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

Abstract

In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series.

1. Introduction

Let us consider the classical logistic differential equation

$$x^{\prime }\left(t\right)=x\left(t\right)(1-x\left(t\right)),$$

(1)

which can be explicitly solved. The constant solutions are $x\left(t\right)=0$ and $x\left(t\right)=1$. If another initial condition $x\left(0\right)=x_0$ is imposed, the solution is given by

$$x\left(t\right)=\frac{x_0}{x_0+(1-x_0)exp(-t)}.$$

The solution can be also obtained in terms of formal power series. Let

$$x\left(t\right)=\sum _{n=0}^{\infty }{a}_n{t}^n.$$

(2)

Then,

$$x^{\prime }\left(t\right)=\sum _{n=1}^{\infty }n{a}_n{t}^{n-1}=\sum _{n=0}^{\infty }(n+1)a_{n+1}{t}^n,$$

and

$$x^2\left(t\right)=\sum _{n=0}^{\infty }\left(\sum _{j=0}^n{a}_j{a}_{n-j}\right)t^n$$

(3)

By substituting into (1), we obtain the following recurrence relation for the coefficients

$$a_{n+1}=\frac{1}{n+1}\left[a_n-\sum _{j=0}^n{a}_j{a}_{n-j}\right],n\ge 1,a_0=x\left(0\right),$$

(4)

which provide a solution in a neighbourhood of $t=0$ as described in [1].

It is possible to obtain the same recurrence relation if we further apply the Laplace transform to (1). Let $x\left(t\right)$ be given in (2), so that (3) holds true. Let $\mathcal{L}$ denote the Laplace transform, and as usual, we shall denote $F\left(s\right)$ the Laplace transform of a function $f\left(t\right)$. Then,

$$\mathcal{L}\left[x\left(t\right)(1-x\left(t\right))\right]=\mathcal{L}[x\left(t\right)-x^2\left(t\right)]=\sum _{n=0}^{\infty }(a_n-b_n)\frac{n!}{s^{n+1}},b_n=\sum _{j=0}^n{a}_j{a}_{n-j}.$$

Moreover,

$$\mathcal{L}\left[x^{\prime }\left(t\right)\right]=sX\left(s\right)=\sum _{n=0}^{\infty }{a}_{n+1}(n+1)\frac{n!}{s^{n+1}}$$

Thus,

$$\sum _{n=0}^{\infty }{a}_{n+1}(n+1)\frac{n!}{s^{n+1}}=\sum _{n=0}^{\infty }(a_n-b_n)\frac{n!}{s^{n+1}}$$

(5)

which implies again the recurrence relation (4) for the coefficients of the power series expansion of the solution. We need to impose that $a_0=x\left(0\right)$ to be able to start the latter recurrence relation.

We have included in Figure 1 some plots by using Mathematica [2] of the logistic function, solution to (1) with $x\left(0\right)=1/2$, and approximations of the function by the corresponding Taylor polynomials.

This very classical logistic differential equation has been deeply studied due to its applications in different fields [3]. Recently, it has been used to study the evolution of the COVID-19 pandemic [4,5]. By considering fractional derivatives, the fractional analogue has been analyzed in several works mainly by considering the Liouville–Caputo fractional derivative [1,6,7,8,9,10,11] (see also [12,13,14]). The classical logistic ordinary differential equation has been recently studied from the view of fractional calculus and solved in some particular cases [1,8,10]. In this work, we consider the fractional logistic differential equation by using the Prabhakar fractional calculus [15,16,17].

The main aim of this work is to present the Prabhakar fractional logistic differential equation and, by appropriate limit transitions, recover several logistic differential equations (Liouville–Caputo, Atangana–Baleanu, and Caputo–Fabrizio), providing in each case a representation of the expansion of the solution in formal power series.

For the fractional Prabhakar logistic differential equation, we know the solutions for the Liouville–Caputo fractional derivative [7] (in terms of power series) and for the Caputo–Fabrizio derivative [18] (in implicit form). We emphasize that in this work we present the solution in terms of a power series expansion, as compared with the previous work [18], in which the solution is given in implicit form. It is also important to notice the fact that much more general fractional-calculus operators are available in the literature survey [19].

