# On a System of Equations with General Fractional Derivatives Arising in Diffusion Theory

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## Abstract

**:**

## 1. Introduction

## 2. Two-Compartment System with General Fractional Derivatives

- Case I

- Case II

## 3. Solution of (10), (22) and (10), (23)

- We first treat (22). By using (17), we obtain$$\begin{array}{ccc}& & \left(\right)open="["\; close="]">a\frac{s}{{\left(\right)}^{s}}+b\frac{s}{{\left(\right)}^{s}}\hfill \\ {\tilde{m}}_{1}\left(s\right)\end{array}$$$$\left(\right)open="["\; close="]">{\tilde{m}}_{1}\left(s\right)+{\tilde{m}}_{2}\left(s\right)$$$${m}_{1}\left(t\right)+{m}_{2}\left(t\right)={m}_{10},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t\ge 0.$$

- Applying the Laplace transform to (23) and using (17), we obtain$$\begin{array}{ccc}& & \left(\right)open="["\; close="]">\frac{a}{{\left(\right)}^{a}}1-\beta \hfill & +\frac{b\left(\right)open="("\; close=")">s+{\lambda}_{2}}{-}\left(\right)open="("\; close=")">s+{\lambda}_{2}ln\left(\right)open="["\; close="]">b\left(\right)open="("\; close=")">s+{\lambda}_{2}\end{array}$$$$\begin{array}{ccc}\hfill {\tilde{m}}_{2}\left(s\right)& =& \frac{1}{{V}_{1}}\frac{k\phantom{\rule{0.277778em}{0ex}}{m}_{10}}{s\left(\right)open="["\; close="]">s\frac{a}{{\left(\right)}^{a}}1-\beta}+s\frac{b\left(\right)open="("\; close=")">s+{\lambda}_{2}}{-}\left(\right)open="("\; close=")">s+{\lambda}_{2}ln\left(\right)open="["\; close="]">b\left(\right)open="("\; close=")">s+{\lambda}_{2}\hfill \end{array}$$

## 4. Numerical Results

#### 4.1. Gentamicin Release from Poly(vinyl Alcohol)/Chitosan/Gentamicin (PVA/CHI/Gent) Hydrogel

#### 4.2. Gentamicin Release from Bioceramic Hydroxyapatite/Poly(vinyl Alcohol)/Chitosan/Gentamicin (HAP/PVA/CHI/Gent) Coating on Titanium

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Increase in relative mass of released gentamicin from HAP/PVA/CS/Gent coating, ${m}_{2}$, and decrease in relative mass of gentamicin inside the HAP/PVA/CS/Gent coating, ${m}_{1}$, with time, obtained from (36) with parameters (38) (solid line, together with measured values $\phantom{\rule{0.277778em}{0ex}}\u29eb$.

**Table 1.**Measured values of ${m}_{2}$ and calculated values of ${m}_{1}=1-{m}_{2}$ at different time instants for experiment Section 4.1.

t (Days) | m${}_{1}$ _{calculated} | m${}_{2}$ _{measured} |
---|---|---|

0 | 1 | 0 |

1 | 0.72389 | 0.27611 |

2 | 0.33093 | 0.69067 |

4 | 0.2612 | 0.73880 |

7 | 0.2522 | 0.74771 |

14 | 0.25164 | 0.74836 |

**Table 2.**Measured values of ${m}_{2}$ and calculated values of ${m}_{1}=1-{m}_{2}$ at different time instants for experiment Section 4.2.

t (Days) | m${}_{1}$_{calculated} | m${}_{2}$_{measured} |
---|---|---|

0 | 1 | 0 |

1 | 0.78 | 0.22 |

2 | 0.70 | 0.30 |

7 | 0.68 | 0.32 |

14 | 0.69 | 0.31 |

21 | 0.69 | 0.31 |

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**MDPI and ACS Style**

Miskovic-Stankovic, V.; Atanackovic, T.M.
On a System of Equations with General Fractional Derivatives Arising in Diffusion Theory. *Fractal Fract.* **2023**, *7*, 518.
https://doi.org/10.3390/fractalfract7070518

**AMA Style**

Miskovic-Stankovic V, Atanackovic TM.
On a System of Equations with General Fractional Derivatives Arising in Diffusion Theory. *Fractal and Fractional*. 2023; 7(7):518.
https://doi.org/10.3390/fractalfract7070518

**Chicago/Turabian Style**

Miskovic-Stankovic, Vesna, and Teodor M. Atanackovic.
2023. "On a System of Equations with General Fractional Derivatives Arising in Diffusion Theory" *Fractal and Fractional* 7, no. 7: 518.
https://doi.org/10.3390/fractalfract7070518