# Probabilistic Interpretations of Fractional Operators and Fractional Behaviours: Extensions, Applications and Tribute to Prof. José Tenreiro Machado’s Ideas

## Abstract

**:**

## 1. Introduction

## 2. Professor Tenreiro Machado’s Probabilistic Interpretation of the Fractional Derivative Operator

- -
- the “present”, namely the sample $x\left(0\right)$, is seen with probability one,
- -
- each sample of the past, namely the samples $x\left(-kh\right)$ with $k\in \left[1,\infty \right[$ is weighted with the probability $\gamma \left(\nu ,k\right)$ and the expression $-{\displaystyle \sum}_{k=0}^{\infty}\gamma \left(\nu ,k\right)x\left(t-kh\right)$ can be viewed as the expected value of the random variable $X$, $E\left(X\right)$, such that $P\left(X=x\left(kh\right)\right)=\left|\gamma \left(\nu ,k\right)\right|$, $k\in \left[1,\infty \right[$.

## 3. Probabilistic Interpretation Based on the Spectral Content of Operators with Fractional Behaviours

- -
- generate a random number ${u}_{k}$from the standard uniform distribution in the interval [0, 1];
- -
- compute${x}_{k}={F}_{c}^{-1}\left({u}_{k}\right)$.

## 4. Probabilistic Interpretation Based on the Delay Approximation of Operators with Fractional Behaviours

#### 4.1. Description of Adsorption Phenomena

#### 4.2. Time Delay Distribution for Fitting Fractional Behaviours

#### 4.3. Adsorption Phenomena to Illustrate the Interest of this Probabilistic Delay Interpretation

#### 4.4. Another Example of Possible Physical Interpretation

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The $\mathsf{\Gamma}$ path used for the computation of the impulse response $h\left(t\right)$.

**Figure 3.**Function ${F}_{c}\left(y\right)$ of relation (14) for various values of $\nu $ and $a=10$.

**Figure 4.**Cumulative distribution function provided by relation (14) (

**left**) and the histogram of ${x}_{k}$ obtained (

**right**) in the case of $\nu =0.4$ and $a=10$.

**Figure 5.**Comparison of the gain (

**left**) and phase (

**right**) diagrams of the transfer functions ${F}_{i}\left(s\right)$ and ${F}_{f}\left(s\right)$.

**Figure 7.**Approximation of a fractional integrator step response by a distribution of delayed steps.

**Figure 8.**Frequency response of the approximation of a fractional integrator time response by a distribution of delayed steps.

**Figure 9.**Approximation of a fractional integrator time response by a recursive distribution of delayed steps.

**Figure 10.**Approximation of a fractional integrator step response by a recursive distribution of delayed steps (relation (28)).

**Figure 11.**Comparison of the fractional integrator step response and the approximation (29) (

**left**) and frequency response of the approximation (29) (

**right**).

**Figure 12.**Comparison of the step response of the fractional filter in relation (14) and its approximation.

**Figure 13.**Distribution of coefficients ${M}_{k}/{K}_{f}$ in relation (31) for $k\in \left[1,100\right]$.

**Figure 14.**Surface covered as a function of trials during the RSA process (

**left**) and disk placement at the end of the process (

**right**).

**Figure 16.**Interpretation in terms of time delay distribution of the fractional behaviour produced by diffusion phenomena.

**Table 1.**Table of impulse responses (inverse Laplace transforms) of some transfer functions with fractional behaviours.

${\mathit{H}}_{\mathit{i}}\left(\mathit{s}\right)$ | ${\mathit{h}}_{\mathit{i}}\left(\mathit{t}\right)$ |
---|---|

$\frac{{{\displaystyle \sum}}_{l=0}^{L}{b}_{l}{s}^{{\beta}_{l}}}{{{\displaystyle \sum}}_{k=0}^{K}{a}_{k}{s}^{{\alpha}_{k}}}$ | $\sum _{i=1}^{r}}{\displaystyle \sum _{j=1}^{{n}_{i}}}{r}_{ij}{Y}_{j}\left(t\right){e}^{{s}_{i}t}+\frac{1}{\pi}{{\displaystyle \int}}_{0}^{+\infty}\frac{{{\displaystyle \sum}}_{k=0}^{K}{{\displaystyle \sum}}_{l=0}^{L}{a}_{k}{b}_{l}\mathrm{sin}\left(\left({\alpha}_{k}-{\beta}_{l}\right)\pi \right){x}^{{\alpha}_{k}+{\beta}_{l}}}{{{\displaystyle \sum}}_{k=0}^{K}{a}_{k}^{2}{x}^{2{\alpha}_{k}}+{{\displaystyle \sum}}_{0\le k\le l\le K}{a}_{k}{a}_{l}cos\left(\left({\alpha}_{k}-{\alpha}_{l}\right)\pi \right){x}^{{\alpha}_{k}+{\alpha}_{l}}}{e}^{-xt}dx$ From [15] with demonstration in [17]. ${s}_{i}$ are the poles. |