The structure of this work is the following: in Section 2 basic definitions, notations and results are presented. In Section 3 the Prabhakar fractional logistic equation is presented. For specific values of the parameters we recover the Liouville–Caputo, Atangana–Baleanu, and Caputo–Fabrizio logistic differential equations. For each of these cases, the solution is computed in terms of a formal power series. Some numerical experiments are also presented.

2. Basic Definition and Notations

Let $\alpha \in (0,1)$ and $\sigma \in L^1(0,1)$. The (Riemann–Liouville) fractional integral is defined by

$$I^{\alpha }\sigma \left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}\sigma \left(s\right)ds,t\in (0,1)$$

where $\mathsf{\Gamma }\left(z\right)$ denotes the Euler gamma function [20].

For $z\in \mathbb{C}$, $\alpha ,\beta ,\gamma \in \mathbb{C}$ with $\Re \left(\alpha \right)>0$, the three-parameter Mittag–Leffler function, introduced by Prabhakar in 1971 [16], is defined by

$$E_{\alpha ,\beta }^{\gamma }\left(z\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_n}{\mathsf{\Gamma }(n\alpha +\beta )}\frac{z^n}{n!},$$

which generalizes both the Mittag–Leffler function ($\gamma =1$) as well as the classical exponential function ($\alpha =\beta =\gamma =1$). Additionally, $E_{\alpha ,\beta }^0\left(z\right)=1/\mathsf{\Gamma }\left(\beta \right)$. We would like to emphasize that $E_{\alpha ,\beta }^{\gamma }\left(z\right)$ in an entire function of order $\varrho =1/\Re \left(\alpha \right)$ and type $\sigma =1$ [21].

Let

$$e_{\alpha ,\beta }^{\gamma }(\lambda ;t)=t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right),$$

(6)

be the Prabhakar kernel. In particular,

$$e_{1,1}^1(\lambda ;t)=exp\left(\lambda t\right);e_{\alpha ,\beta }^0(\lambda ;t)=\frac{t^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}.$$

Additionally,

$$e_{\alpha ,\beta }^{\gamma }(0;t)=\frac{t^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}.$$

The Prabhakar fractional integral with base point 0 is defined by

$${\mathbb{P}}_{\alpha ,\beta ,\lambda }^{\gamma }\sigma \left(t\right)={\int }_0^t{e}_{\alpha ,\beta }^{\gamma }(\lambda ;t-s)\sigma \left(s\right)ds.$$

(7)

for $\sigma \in L^1(0,1)$. For $\sigma \in L^1(0,1)$,

$${\mathbb{P}}_{\alpha ,\beta ,\lambda }^{\gamma }\sigma \left(t\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_n{\lambda }^n}{n!}{I}^{\alpha n+\beta }\sigma \left(t\right).$$

Thus, the Prabhakar fractional integral ${\mathbb{P}}_{\alpha ,\beta ,\lambda }^{\gamma }$ is linear and bounded from $L^p(0,1)$ into $L^p(0,1)$ for any $1\le p\le \infty$.

Recall that taking $\lambda =-1$

$$e_{\alpha ,\beta }^{\gamma }(-1,t)=t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }(-t^{\alpha })$$

is completely monotone if $0<\alpha \gamma \le b\le 1$ [22]. For example, $e_{1,1}^1(-1,t)=exp(-t)$.

Moreover, Ref. [15]

$${\mathbb{P}}_{\alpha ,\beta ,\lambda }^{\gamma }{e}_{\alpha ,\mu }^{\omega }(\lambda ;t)=e_{\alpha ,\beta +\mu }^{\gamma +\omega }(\lambda ;t).$$

Let $\sigma \in L^1(0,1)$, $\Re \left(\alpha \right)>0$, $\Re \left(\beta \right)>0$. The Prabhakar fractional derivative in the Riemann–Liouville sense is defined by

$${\mathbb{D}}_{\alpha ,\beta ,\lambda }^{\gamma }\sigma \left(t\right)=\frac{d}{dt}{\mathbb{P}}_{\alpha ,1-\beta ,\lambda }^{-\gamma }\sigma \left(t\right).$$

