Draft:Fractional calculus of sets

The Fractional Calculus of Sets (FCS), first introduced 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 primary concept behind FCS is the characterization of fractional calculus elements using sets due to the plethora of fractional operators available. This methodology originated from the development of the Fractional Newton-Raphson method and subsequent related works .

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:

$ {}_mO_{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 $$h$$.

Set $$ {}_mMO_{x,\alpha}^{\infty,u}(h)$$ of Fractional Operators
The set of fractional operators considering infinite orders is defined as:

$ {}_mMO_{x,\alpha}^{\infty,u}(h) := \bigcap_{k \in \mathbb{Z}} {}_mO_{x,\alpha}^{k,u}(h), $|undefined

where under classic Hadamard product it holds that:

$ o_x^0 \circ h(x) := h(x) \quad \forall o_x^\alpha \in {}_mMO_{x,\alpha}^{\infty,u}(h). $

Fractional Matrix Operators
For each operator $$o_x^\alpha$$, the fractional matrix operator is defined as:

$ A_\alpha(o_x^\alpha) = \left([A_\alpha(o_x^\alpha)]_{jk}\right) = \left(o_k^\alpha\right), $

and for each operator $$o_x^\alpha \in {}_mMO_{x,\alpha}^{\infty,u}(h)$$, the following matrix, corresponding to a generalization of the Jacobian matrix, can be defined:

$ A_{h,\alpha} := A_\alpha(o_x^\alpha) \circ A_\alpha^T(h), $

where $$A_\alpha(h) := \left([A_\alpha(h)]_{jk}\right) = \left([h]_k\right)$$.

Modified Hadamard Product
Considering that, in general, $$o_x^{p\alpha} \circ o_{x}^{q\alpha} \neq o_x^{(p+q)\alpha}$$, the following modified Hadamard product is defined:

$$ o_{i,x}^{p\alpha} \circ o_{j,x}^{q\alpha} := \left\{ \begin{array}{cl} o_{i,x}^{p\alpha} \circ o_{j,x}^{q\alpha}, & \text{if } i \neq j \ (\text{horizontal type Hadamard product}) \\ o_{i,x}^{(p+q)\alpha}, & \text{if } i = j \ (\text{vertical type Hadamard product}) \end{array}\right., $$

with which the following theorem is obtained:

Theorem: Abelian Group of Fractional Matrix Operators
Let $$o_x^\alpha$$ be a fractional operator such that $$o_x^\alpha \in {}_mMO_{x,\alpha}^{\infty,u}(h)$$. Considering the modified Hadamard product, the following set of fractional matrix operators is defined:

$$ \begin{array}{c} {}_mG(A_\alpha(o_x^\alpha)) := \left\{ A_\alpha^{\circ r} = A_\alpha(o_x^{r\alpha}) : r \in \mathbb{Z} \ \text{ and } \ A_{\alpha}^{\circ r} = \left([A_{\alpha}^{\circ r}]_{jk}\right) := \left(o_k^{r\alpha}\right) \right\}, \end{array} \quad (1) $$

which corresponds to the Abelian group generated by the operator $$A_\alpha(o_x^\alpha)$$.

Proof
Since the set in equation (1) is defined by applying only the vertical type Hadamard product between its elements, for all $$A_\alpha^{\circ p}, A_\alpha^{\circ q} \in {}_mG(A_\alpha(o_x^\alpha))$$ it holds that:

$$ A_\alpha^{\circ p} \circ A_\alpha^{\circ q} = \left([A_\alpha^{\circ p}]_{jk}\right) \circ \left([A_\alpha^{\circ q}]_{jk}\right) = \left(o_k^{(p+q)\alpha}\right) = \left([A_\alpha^{\circ(p+q)}]_{jk}\right) = A_\alpha^{\circ (p+q)}, $$

with which it is possible to prove that the set (1) satisfies the following properties of an Abelian group:

$$ \left\{ \begin{array}{l} \forall A_\alpha^{\circ p}, A_\alpha^{\circ q}, A_\alpha^{\circ r} \in {}_mG(A_\alpha(o_x^\alpha)), \ \left(A_\alpha^{\circ p} \circ A_\alpha^{\circ q}\right) \circ A_\alpha^{\circ r} = A_\alpha^{\circ p} \circ \left(A_\alpha^{\circ q} \circ A_\alpha^{\circ r}\right) \\ \exists A_\alpha^{\circ 0} \in {}_mG(A_\alpha(o_x^\alpha)) \ \text{such that} \ \forall A_\alpha^{\circ p} \in {}_mG(A_\alpha(o_x^\alpha)), \ A_\alpha^{\circ 0} \circ A_\alpha^{\circ p} = A_\alpha^{\circ p} \\ \forall A_\alpha^{\circ p} \in {}_mG(A_\alpha(o_x^\alpha)), \ \exists A_\alpha^{\circ -p} \in {}_mG(A_\alpha(o_x^\alpha)) \ \text{such that} \ A_\alpha^{\circ p} \circ A_\alpha^{\circ -p} = A_\alpha^{\circ 0} \\ \forall A_\alpha^{\circ p}, A_\alpha^{\circ q} \in {}_mG(A_\alpha(o_x^\alpha)), \ A_\alpha^{\circ p} \circ A_\alpha^{\circ q} = A_\alpha^{\circ q} \circ A_\alpha^{\circ p} \end{array} \right. . $$

Set $${}_m S_{x,\alpha}^{n,\gamma}(h)$$ of Fractional Operators
Let $$\mathbb{N}_0$$ be the set $$\mathbb{N} \cup \{0\}$$. If $$\gamma \in \mathbb{N}_0^m$$ and $$x \in \mathbb{R}^m$$, then the following multi-index notation can be defined:

$$ \left\{ \begin{array}{c} \begin{array}{ccc} \displaystyle \gamma!:= \prod_{k=1}^m [\gamma]_k !,& |\gamma|:= \displaystyle \sum_{k=1}^m [\gamma]_k, & \displaystyle x^\gamma:= \prod_{k=1}^m [x]_k^{[\gamma]_k} \end{array} \\ \displaystyle \frac{\partial^\gamma}{\partial x^\gamma}:= \frac{\partial^{[\gamma]_1}}{\partial [x]_1} \frac{\partial^{[\gamma]_2}}{\partial [x]_2}\cdots \frac{\partial^{[\gamma]_m}}{\partial [x]_m} \end{array}\right. . $$

Then, considering a function $$h: \Omega \subset \mathbb{R}^m \to \mathbb{R}$$ and the fractional operator:

$$ s_x^{\alpha\gamma}\left( o_x^\alpha \right):=o_1^{\alpha[\gamma]_1}o_2^{\alpha[\gamma]_2}\cdots o_m^{\alpha[\gamma]_m}, $$

the following set of fractional operators is defined:

$$ S_{x,\alpha}^{n,\gamma}(h):=\left\{ s_x^{\alpha \gamma}=s_x^{\alpha\gamma}\left( o_x^\alpha \right) \ : \ \exists s_x^{\alpha \gamma}h(x) \ \text{ such that } \ o_x^\alpha \in O_{x,\alpha}^{s}(h) \ \forall s\leq n^2 \ \text{ and } \ \lim_{\alpha \to k} s_x^{\alpha \gamma}h(x)=\frac{\partial^{k\gamma}}{\partial x^{k\gamma}}h(x) \ \forall \alpha, |\gamma| \leq n    \right\}. $$

From which the following results are obtained:

$$ \text{If } s_x^{\alpha \gamma} \in S_{x,\alpha}^{n,\gamma}(h) \ \Rightarrow \ \left\{ \begin{array}{l} \displaystyle \lim_{\alpha \to 0} s_x^{\alpha \gamma} h(x)=o_1^{0} o_2^{0}\cdots o_m^{0} h(x)=h(x) \\ \displaystyle \lim_{\alpha \to 1} s_x^{\alpha \gamma} h(x)= o_1^{[\gamma]_1} o_2^{[\gamma]_2}\cdots o_m^{[\gamma]_m} h(x)=\frac{\partial^\gamma}{\partial x^\gamma} h(x) \ \forall |\gamma|\leq n \\ \displaystyle \lim_{\alpha \to q} s_x^{\alpha \gamma} h(x)= o_1^{q[\gamma]_1} o_2^{q[\gamma]_2}\cdots o_m^{q[\gamma]_m} h(x)=\frac{\partial^{q\gamma}}{\partial x^{q\gamma}} h(x) \ \forall q |\gamma|\leq qn \\ \displaystyle \lim_{\alpha \to n} s_x^{\alpha \gamma} h(x)= o_1^{n[\gamma]_1} o_2^{n[\gamma]_2}\cdots o_m^{n[\gamma]_m} h(x)=\frac{\partial^{n\gamma}}{\partial x^{n\gamma}} h(x) \ \forall n |\gamma| \leq n^2 \end{array}\right. . $$

