# A Discretization Approach for the Nonlinear Fractional Logistic Equation

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## Abstract

**:**

## 1. Introduction

## 2. Fractional Calculus

**Definition**

**1.**

- (1)
- ${\mathcal{I}}^{\nu}{\mathcal{I}}^{\beta}f\left(t\right)={\mathcal{I}}^{\nu +\beta}f\left(t\right),$
- (2)
- ${\mathcal{I}}^{\nu}\left({c}_{1}f\left(t\right)+{c}_{2}g\left(t\right)\right)={c}_{1}{\mathcal{I}}^{\nu}f\left(t\right)+{c}_{2}{\mathcal{I}}^{\nu}g\left(t\right),\phantom{\rule{1.em}{0ex}}{c}_{1},{c}_{2}\in \mathbb{R},$
- (3)
- ${\mathcal{I}}^{\nu}{t}^{\gamma}=\frac{\Gamma \left(\gamma +1\right)}{\Gamma \left(\gamma +\nu +1\right)}{t}^{\nu +\gamma},\phantom{\rule{1.em}{0ex}}\gamma >-1.$

**Definition**

**2.**

## 3. Discretized LDG Formulation

#### Algebraic Formulation

## 4. Numerical Stability and Error Estimates

**Lemma**

**1.**

**Lemma**

**2.**

## 5. Numerical Results and Discussions

#### 5.1. Linear Model

#### 5.2. Nonlinear Model

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The approximated LDG with exact solutions (

**left**) and the corresponding absolute errors (

**right**) for $J=1$, $\nu =1$, $\sigma =1,\phantom{\rule{3.33333pt}{0ex}}{X}_{0}=0.75$, and different $r=3,6,9$.

**Figure 2.**The approximated LDG with exact solutions (

**left**) and the corresponding absolute errors (

**right**) for $J=1$, $r=5$, $\sigma =1,{X}_{0}=0.75$, and various values of $\nu =0.65,0.75,0.85,0.95$.

**Figure 3.**Numerical solutions of LDG scheme using P.A. and D.C. approaches with $h=0.2$, $\sigma =0.5$, ${X}_{0}=0.5$, and $\nu =1.0$. The magnification of solutions at time $t=0.4$ is plotted in the box. The exact solution is displayed by a solid line.

**Figure 4.**Absolute errors of LDG versus time using D.C. (

**left**) and P.A. (

**right**) approaches with $h=0.2$, $\sigma =0.5$, ${X}_{0}=0.5$, $\nu =1.0$, and $r=2$. In the left and right plots, the upwind and downwind points are highlighted by black pentagon.

**Figure 5.**Absolute-errors versus polynomial degrees r for $J=1,2,4$ (

**left**) and against the number of elements J for $r=0,1,2,3$ (

**right**) evaluated at $T=1.0$ and for $\nu =1$.

**Figure 6.**The approximated LDG solutions versus time using P.A. (

**left**) and D.C. (

**right**) approaches with $J=4$, $r=3$, $\sigma ,{X}_{0}=0.5$, and various values of $\nu =0.65,0.75,0.85,0.95,1.0$.

**Table 1.**Comparison of absolute errors in LDG with $r=2$ and PECE for different number of interval J and $\nu =1$. Numbers in bold show that the correct digits are obtained by the LDG.

LDG | PECE | |||||
---|---|---|---|---|---|---|

$\mathit{J}$ | ${\mathcal{Z}}_{\mathbf{0}}\left(\mathbf{2}\right)$ | $|\mathit{X}\left(\mathbf{2}\right)-{\mathcal{Z}}_{\mathbf{0}}\left(\mathbf{2}\right)|$ | EOC | Numerical | Error | EOC |

1 | 5.625000000000 | ${8.3208}_{-2}$ | − | $3.750000000000$ | ${1.7918}_{+0}$ | − |

2 | 5.543701171875 | ${1.9091}_{-3}$ | $5.45$ | $4.687500000000$ | ${0.8543}_{+0}$ | $1.07$ |

4 | $\mathbf{5.541}845071676$ | ${5.2998}_{-5}$ | $5.17$ | 5.229675292969 | ${0.3121}_{+0}$ | $1.45$ |

8 | $\mathbf{5.54179}3647744$ | ${1.5735}_{-6}$ | $5.07$ | 5.446685392454 | ${9.5107}_{-2}$ | $1.71$ |

16 | $\mathbf{5.541792}122228$ | ${4.8030}_{-8}$ | $5.03$ | $\mathbf{5.5}15562177333$ | ${2.6230}_{-2}$ | $1.86$ |

32 | $\mathbf{5.54179207}5682$ | ${1.4842}_{-9}$ | $5.02$ | $\mathbf{5.5}34910274764$ | ${6.8817}_{-3}$ | $1.93$ |

64 | $\mathbf{5.541792074}244$ | ${4.6126}_{-11}$ | $5.01$ | $\mathbf{5.54}0030137766$ | ${1.7619}_{-3}$ | $1.97$ |

128 | $\mathbf{5.54179207419}9$ | ${1.4380}_{-12}$ | $5.00$ | $\mathbf{5.541}346351966$ | ${4.4572}_{-4}$ | $1.98$ |

**Table 2.**Comparison of numerical solutions in LDG with $r=5$, $h=1$ and L1/fast L1 schemes with $h={10}^{-3}$ for some $t\in [0,1]$ and $\nu =0.75,0.5$.