In doing so, it is required some regularity for $\sigma$, for example, that $\sigma \star e_{\alpha ,1-\beta ,\lambda }^{-\gamma }\in W^{1,1}(0,1)$, where for $\Omega \subset {\mathbf{R}}^n$ the Sobolev space $W^{m,p}(\Omega )$ is defined by

$$W^{m,p}(\Omega )=\{f\in L^p(\Omega )|D^{\alpha }\in L^p(\Omega ),\forall \alpha \in {\mathbf{N}}^n:\left|\alpha \right|\le m\}.$$

The Prabhakar fractional derivative in the Liouville–Caputo sense is

$${\mathbb{D}}_{\alpha ,\beta ,\lambda }^{\gamma }\sigma \left(t\right)={\mathbb{P}}_{\alpha ,1-\beta ,\lambda }^{-\gamma }{\sigma }^{\prime }\left(t\right).$$

We note that

$${\mathbb{P}}_{\alpha ,\beta ,1}^0\sigma \left(t\right)={\int }_0^t{e}_{\alpha ,\beta }^0(\lambda ,t-s)\sigma \left(s\right)ds=\frac{1}{\mathsf{\Gamma }\left(\beta \right)}{\int }_0^t{(t-s)}^{\beta -1}\sigma \left(s\right)ds=I^{\beta }\sigma \left(t\right).$$

Additionally,

$${\mathbb{P}}_{\alpha ,\beta ,0}^{\gamma }\sigma \left(t\right)={\int }_0^t{e}_{\alpha ,\beta }^{\gamma }(0,t-s)\sigma \left(s\right)ds=\frac{1}{\mathsf{\Gamma }\left(\beta \right)}{\int }_0^t{(t-s)}^{\beta -1}\sigma \left(s\right)ds=I^{\beta }\sigma \left(t\right).$$

In both cases, the classical Riemann-Liouville integral of order $\beta >0$ is a particular case of the Prabhakar operator.

The Laplace transform of the Prabhakar fractional derivative in the Liouville–Caputo sense is [15] (page 27, Section 5.1, Equation (5.13))

$$\mathcal{L}[{\mathbb{D}}_{\alpha ,\beta ,\lambda }^{\gamma }x\left(t\right)]\left(s\right)=s^{\beta -\alpha \gamma }{(s^{\alpha }-\lambda )}^{\gamma }\left\{\mathcal{L}[x\left(t\right)]\left(s\right)-\sum _{k=0}^{m-1}{s}^{-k-1}{f}^{\left(k\right)}(0+)\right\},$$

(8)

where m denotes the integer part of $\beta$. In particular, if $m=0$ or $m=1$, we have

$$\mathcal{L}[{\mathbb{D}}_{\alpha ,\beta ,\lambda }^{\gamma }{t}^n]\left(s\right)=s^{\beta -\alpha \gamma }{(s^{\alpha }-\lambda )}^{\gamma }\frac{\mathsf{\Gamma }(n+1)}{s^{n+1}}.$$

(9)

Let us consider the Liouville–Caputo fractional derivative [23] for an absolutely continuous function $f:[0,T]\mapsto \mathbb{R}$

$${}^{\mathrm{C}}{\mathbb{D}}^{\alpha }f\left(t\right)=\frac{1}{\mathsf{\Gamma }(1-\alpha )}{\int }_0^t{(t-s)}^{-\alpha }{f}^{\prime }\left(s\right)ds,t\in [0,T].$$

(10)

We have that

$$\mathcal{L}\left[{}^{\mathrm{C}}{\mathbb{D}}^{\alpha }{t}^n\right]\left(s\right)=\frac{\mathsf{\Gamma }(n+1)}{s^{n-\alpha +1}},\alpha >0.$$

(11)

Since

$$\mathcal{L}[{\mathbb{D}}_{\alpha ,\beta ,0}^{\gamma }{t}^n]\left(s\right)=\mathcal{L}[{\mathbb{D}}_{\alpha ,\beta ,\lambda }^0{t}^n]\left(s\right)=\frac{\mathsf{\Gamma }(n+1)}{s^{n-\beta +1}}=\mathcal{L}\left[{}^{\mathrm{C}}{\mathbb{D}}^{\beta }{t}^n\right]\left(s\right).$$

the Liouville–Caputo fractional derivative is a particular case of the Prabhakar fractional derivative.

