# Modeling Alcohol Concentration in Blood via a Fractional Context

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**3.**

- If there exists ${D}^{\lceil \alpha \rceil}f$ at the point $t\in I$, then f is ${G}_{T}^{\alpha}$-differentiable at t and$${G}_{T}^{\alpha}f(t)=T{(t,\alpha )}^{\lceil \alpha \rceil}{D}^{\lceil \alpha \rceil}f(t).$$
- If $\alpha \in (0,1]$, then f is ${G}_{T}^{\alpha}$-differentiable at $t\in I$ if and only if f is differentiable at t; in this case, we have$${G}_{T}^{\alpha}f(t)=T(t,\alpha ){f}^{\prime}(t).$$

**Theorem**

**4.**

- $af+b\phantom{\rule{0.166667em}{0ex}}g$ is ${G}_{T}^{\alpha}$-differentiable at t for every $a,b\in \mathbb{R}$, and$${G}_{T}^{\alpha}(af+b\phantom{\rule{0.166667em}{0ex}}g)(t)=a\phantom{\rule{0.166667em}{0ex}}{G}_{T}^{\alpha}f(t)+b\phantom{\rule{0.166667em}{0ex}}{G}_{T}^{\alpha}g(t).$$
- If $\alpha \in (0,1]$, then $fg$ is ${G}_{T}^{\alpha}$-differentiable at t and$${G}_{T}^{\alpha}(fg)(t)=f(t){G}_{T}^{\alpha}g(t)+g(t){G}_{T}^{\alpha}f(t).$$
- If $\alpha \in (0,1]$ and $g(t)\ne 0$, then $f/g$ is ${G}_{T}^{\alpha}$-differentiable at t and$${G}_{T}^{\alpha}(\frac{f}{g})(t)=\frac{g(t){G}_{T}^{\alpha}f(t)-f(t){G}_{T}^{\alpha}g(t)}{g{(t)}^{2}}.$$
- ${G}_{T}^{\alpha}(\lambda )=0$, for every $\lambda \in \mathbb{R}.$
- ${G}_{T}^{\alpha}({t}^{p})=\frac{\mathsf{\Gamma}(p+1)}{\mathsf{\Gamma}(p-\lceil \alpha \rceil +1)}{t}^{p-\lceil \alpha \rceil}T{(t,\alpha )}^{\lceil \alpha \rceil}$, for every$p\in \mathbb{R}\backslash {\mathbb{Z}}^{-}\phantom{\rule{-0.166667em}{0ex}}.$
- ${G}_{T}^{\alpha}({t}^{-n})={(-1)}^{\lceil \alpha \rceil}\frac{\mathsf{\Gamma}(n+\lceil \alpha \rceil )}{\mathsf{\Gamma}(n)}\phantom{\rule{0.166667em}{0ex}}{t}^{-n-\lceil \alpha \rceil}T{(t,\alpha )}^{\lceil \alpha \rceil}$, for every$n\in {\mathbb{Z}}^{+}.$

**Theorem**

**5.**

**Theorem**

**6**(Chain Rule)

**.**

## 2. Results

**Proposition**

**7.**

**Proof.**

## 3. Estimation for Comformable Gaussian Models

**0.01756749**), ${G}_{T}^{\alpha}$ with $T(t,\alpha )={e}^{(\alpha -1)t}$ (0.02370243), Grünwald–Letnikov derivative (0.06573368) and ordinary derivative (0.03790216).

**0.01563364**), Grünwald–Letnikov derivative (1.74848946), and ordinary derivative (0.52879824).

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Derivative | $\mathit{\alpha}$ | a | b | c | $\mathit{\tau}$ |
---|---|---|---|---|---|

Ordinary | $\phantom{\rule{2.em}{0ex}}-$ | 0.8951823 | 0.8119931 | 2.6501296 | 0.1195180 |

${G}_{T}^{\alpha}$; $T(t,\alpha )={e}^{(1-\alpha )t}$ | 0.6728009 | 0.9622524 | 0.8831519 | 1.5542520 | 0.1033001 |

Khalil et al. | 0.4848604 | 0.9894414 | 0.6424266 | 2.0477500 | 0.0974800 |

Grünwald–Letnikov | 0.7027876 | - | 1.7451992 | 2.2566254 | 0.1232454 |

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**MDPI and ACS Style**

Rosario Cayetano, O.; Fleitas Imbert, A.; Gómez-Aguilar, J.F.; Sarmiento Galán, A.F.
Modeling Alcohol Concentration in Blood via a Fractional Context. *Symmetry* **2020**, *12*, 459.
https://doi.org/10.3390/sym12030459

**AMA Style**

Rosario Cayetano O, Fleitas Imbert A, Gómez-Aguilar JF, Sarmiento Galán AF.
Modeling Alcohol Concentration in Blood via a Fractional Context. *Symmetry*. 2020; 12(3):459.
https://doi.org/10.3390/sym12030459

**Chicago/Turabian Style**

Rosario Cayetano, Omar, Alberto Fleitas Imbert, José Francisco Gómez-Aguilar, and Antonio Fernando Sarmiento Galán.
2020. "Modeling Alcohol Concentration in Blood via a Fractional Context" *Symmetry* 12, no. 3: 459.
https://doi.org/10.3390/sym12030459