# Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Caputo and Fabrizio Fractional Order Derivative

**Definition 1.**Let$f\in {H}^{1}\left(a,b\right),\phantom{\rule{0.2em}{0ex}}b>a,\phantom{\rule{0.2em}{0ex}}\alpha \in [0,1]$ then, the new Caputo derivative of a fractional derivative is defined as:

**Remark 1.**The instigators observed that, if$\sigma =\frac{1-\alpha}{\alpha}\in \left[0,\infty \right),\alpha =\frac{1}{1+\sigma}\in [0,1]$, then Equation (2) assumes the form:

**Definition 2.**Let$0<\alpha <1.$ The fractional integral of order$\alpha $ of a function$f$ is defined as:

**Remark 2.**Note that, according to the above definition, the fractional integral of Caputo type of function of order$0<\alpha <1$ is an average between function f and its integral of order one. This therefore imposes the condition [21]:

**Theorem 1.**For the new Caputo fractional order derivative, if the function$f(t)$ is such that:

## 3. Chemotaxis Model Proposed by Keller and Segel

#### 3.1. Existence of Coupled Solutions

**Theorem 2.**${K}_{1}$ and${K}_{2}$ satisfy the Lipschiz condition and contraction if the following inequality holds:

**Proof.**We shall start with ${K}_{1}$. Let $u\phantom{\rule{0.2em}{0ex}}$and $v$ be two functions, then we evaluate the following:

**Theorem 3.**Since the concentration of a chemical substance and concentration of amoebae are taking place in a confined medium, then, Equation (9) has a coupled-solution.

**Proof.**We have that, both $u\left(x,t\right)$ and $\rho \left(x,y\right)$ are bounded, in addition, we have proved that both kernels satisfy the Lipschiz condition, therefore following the results obtained in Equations (21) and (22), using the recursive technique, we obtain the following relation:

#### 3.2. Uniqueness of the Coupled Solutions

## 4. Derivation of Approximate Coupled-Solutions

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Reference

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Atangana, A.; Alkahtani, B.S.T.
Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel. *Entropy* **2015**, *17*, 4439-4453.
https://doi.org/10.3390/e17064439

**AMA Style**

Atangana A, Alkahtani BST.
Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel. *Entropy*. 2015; 17(6):4439-4453.
https://doi.org/10.3390/e17064439

**Chicago/Turabian Style**

Atangana, Abdon, and Badr Saad T. Alkahtani.
2015. "Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel" *Entropy* 17, no. 6: 4439-4453.
https://doi.org/10.3390/e17064439