The Atangana–Baleanu operator in the sense of Caputo for $u\in \mathrm{AC}(0,1)=W^{1,1}(0,1)$ is defined by [24]

$${}^{\mathrm{AB}}{\mathbb{D}}^{\alpha }u\left(t\right)=\frac{1}{1-\alpha }{\int }_0^t{E}_{\alpha }\left(-\frac{\alpha }{1-\alpha }{(t-s)}^{\alpha }\right)u^{\prime }\left(s\right)ds.$$

(12)

It yields

$$\mathcal{L}\left[{}^{\mathrm{AB}}{\mathbb{D}}^{\alpha }{t}^n\right]\left(s\right)=\frac{B\left(\alpha \right)}{1-\alpha }\frac{1}{s^{\alpha }+\frac{\alpha }{1-\alpha }}\frac{\mathsf{\Gamma }(n+1)}{s^{n-\alpha +1}}.$$

(13)

where $B\left(\alpha \right)$ is a normalizing function satisfying $B\left(0\right)=B\left(1\right)=1$. Let $\beta =0$, $\gamma =-1$, $\lambda =\alpha /(\alpha -1)$, so that

$$\mathcal{L}[{\mathbb{D}}_{\alpha ,0,\alpha /(\alpha -1)}^{-1}{t}^n]\left(s\right)=\frac{\mathsf{\Gamma }(n+1)s^{\alpha -n-1}}{s^{\alpha }+\frac{\alpha }{1-\alpha }}=\frac{1-\alpha }{B\left(\alpha \right)}\mathcal{L}[{}^{\mathrm{AB}}{\mathbb{D}}^{\alpha }{t}^n]\left(s\right).$$

As a consequence, the Atangana–Baleanu derivative is a particular case of the Prabhakar fractional derivative.

For $u\in \mathrm{AC}(0,1)=W^{1,1}(0,1)$, the Caputo–Fabrizio fractional derivative is defined by [25] (see also [26])

$${}^{\mathrm{CF}}{\mathbb{D}}^{\alpha }u\left(t\right)=\frac{1}{1-\alpha }{\int }_0^{t}exp\left(-\frac{\alpha }{1-\alpha }(t-s)\right)u^{\prime }\left(s\right)ds.$$

(14)

We have that

$$\mathcal{L}\left[{}^{\mathrm{CF}}{\mathbb{D}}^{\alpha }{t}^n\right]\left(s\right)=\frac{\mathsf{\Gamma }(n+1)}{\alpha +(1-\alpha )s}\frac{1}{s^n}.$$

(15)

Let $\alpha =1$, $\beta =0$, $\gamma =-1$, $\lambda =\alpha /(\alpha -1)$, so that

$$\mathcal{L}[{\mathbb{D}}_{1,0,\alpha /(\alpha -1)}^1{t}^n]\left(s\right)=\frac{s^{-n}\mathsf{\Gamma }(n+1)}{s-\frac{\alpha }{\alpha -1}}=(1-\alpha )\mathcal{L}[{}^{\mathrm{CF}}{\mathbb{D}}^{\alpha }{t}^n]\left(s\right),$$

revealing that the Caputo–Fabrizio derivative [25] is also a particular case of the Prabhakar fractional derivative.

