development of the fractional Newton-Raphson method and subsequent related works.

Illustration of some lines generated by the fractional Newton-Raphson method for the same initial condition $x_{0}$ but with different orders $\alpha$ of the fractional operator implemented. Source: Applied Mathematics and Computation

Fractional calculus, a branch of mathematics dealing with derivatives of non-integer order, emerged almost simultaneously with traditional calculus. This emergence was partly due to Leibniz's notation for integer-order derivatives: ${\frac {d^{n}}{dx^{n}}}$. Thanks to this notation, L’Hopital was able to ask Leibniz in a letter about the interpretation of taking $n={\frac {1}{2}}$ in a derivative. At the time, Leibniz could not provide a physical or geometrical interpretation for this question, so he simply responded to L’Hopital in a letter that "... it 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 it is not necessary to explicitly specify the form of a fractional derivative, it is typically denoted as follows:

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

Fractional operators have several 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 :

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

whose complement is:

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

As a consequence, 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 Vector 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 $h$.

The use of fractional operators in fixed-point methods has been widely studied and cited in various academic sources. Examples of this can be found in several articles published in renowned journals, such as those appearing in ScienceDirect , , Springer , World Scientific , and MDPI , , , , , , , . Studies from Taylor & Francis (Tandfonline) , Cubo , Revista Mexicana de Ciencias Agrícolas , Journal of Research and Creativity , MQR , and Актуальные вопросы науки и техники . These works highlight the relevance and applicability of fractional operators in problem-solving.

See also

• Initial conditions
• Dynamical systems

References

• Lorenzo, Carl F.; Hartley, Tom T. (2000), Initialized Fractional Calculus (PDF), NASA (technical report).