$\mathit{\nu}=0.75$ | $\mathit{\nu}=0.5$ | |||||||
---|---|---|---|---|---|---|---|---|

$\mathit{t}$ | LDG | L1 | Fast L1 | Exact | LDG | L1 | Fast L1 | Exact |

$0.2$ | $1.0536$ | $1.0524$ | $1.0524$ | $1.053507$ | $1.3420$ | $1.3459$ | $1.3345$ | $1.349263$ |

0.4 | $1.3512$ | $1.3486$ | $1.3486$ | $1.350342$ | $1.8370$ | $1.8176$ | $1.8176$ | $1.822532$ |

$0.6$ | $1.6963$ | $1.6945$ | $1.6945$ | $1.697186$ | $2.3489$ | $2.3525$ | $2.3525$ | $2.359660$ |

$0.8$ | $2.1128$ | $2.1087$ | $2.1087$ | $2.112499$ | $2.9957$ | $2.9845$ | $2.9845$ | $2.994627$ |

$1.0$ | $2.6134$ | $2.6091$ | $2.6091$ | $2.614400$ | $3.7385$ | $3.7427$ | $3.7427$ | $3.756735$ |

**Table 3.**Comparison of absolute errors in LDG with $r=1$ using P.A. and D.C. for different number of interval J and $\nu =1$. Numbers in bold show that the correct digits are obtained by the LDG.

P.A. | D.C. | |||||
---|---|---|---|---|---|---|

$\mathit{J}$ | ${\mathcal{Z}}_{\mathbf{0}}\left(\mathbf{1}\right)$ | $|\mathit{X}\left(\mathbf{1}\right)-{\mathcal{Z}}_{\mathbf{0}}\left(\mathbf{1}\right)|$ | EOC | ${\mathcal{Z}}_{\mathbf{0}}\left(\mathbf{1}\right)$ | $|\mathit{X}\left(\mathbf{1}\right)-{\mathcal{Z}}_{\mathbf{0}}\left(\mathbf{1}\right)|$ | EOC |

1 | 0.6234038976 | ${0.9445664060}_{-3}$ | − | $\mathbf{0.6224}742460$ | ${0.1491482269}_{-4}$ | − |

2 | 0.6226973939 | ${0.2380627190}_{-3}$ | $1.99$ | $\mathbf{0.6224}610781$ | ${0.1746857403}_{-5}$ | $3.09$ |

4 | $\mathbf{0.622}5290166$ | ${0.6968541429}_{-4}$ | $1.77$ | $\mathbf{0.622459}5421$ | ${0.2108842001}_{-6}$ | $3.05$ |

**Table 4.**Comparison of absolute errors in LDG with $r=2$ using P.A. and D.C. for different number of interval J and $\nu =1$. Numbers in bold show that the correct digits are obtained by the LDG.

P.A. | D.C. | |||||
---|---|---|---|---|---|---|

$\mathit{J}$ | ${\mathcal{Z}}_{\mathbf{0}}\left(\mathbf{1}\right)$ | $|\mathit{X}\left(\mathbf{1}\right)-{\mathcal{Z}}_{\mathbf{0}}\left(\mathbf{1}\right)|$ | EOC | ${\mathcal{Z}}_{\mathbf{0}}\left(\mathbf{1}\right)$ | $|\mathit{X}\left(\mathbf{1}\right)-{\mathcal{Z}}_{\mathbf{0}}\left(\mathbf{1}\right)|$ | EOC |

1 | 0.6233820141 | ${0.9226828763}_{-3}$ | − | $\mathbf{0.6224593}588$ | ${0.2759267670}_{-7}$ | − |

2 | 0.6226943815 | ${0.2350503824}_{-3}$ | $1.97$ | $\mathbf{0.62245933}21$ | ${0.9149985214}_{-9}$ | $4.91$ |

4 | $\mathbf{0.622}5286311$ | ${0.6929984936}_{-4}$ | $1.76$ | $\mathbf{0.6224593312}$ | ${0.2863453918}_{-10}$ | $5.00$ |

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Izadi, M.; Srivastava, H.M.
A Discretization Approach for the Nonlinear Fractional Logistic Equation. *Entropy* **2020**, *22*, 1328.
https://doi.org/10.3390/e22111328

**AMA Style**

Izadi M, Srivastava HM.
A Discretization Approach for the Nonlinear Fractional Logistic Equation. *Entropy*. 2020; 22(11):1328.
https://doi.org/10.3390/e22111328

**Chicago/Turabian Style**

Izadi, Mohammad, and Hari M. Srivastava.
2020. "A Discretization Approach for the Nonlinear Fractional Logistic Equation" *Entropy* 22, no. 11: 1328.
https://doi.org/10.3390/e22111328