3. Prabhakar Fractional Logistic Equation and Its Limiting Cases

Let ${\mathbb{D}}_{\alpha ,\beta ,\lambda }^{\gamma }x\left(t\right)$ be the Prabhakar fractional derivative of a given function $x\left(t\right)$. Let us now consider the Prabhakar fractional logistic differential equation

$$\mathsf{\Lambda }(\alpha ,\beta ,\gamma ,\lambda ){\mathbb{D}}_{\alpha ,\beta ,\lambda }^{\gamma }x\left(t\right)=x\left(t\right)(1-x\left(t\right)),$$

(16)

where the constant $\mathsf{\Lambda }(\alpha ,\beta ,\gamma ,\lambda )$ is defined by

$$\mathsf{\Lambda }(\alpha ,\beta ,\gamma ,\lambda )=\left\{\begin{array}{cc}{\displaystyle 1-\frac{\lambda }{\lambda -1},}\hfill & \alpha =1,\hfill \\ {\displaystyle {\left(\frac{B\left(\alpha \right)}{1-\alpha }\right)}^{\frac{(1-\alpha )\gamma \lambda }{\alpha }},}\hfill & \alpha \ne 1.\hfill \end{array}\right.$$

Let

$$x\left(t\right)=\sum _{n=0}^{\infty }{a}_n{t}^{n\xi },$$

(17)

so that

$$x\left(t\right)(1-x\left(t\right))=\sum _{n=0}^{\infty }(a_n-b_n)t^{n\xi },b_n=\sum _{j=0}^n{a}_j{a}_{n-j}.$$

(18)

If we apply the Laplace transform, we obtain

$$\mathsf{\Lambda }(\alpha ,\beta ,\gamma ,\lambda )s^{\beta -\alpha \gamma }{(s^{\alpha }-\lambda )}^{\gamma }\sum _{n=0}^{\infty }{a}_n\frac{\mathsf{\Gamma }(\xi n+1)}{s^{\xi n+1}}=\sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(\xi n+1)}{s^{\xi n+1}},$$

or equivalently

$$\mathsf{\Lambda }(\alpha ,\beta ,\gamma ,\lambda ){(s^{\alpha }-\lambda )}^{\gamma }\sum _{n=0}^{\infty }{a}_n\frac{\mathsf{\Gamma }(\xi n+1)}{s^{\xi n+1-\beta +\alpha \gamma }}=\sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(\xi n+1)}{s^{\xi n+1}}.$$

(19)

By using the binomial theorem,

$$\mathsf{\Lambda }(\alpha ,\beta ,\gamma ,\lambda ){(s^{\alpha }-\lambda )}^{\gamma }=\sum _{n=0}^{\infty }\frac{\mathsf{\Gamma }(\gamma +1)}{\mathsf{\Gamma }(k+1)\mathsf{\Gamma }(\gamma -n+1)}{s}^{\alpha n}{(-\lambda )}^{\gamma -n}=\sum _{n=0}^{\infty }{c}_n{s}^{\alpha n},$$

where

$$c_n=\frac{\mathsf{\Gamma }(\gamma +1)}{\mathsf{\Gamma }(n+1)\mathsf{\Gamma }(\gamma -n+1)}{(-\lambda )}^{\gamma -n}$$

and then

$$\mathsf{\Lambda }(\alpha ,\beta ,\gamma ,\lambda )\sum _{n=0}^{\infty }\frac{c_n}{s^{-\alpha n}}\sum _{n=0}^{\infty }{a}_n\frac{\mathsf{\Gamma }(\xi n+1)}{s^{\xi n+1-\beta +\alpha \gamma }}=\sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(\xi n+1)}{s^{\xi n+1}}.$$

Thus, we have

$$\begin{array}{c}\hfill \mathsf{\Lambda }(\beta ,\alpha ,\gamma ,0){\mathbb{D}}_{\beta ,\alpha ,0}^{\gamma }={}^{\mathrm{C}}{\mathbb{D}}^{\alpha }\left(\mathrm{Liouville}–\mathrm{Caputo}\right),\\ \hfill \mathsf{\Lambda }(\beta ,\alpha ,0,\lambda ){\mathbb{D}}_{\beta ,\alpha ,\lambda }^0={}^{\mathrm{C}}{\mathbb{D}}^{\alpha }\left(\mathrm{Liouville}–\mathrm{Caputo}\right),\\ \hfill \mathsf{\Lambda }(\alpha ,0,-1,\alpha /(\alpha -1)){\mathbb{D}}_{\alpha ,0,\alpha /(\alpha -1)}^{-1}={}^{\mathrm{AB}}{\mathbb{D}}^{\alpha }\left(\mathrm{Atangana}–\mathrm{Baleanu}\right),\\ \hfill \mathsf{\Lambda }(1,0,-1,\alpha /(\alpha -1)){\mathbb{D}}_{\alpha ,0,\alpha /(\alpha -1)}^{-1}={}^{\mathrm{CF}}{\mathbb{D}}^{\alpha }\left(\mathrm{Caputo}–\mathrm{Fabrizio}\right).\end{array}$$

