# Fractional Scale Calculus: Hadamard vs. Liouville

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## Abstract

**:**

## 1. Introduction

**Remark**

**1.**

## 2. On the Linear Systems

**Definition**

**1.**

**Definition**

**2.**

- piecewise continuous,
- with bounded variation.

**Remark**

**2.**

## 3. Shift-Invariant Systems: The Liouville Derivatives

- 1.
- The exponentials are the eigenfunctions of LTIS$$y\left(t\right)=G\left(s\right){e}^{st},\phantom{\rule{1.em}{0ex}}t\in \mathbb{R},$$$$G\left(s\right)={\int}_{-\infty}^{+\infty}g\left(t\right){e}^{-st}dt$$
- 2.
- If the region of convergence (ROC) of $G\left(s\right)$ contains the imaginary axis, we can set $s=i\omega ,\phantom{\rule{0.277778em}{0ex}}\omega \in \mathbb{R}$, in such a way that the response of an LS to a sinusoid is also a sinusoid with the same frequency. In such a situation, the LT degenerates into the Fourier transform and we say that the system is stable.

**Definition**

**3.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 4. On the Scale-Invariant Systems: Hadamard Derivatives

#### 4.1. From the System to the Derivative

**Remark**

**6.**

**Definition**

**4.**

**Theorem**

**1.**

**Proof.**

- 1.
- From (27), we obtain$${\mathfrak{D}}_{s+}x\left(\tau \right)=\underset{q\to {1}^{+}}{lim}\frac{x\left(\tau \right)-x\left(\tau {q}^{-1}\right)}{lnq}$$$${\mathfrak{D}}_{s+}^{-1}x\left(\tau \right)=\underset{q\to {1}^{+}}{lim}lnq\sum _{n=0}^{\infty}x\left(\tau {q}^{-n}\right),$$
- 2.
- Ref. (28) gives$${\mathfrak{D}}_{s-}x\left(\tau \right)=\underset{q\to {1}^{+}}{lim}\frac{x\left(\tau q\right)-x\left(t\right)}{lnq}$$$${\mathfrak{D}}_{s-}^{-1}x\left(\tau \right)=-\underset{q\to {1}^{+}}{lim}lnq\sum _{n=0}^{\infty}x\left(\tau {q}^{n}\right),$$

**Example**

**1.**

- 1.
- Power functions: ${\tau}^{a}$We have$${\mathfrak{D}}_{s+}{\tau}^{a}=\underset{q\to {1}^{+}}{lim}\frac{{\tau}^{a}-{\tau}^{a}{q}^{-a}}{lnq}={\tau}^{a}\underset{q\to {1}^{+}}{lim}\frac{1-\left(1-aln\left(q\right)+{a}^{2}a{ln}^{2}\left(q\right)/2-\cdots \right)}{lnq}=a{\tau}^{a}$$$${\mathfrak{D}}_{s-}{\tau}^{a}=\underset{q\to {1}^{+}}{lim}\frac{{\tau}^{a}{q}^{a}-{\tau}^{a}}{lnq}={\tau}^{a}\underset{q\to {1}^{+}}{lim}\frac{{q}^{a}-1}{lnq}=a{\tau}^{a}$$
- 2.
- Logarithm: ${ln}^{a}\left(\tau \right)$As above, we obtain$$\begin{array}{cc}\hfill {\mathfrak{D}}_{s+}{ln}^{a}\left(\tau \right)& =\underset{q\to {1}^{+}}{lim}\frac{{ln}^{a}\left(\tau \right)-{\left(ln\left(\tau \right)-ln\left(q\right)\right)}^{a}}{ln\left(q\right)}\hfill \\ & ={ln}^{a}\left(\tau \right)\underset{q\to {1}^{+}}{lim}{\displaystyle \frac{a\frac{ln\left(q\right)}{ln\left(\tau \right)}-\frac{{a}^{2}}{2}{\left(\frac{ln\left(q\right)}{ln\left(\tau \right)}\right)}^{2}+\cdots}{ln\left(q\right)}}=a{ln}^{a-1}\left(\tau \right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathfrak{D}}_{s-}{ln}^{a}\left(\tau \right)& =\underset{q\to {1}^{+}}{lim}\frac{{\left(ln\left(\tau \right)+ln\left(q\right)\right)}^{a}-{ln}^{a}\left(\tau \right)}{ln\left(q\right)}\hfill \\ & ={ln}^{a}\left(\tau \right)\underset{q\to {1}^{+}}{lim}{\displaystyle \frac{a\frac{ln\left(q\right)}{ln\left(\tau \right)}+\frac{{a}^{2}}{2}{\left(\frac{ln\left(q\right)}{ln\left(\tau \right)}\right)}^{2}+\cdots}{ln\left(q\right)}}=a{ln}^{a-1}\left(\tau \right)\hfill \end{array}$$

