# Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel

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## Abstract

**:**

## 1. Introduction

## 2. Basic Definition and Notations

## 3. Prabhakar Fractional Logistic Equation and Its Limiting Cases

#### 3.1. Fractional Liouville–Caputo Logistic Differential Equation

#### 3.2. Atangana–Baleanu Logistic Differential Equation

#### 3.3. Caputo–Fabrizio Logistic Differential Equation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Logistic function solution to Equation (1) with $x\left(0\right)=1/2$, in blue, as well approximations of the function by the corresponding Taylor polynomials in $[1,2]$: $n=3$ in orange color, $n=5$ in green color, $n=7$ in red color, and $n=9$ in grey color.

**Figure 2.**Logistic function solution to Equation (1) with $x\left(0\right)=1/2$, in blue, as well as some approximations of the solution to the Caputo fractional logistic differential Equation (20) in $[0,2]$ for $\alpha =0.75$, in orange. From left to right and top to bottom the approximations are shown for $n=3$, $n=5$, $n=7$, and $n=9$. From these figures, one must use $\alpha $ closer to one as shown in Figure 3 in order to approximate the classical solution.

**Figure 3.**Logistic function solution to (1) with $x\left(0\right)=1/2$, in blue, as well as some approximations of the solution to the Caputo fractional logistic differential Equation (20) in $[0,2]$ for $\alpha =0.95$, in orange. From left to right and top to bottom the approximations are shown for $n=3$, $n=5$, $n=7$, and $n=9$.

**Figure 4.**Logistic function solution to (1) with $x\left(0\right)=1/2$, in blue, as well as some approximations of the solution to the Atangana–Baleanu logistic differential equation in $[0,2]$ for $\alpha =0.75$, in orange. From left to right and top to bottom the approximations are shown for $n=3$, $n=5$, $n=7$, and $n=9$. From these figures, one must use $\alpha $ closer to one as shown in Figure 5 in order to approximate the classical solution.

**Figure 5.**Logistic function solution to (1) with $x\left(0\right)=1/2$, in blue, as well as some approximations of the solution to the Atangana–Baleanu logistic differential equation in $[0,2]$ for $\alpha =0.95$, in orange. From left to right and top to bottom the approximations are shown for $n=3$, $n=5$, $n=7$, and $n=9$.

**Figure 6.**Logistic function solution to (1) with $x\left(0\right)=1/2$, in blue, as well as some approximations of the solution to the Caputo–Fabrizio logistic differential Equation (29) in $[0,2]$ for $\alpha =0.75$, in orange. From left to right and top to bottom the approximations are shown for $n=3$, $n=5$, $n=7$, and $n=9$. As in the previous cases, from these figures one must use $\alpha $ closer to one as shown in Figure 7 in order to approximate the classical solution.

**Figure 7.**Logistic function solution to (1) with $x\left(0\right)=1/2$, in blue, as well as some approximations of the solution to the Caputo–Fabrizio logistic differential Equation (29) in $[0,2]$ for $\alpha =0.95$, in orange. From left to right and top to bottom the approximations are shown for $n=3$, $n=5$, $n=7$, and $n=9$.

**Figure 8.**In $[0,2]$ for $\alpha =0.9$, logistic function solution to (1) with $x\left(0\right)=1/2$, in blue, some approximations of the solution to the Caputo–Fabrizio logistic differential Equation (29), as well as the solution given in [18] in orange. From left to right and top to bottom the approximations are shown for $n=3$, $n=5$, $n=7$, and $n=9$.

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**MDPI and ACS Style**

Area, I.; Nieto, J.J.
Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel. *Fractal Fract.* **2021**, *5*, 273.
https://doi.org/10.3390/fractalfract5040273

**AMA Style**

Area I, Nieto JJ.
Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel. *Fractal and Fractional*. 2021; 5(4):273.
https://doi.org/10.3390/fractalfract5040273

**Chicago/Turabian Style**

Area, Iván, and Juan J. Nieto.
2021. "Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel" *Fractal and Fractional* 5, no. 4: 273.
https://doi.org/10.3390/fractalfract5040273