3.1. Fractional Liouville–Caputo Logistic Differential Equation

Let us consider the Liouville–Caputo fractional logistic differential equation

$${}^{\mathrm{C}}{\mathbb{D}}^{\alpha }x\left(t\right)=x\left(t\right)(1-x\left(t\right)),$$

(20)

where ${}^{\mathrm{C}}{\mathbb{D}}^{\alpha }$ is defined in (10). Since

$$\begin{array}{cc}\hfill & \mathsf{\Lambda }(\beta ,\alpha ,\gamma ,0){\mathbb{D}}_{\beta ,\alpha ,0}^{\gamma }={}^{\mathrm{C}}{\mathbb{D}}^{\alpha },\hfill \\ \hfill & \mathsf{\Lambda }(\beta ,\alpha ,0,\lambda ){\mathbb{D}}_{\beta ,\alpha ,\lambda }^0={}^{\mathrm{C}}{\mathbb{D}}^{\alpha },\hfill \end{array}$$

by applying the Laplace transform, taking into account (19) and (18), we obtain

$$\sum _{n=1}^{\infty }\frac{\mathsf{\Gamma }(\xi n+1)}{s^{n\xi -\alpha +1}}{a}_n=\sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(n\xi +1)}{s^{n\xi +1}}.$$

For $\xi =\alpha$, equating coefficients we have the following recurrence relation for the coefficients in the power series expansion (19)

$$a_1=\frac{a_0-b_0}{\mathsf{\Gamma }(1+\alpha )},a_n=\frac{\mathsf{\Gamma }\left(\right(n-1)\alpha +1)}{\mathsf{\Gamma }(n\alpha +1)}(a_{n-1}-b_{n-1}).$$

(21)

We have included in Figure 2 and Figure 3 some plots of the logistic function, solution to (1) with $x\left(0\right)=1/2$, as well as some approximations of the solution to the Liouville–Caputo fractional logistic differential Equation (20).

3.2. Atangana–Baleanu Logistic Differential Equation

Let us consider

$${}^{\mathrm{AB}}{\mathbb{D}}^{\alpha }x\left(t\right)=x\left(t\right)(1-x\left(t\right)),$$

(22)

where ${}^{\mathrm{AB}}{\mathbb{D}}^{\alpha }f\left(t\right)$ is the Atangana–Baleanu derivative defined in (12).

Since

$$\mathsf{\Lambda }(\alpha ,0,-1,\alpha /(\alpha -1)){\mathbb{D}}_{\alpha ,0,\alpha /(\alpha -1)}^{-1}={}^{\mathrm{AB}}{\mathbb{D}}^{\alpha },$$

if we apply the Laplace transform to (22), by using (19) and (18) we obtain

$$\mathcal{L}\left[{}^{\mathrm{AB}}{\mathbb{D}}^{\alpha }x\left(t\right)\right]\left(s\right)=\sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(n\xi +1)}{s^{n\xi +1}}.$$

(23)

Thus, for $\xi =\alpha$,

$$\frac{B\left(\alpha \right)}{(1-\alpha )(s^{\alpha }+\frac{\alpha }{1-\alpha })}\sum _{n=1}^{\infty }{a}_n\frac{\mathsf{\Gamma }(n\alpha +1)}{s^{(n-1)\alpha +1}}=\sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(n\alpha +1)}{s^{n\alpha +1}}.$$

(24)