#### 4.2. Properties of the Scale Derivatives

- LinearityIt is obvious from (26).
- Additivity and Commutativity of the orders$${\mathfrak{D}}_{s}^{\alpha}{\mathfrak{D}}_{s}^{\beta}x\left(\tau \right)={\mathfrak{D}}_{s}^{\alpha +\beta}x\left(\tau \right).$$This comes from (25).
- Neutral and inverse elementsLet $\alpha =-\beta .$ Then,$${\mathfrak{D}}_{s}^{\alpha}{\mathfrak{D}}_{s}^{-\alpha}x\left(\tau \right)={\mathfrak{D}}_{s-}^{0}x\left(\tau \right)=x\left(\tau \right).$$From (36), we conclude that there is always an inverse element—that is, for every $\alpha $ there is always the $-\alpha $ order that we call anti-derivative.
- The generalized Leibniz ruleThis rule gives the FD of the product of two functions and assumes the format of other fractional derivatives [31]$${\mathfrak{D}}_{s}^{\alpha}\left[x\left(\tau \right)y\left(\tau \right)\right]=\sum _{k=0}^{\infty}\left(\genfrac{}{}{0pt}{}{\alpha}{k}\right){\mathfrak{D}}_{s}^{k}x\left(\tau \right){\mathfrak{D}}_{s}^{\alpha -k}y\left(\tau \right).$$To prove this relation, we note first that$$\mathcal{M}\left[x\left(\tau \right)y\left(\tau \right)\right]=X\left(v\right)\u2605Y\left(v\right).$$Using the Bromwich inverse Mellin transform, we can write$${\mathfrak{D}}_{s}^{\alpha}\left[x\left(\tau \right)y\left(\tau \right)\right]={\displaystyle \frac{1}{2\pi i}}{\int}_{{\gamma}_{1}}{v}^{\alpha}{\int}_{{\gamma}_{2}}X\left(u\right)Y(v-u)du{\tau}^{v}dv,$$$${v}^{\alpha}={(v-u+u)}^{\alpha}={(v-u)}^{\alpha}{\left[1+{\displaystyle \frac{u}{v-u}}\right]}^{\alpha}=\sum _{k=0}^{\infty}\left(\genfrac{}{}{0pt}{}{\alpha}{k}\right){k}^{k}{(v-u)}^{\alpha -k},$$$${\mathfrak{D}}_{s}^{\alpha}\left[x\left(\tau \right)y\left(\tau \right)\right]=\sum _{k=0}^{\infty}\left(\genfrac{}{}{0pt}{}{\alpha}{k}\right){\displaystyle \frac{1}{2\pi i}}{\int}_{{\gamma}_{1}}{\int}_{{\gamma}_{2}}{u}^{k}X\left(u\right){(v-u)}^{\alpha -k}Y(v-u)du{\tau}^{v}dv,$$

#### 4.3. Relation with Classic and Quantum Derivatives

#### 4.4. Scale Conversion: Logarithmic Series

**Theorem**

**2.**

**Proof.**

**Example**

**2.**

**Corollary**

**1.**

#### 4.5. Hadamard Derivatives

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**4.**

- 1.
- 2.
- 3.
- Hadamard–Liouville right derivative$${\mathfrak{D}}_{s+}^{\alpha}x\left(\tau \right)={\displaystyle \frac{1}{\Gamma (N-\alpha )}}{\int}_{1}^{\infty}\left[{\mathfrak{D}}_{s+}^{N}x\left(\tau \right)\right]{ln}^{N-\alpha -1}(\tau /u){\displaystyle \frac{du}{u}}$$
- 4.
- Hadamard–Liouville left derivative$${\mathfrak{D}}_{s-}^{\alpha}x\left(\tau \right)={\displaystyle \frac{1}{\Gamma (N-\alpha )}}{\int}_{0}^{1}\left[{\mathfrak{D}}_{s-}^{N}x\left(\tau \right)\right]{ln}^{N-\alpha -1}(u/\tau ){\displaystyle \frac{du}{u}}$$

**Remark**

**7.**

**Definition**

**5.**

#### 4.6. Tempered Scale-Invariant Derivatives

- 1.
- Forward Grünwald–Letnikov$${D}_{\lambda ,f}^{\alpha}f\left(t\right)=\underset{h\to {0}^{+}}{lim}{h}^{-\alpha}\sum _{n=0}^{\infty}\frac{{(-\alpha )}_{n}}{n!}{e}^{-n\lambda h}f(t-nh),$$
- 2.
- Forward regularized derivative$${D}_{\lambda ,f}^{\alpha}f\left(t\right)={\int}_{0}^{\infty}\left[f(t-\tau )-\sum _{0}^{N-1}\frac{{(-1)}^{m}{f}^{\left(m\right)}\left(t\right)}{m!}{\tau}^{m}\right]{e}^{-\lambda \tau}\frac{{\tau}^{-\alpha -1}}{\Gamma (-\alpha )}d\tau $$

**Definition**

**6.**

**Theorem**

**5.**

## 5. Scale-Invariant Systems

**Definition**

**7.**

**Example**

**3.**

**Remark**

**8.**

**Example**

**4.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Ortigueira, M.D.; Bohannan, G.W.
Fractional Scale Calculus: Hadamard vs. Liouville. *Fractal Fract.* **2023**, *7*, 296.
https://doi.org/10.3390/fractalfract7040296

**AMA Style**

Ortigueira MD, Bohannan GW.
Fractional Scale Calculus: Hadamard vs. Liouville. *Fractal and Fractional*. 2023; 7(4):296.
https://doi.org/10.3390/fractalfract7040296

**Chicago/Turabian Style**

Ortigueira, Manuel D., and Gary W. Bohannan.
2023. "Fractional Scale Calculus: Hadamard vs. Liouville" *Fractal and Fractional* 7, no. 4: 296.
https://doi.org/10.3390/fractalfract7040296