# Fractal Logistic Equation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Basic Tools

#### 2.1. Middle-$\kappa $ Cantor Set

- 1-Step 1.
- Pick up an open interval of length $0<\kappa <1$ from the middle of the $I=[0,1]$.$${C}_{1}^{\kappa}=[0,\frac{1}{2}(1-\kappa )]\cup [\frac{1}{2}(1+\kappa ),1].$$
- 2-Step 2.
- Delete disjoint open intervals of length $\kappa $ from the middle of the remaining closed intervals of step 1.$$\begin{array}{c}{C}_{2}^{\kappa}=[0,\frac{1}{4}{(1-\kappa )}^{2}]\cup [\frac{1}{4}(1-{\kappa}^{2}),\frac{1}{2}(1-\kappa )]\cup [\frac{1}{2}(1+\kappa )+\frac{1}{2}((1+\kappa )\hfill \\ +\frac{1}{2}{(1-\kappa )}^{2})]\cup [\frac{1}{2}(1+\kappa )(1+\frac{1}{2}(1-\kappa )),1].\hfill \end{array}$$
- 3-Step m.
- Remove disjoint open intervals of length $\kappa $ from the middle of the remaining closed intervals of step m-1.$${C}^{\kappa}=\bigcap _{m=1}^{\infty}{C}_{m}^{\kappa}.$$

#### 2.2. Local Fractal Calculus

**mass function**${\gamma}^{\alpha}({C}^{\kappa},{b}_{1},{b}_{2})$ is defined in [57,58,60] by

**integral staircase function**${S}_{{C}^{\kappa}}^{\alpha}(t)$ is defined in [57,58] by

**$\gamma $-dimension**of a set ${C}^{\kappa}\cap [{b}_{1},{b}_{2}]$ is defined

**limit**of a function $g:\Re \to \Re $ is given by

**${C}^{\alpha}$-continuity**of a function $k:\Re \to \Re $ is defined by

**${C}^{\alpha}$-derivative**of $f(t)$ at t is defined [57]

**${C}^{\alpha}$-integral**of $k(t)$ on $J=[{b}_{1},{b}_{2}]$ is defined in [57,58,60] and approximately given by

**Characteristic function of the middle-$\kappa $ Cantor set**is defined in [60] by

**Some important formulas:**

## 3. Fractal Finite Difference and Fractal Derivative

**Fractal difference operator**is denoted by ${\Delta}_{K}$ and defined by

**Example**

**1.**

## 4. Fractal Difference and Differential Equations

## 5. Numerical Method for Solving Fractal Differential Equation

**fractal Euler method**.

**fractal local truncation error**(FLTE) is given by

**fractal global truncation error**is given by

**fractal Lipschitz continuous**, namely

**Example**

**2.**

**Example**

**3.**

## 6. Fractal Logistic Equation

**fractal growth parameter**and ${r}_{K}^{\prime}$ is called

**fractal carrying capacity**. Applying conjugacy of fractal calculus and standard calculus we obtain the solution of Equation (52) as follows:

**fractal Logistic function**. In Figure 3 we have plotted Equation (53).

**Remark:**

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Figures for the Section 2.

**Figure 2.**Exact solution and approximate solution using numerical fractal Euler method where step sizes $h=1$ and $\alpha =0.63$.

**Figure 3.**Fractal logistic curve with $\alpha =0.5$ where $\mu =1/2$. $z(0)=3,\phantom{\rule{3.33333pt}{0ex}}{r}_{K}=100,\phantom{\rule{3.33333pt}{0ex}}{r}_{K}^{\prime}=0.2$.

**Figure 5.**The inflection point as function of $\alpha $ by choosing $z(0)=3,\phantom{\rule{3.33333pt}{0ex}}{r}_{K}=100,\phantom{\rule{3.33333pt}{0ex}}{r}_{K}^{\prime}=0.2$.

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**MDPI and ACS Style**

Khalili Golmankhaneh, A.; Cattani, C.
Fractal Logistic Equation. *Fractal Fract.* **2019**, *3*, 41.
https://doi.org/10.3390/fractalfract3030041

**AMA Style**

Khalili Golmankhaneh A, Cattani C.
Fractal Logistic Equation. *Fractal and Fractional*. 2019; 3(3):41.
https://doi.org/10.3390/fractalfract3030041

**Chicago/Turabian Style**

Khalili Golmankhaneh, Alireza, and Carlo Cattani.
2019. "Fractal Logistic Equation" *Fractal and Fractional* 3, no. 3: 41.
https://doi.org/10.3390/fractalfract3030041