$\frac{1}{{s}^{\nu}}$ | $\frac{\mathrm{sin}\left(\nu \pi \right)}{\pi}\underset{0}{\overset{+\infty}{{\displaystyle \int}}}\frac{1}{{x}^{\nu}}{e}^{-xt}dx$ |

$\frac{1}{{\left(\frac{s}{{\omega}_{l}}+1\right)}^{\nu}}$ | $\frac{\mathrm{sin}\left(\nu \pi \right)}{\pi}\underset{{\omega}_{l}}{\overset{+\infty}{{\displaystyle \int}}}\frac{{\omega}_{l}{}^{\nu}}{{\left(x-{\omega}_{l}\right)}^{\nu}}{e}^{-xt}dx$ |

$\frac{{\left(\frac{s}{{\omega}_{h}}+1\right)}^{\nu -1}}{{\left(\frac{s}{{\omega}_{l}}+1\right)}^{\nu}}$ | $\frac{\mathrm{sin}\left(\nu \pi \right)}{\pi}\frac{{\omega}_{l}{}^{\nu}}{{\omega}_{h}{}^{\nu -1}}\underset{{\omega}_{l}}{\overset{{\omega}_{h}}{{\displaystyle \int}}}\frac{{\left({\omega}_{h}-x\right)}^{\nu -1}}{{\left(x-{\omega}_{l}\right)}^{\nu}}{e}^{-xt}dx$ |

$\frac{{\left(\frac{s}{{\omega}_{h}}+1\right)}^{\nu}}{{\left(\frac{s}{{\omega}_{l}}+1\right)}^{\nu}}$ | ${\left(\frac{{\omega}_{l}}{{\omega}_{h}}\right)}^{\nu}\left(\delta \left(t\right)+\frac{\mathrm{sin}\left(\nu \pi \right)}{\pi}\underset{{\omega}_{b}}{\overset{{\omega}_{h}}{{\displaystyle \int}}}\frac{{\left({\omega}_{h}-x\right)}^{\nu}}{{\left(x-{\omega}_{l}\right)}^{\nu}\left(s+x\right)}dx\right)$ $\delta \left(t\right)$: Dirac impulse |

$\frac{1}{s}\frac{{\left(\frac{s}{{\omega}_{l}}+1\right)}^{1-\nu}}{{\left(\frac{s}{{\omega}_{h}}+1\right)}^{1-\nu}}$ | ${\left(\frac{{\omega}_{h}}{{\omega}_{l}}\right)}^{1-\nu}\left({H}_{e}\left(t\right)+\frac{\mathrm{sin}\left(\left(1-\nu \right)\pi \right)}{\pi}\underset{{\omega}_{l}}{\overset{{\omega}_{h}}{{\displaystyle \int}}}\frac{{\left(x-{\omega}_{l}\right)}^{1-\nu}}{x{\left({\omega}_{h}-x\right)}^{1-\nu}}{e}^{-xt}dx\right)$ ${H}_{e}\left(t\right)$: Heaviside step function |

$\frac{{e}^{-z\sqrt{as+b}}}{\sqrt{as+b}}a0,b0,z0$ | $\frac{1}{\pi}\underset{\raisebox{1ex}{$\mathrm{b}$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}{\overset{+\infty}{{\displaystyle \int}}}\frac{\mathrm{cos}\left(z\sqrt{ax-b}\right)}{\sqrt{ax-b}}{e}^{-xt}dx$ |

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**MDPI and ACS Style**

Sabatier, J.
Probabilistic Interpretations of Fractional Operators and Fractional Behaviours: Extensions, Applications and Tribute to Prof. José Tenreiro Machado’s Ideas. *Mathematics* **2022**, *10*, 4184.
https://doi.org/10.3390/math10224184

**AMA Style**

Sabatier J.
Probabilistic Interpretations of Fractional Operators and Fractional Behaviours: Extensions, Applications and Tribute to Prof. José Tenreiro Machado’s Ideas. *Mathematics*. 2022; 10(22):4184.
https://doi.org/10.3390/math10224184

**Chicago/Turabian Style**

Sabatier, Jocelyn.
2022. "Probabilistic Interpretations of Fractional Operators and Fractional Behaviours: Extensions, Applications and Tribute to Prof. José Tenreiro Machado’s Ideas" *Mathematics* 10, no. 22: 4184.
https://doi.org/10.3390/math10224184