As a consequence, considering a function $$h: \Omega \subset \mathbb{R}^m \to \mathbb{R}^m$$, the following set of fractional operators is defined:

$$ {}_m S_{x,\alpha}^{n,\gamma}(h):=\left\{ s_x^{\alpha \gamma} \ : \ s_x^{\alpha \gamma} \in S_{x,\alpha}^{n,\gamma}\left( [h]_k \right) \ \forall k \leq m \right\}. $$

Set $${}_m T_{x,\alpha}^{\infty,\gamma}(a,h)$$ of Fractional Operators
Considering a function $$h: \Omega \subset \mathbb{R}^m \to \mathbb{R}^m$$ and the following set of fractional operators:

$$ {}_m S_{x,\alpha}^{\infty,\gamma}(h):= \lim_{n \to \infty} {}_m S_{x,\alpha}^{n,\gamma}(h). $$

Then, taking a ball $$B(a; \delta) \subset \Omega$$, it is possible to define the following set of fractional operators:

$$ {}_m T_{x,\alpha}^{\infty,\gamma}(a,h):=\left\{ t_{x}^{\alpha,\infty}=t_x^{\alpha,\infty}\left(s_x^{\alpha \gamma} \right) \ : \ s_x^{\alpha \gamma}\in {}_m S_{x,\alpha}^{\infty,\gamma}(h) \ \text{ and } \ t_{x}^{\alpha,\infty} h(x):= \sum_{|\gamma| =0}^\infty \frac{1}{\gamma !}\hat{e}_j s_x^{\alpha \gamma} [h]_j(a)(x-a)^\gamma \right\}, $$

which allows generalizing the expansion in Taylor series of a vector-valued function in multi-index notation. As a consequence, the following result can be obtained:

$$ \text{If } t_{x}^{\alpha,\infty}\in {}_m T_{x,\alpha}^{\infty,\gamma}(a,h) \Rightarrow \left\{ \begin{array}{rl} t_{x}^{\alpha,\infty} h(x) =& \displaystyle \hat{e}_j [h]_j(a) +\sum_{|\gamma| =1} \frac{1}{\gamma !}\hat{e}_j s_x^{\alpha \gamma} [h]_j(a)(x-a)^\gamma + \sum_{|\gamma| =2}^\infty \frac{1}{\gamma !}\hat{e}_j s_x^{\alpha \gamma} [h]_j(a)(x-a)^\gamma\\ =& \displaystyle h(a) + \sum_{k=1}^n \hat{e}_j o_k^{\alpha} [h]_j(a) \left[ (x-a) \right]_k + \sum_{|\gamma| =2}^\infty \frac{1}{\gamma !}\hat{e}_j s_x^{\alpha \gamma} [h]_j(a)(x-a)^\gamma \end{array}\right. . $$

Fractional Newton-Raphson Method
Let $$f: \Omega \subset \mathbb{R}^m \to \mathbb{R}^m$$ be a function with a point $$\xi \in \Omega$$ such that $$\|f(\xi)\|=0$$. Then, for some $$x_i \in B(\xi; \delta) \subset \Omega$$ and a fractional operator $$t_{x}^{\alpha,\infty} \in {}_m T_{x,\alpha}^{\infty,\gamma}(x_i,f)$$, it is possible to define a type of linear approximation of the function $$f$$ around $$x_i$$ as follows:

$$ t_{x}^{\alpha,\infty} f(x) \approx f(x_i) + \sum_{k=1}^m \hat{e}_j o_k^{\alpha} [f]_j(x_i) \left[ (x-x_i) \right]_k, $$

which can be expressed more compactly as:

$$ t_{x}^{\alpha,\infty} f(x) \approx f(x_i) + \left( o_k^\alpha [f]_j(x_i) \right) (x - x_i), $$

where $$\left( o_k^\alpha [f]_j(x_i) \right)$$ denotes a square matrix. On the other hand, as $$x \to \xi$$ and given that $$\|f(\xi)\| = 0$$, the following is inferred:

$$ 0 \approx f(x_i) + \left( o_k^\alpha [f]_j(x_i) \right) (\xi - x_i) \quad \Rightarrow  \quad \xi \approx x_i - \left( o_k^\alpha [f]_j(x_i) \right)^{-1} f(x_i). $$

As a consequence, defining the matrix:

$$ A_{f,\alpha}(x) = \left( [A_{f,\alpha}]_{jk}(x) \right) := \left( o_k^\alpha [f]_j(x) \right)^{-1}, $$

the following fractional iterative method can be defined:

$$ x_{i+1} := \Phi(\alpha, x_i) = x_i - A_{f,\alpha}(x_i) f(x_i), \quad i = 0, 1, 2, \cdots, $$

which corresponds to the most general case of the fractional Newton-Raphson method.