Hence,

$$\begin{array}{c}\frac{B\left(\alpha \right)}{(1-\alpha )}\sum _{n=1}^{\infty }{a}_n\frac{\mathsf{\Gamma }(n\alpha +1)}{s^{(n-1)\alpha +1}}\hfill \\ =\sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(n\alpha +1)}{s^{\alpha (n-1)+1}}+\frac{\alpha }{1-\alpha }\sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(n\alpha +1)}{s^{n\alpha +1}}.\hfill \end{array}$$

(25)

Equating the coefficients, we obtain

$$\begin{array}{cc}\hfill a_1& =\frac{a_0-b_0+b_1(\alpha -1)\mathsf{\Gamma }\left(\alpha \right)}{\left(B\right(\alpha )+\alpha -1)\mathsf{\Gamma }\left(\alpha \right)},\hfill \\ \hfill a_n& =\frac{(\alpha -1)b_n+\frac{\alpha (a_{n-1}-b_{n-1})\mathsf{\Gamma }((n-1)\alpha +1)}{\mathsf{\Gamma }(n\alpha +1)}}{B\left(\alpha \right)+\alpha -1}.\hfill \end{array}$$

By using

$$b_n=2a_0{a}_n+\sum _{j=1}^{n-1}{a}_j{a}_{n-j},$$

(26)

we finally obtain the initial step in terms of the initial condition

$$a_1=\frac{\left(a_0-1\right)a_0}{\mathsf{\Gamma }\left(\alpha \right)\left(\left(2a_0-1\right)(\alpha -1)-B\left(\alpha \right)\right)},$$

(27)

as well the recurrence relation for the coefficients

$$a_n=\frac{(1-\alpha ){\sum }_{j=1}^{n-1}{a}_j{a}_{n-j}+\frac{\alpha \mathsf{\Gamma }(\alpha (n-1)+1)\left({\sum }_{j=1}^{n-1}{a}_j{a}_{n-j}+\left(2a_0-1\right)a_{n-1}\right)}{\mathsf{\Gamma }(\alpha n+1)}}{\left(2a_0-1\right)(\alpha -1)-B\left(\alpha \right)},$$

(28)

which in the limit as $\alpha \to 1$ converge to (21).

We have included in Figure 4 and Figure 5 some plots of the logistic function, solution to (1) with $x\left(0\right)=1/2$, as well as some approximations of the solution to the Atangana–Baleanu logistic differential Equation (22).

3.3. Caputo–Fabrizio Logistic Differential Equation

Let us consider

$${}^{\mathrm{CF}}{\mathbb{D}}^{\alpha }x\left(t\right)=x\left(t\right)(1-x\left(t\right)),$$

(29)

where ${}^{\mathrm{CF}}{\mathbb{D}}^{\alpha }$ is the Caputo–Fabrizio derivative introduced in (14).

Since

$$\mathsf{\Lambda }(1,0,-1,\alpha /(\alpha -1)){\mathbb{D}}_{\alpha ,0,\alpha /(\alpha -1)}^{-1}={}^{\mathrm{CF}}{\mathbb{D}}^{\alpha },$$

if we apply the Laplace transform to (29), we obtain

$$\sum _{n=1}^{\infty }\frac{\mathsf{\Gamma }(n\xi +1)}{\left(\alpha \right(1-s)+s)}\frac{a_n}{s^{n\xi }}=\sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(n\xi +1)}{s^{n\xi +1}},$$

which, if we fix $\xi =\alpha$, in the limit as $\alpha \to 1^{-}$, gives (5). Equivalently,

$$\sum _{n=1}^{\infty }\mathsf{\Gamma }(n\xi +1)\frac{a_n}{s^{n\xi }}=(\alpha +(1-\alpha )s)\sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(n\xi +1)}{s^{n\xi +1}},$$

which can be rewritten as

$$\sum _{n=1}^{\infty }\mathsf{\Gamma }(n\xi +1)\frac{a_n}{s^{n\xi }}=\alpha \sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(n\xi +1)}{s^{n\xi +1}}+(1-\alpha )\sum _{n=0}^{\infty }(a_n-b_n)\frac{\mathsf{\Gamma }(n\xi +1)}{s^{n\xi }}.$$

Let $\xi =1$. If we equate the coefficients, we obtain

$$a_n=\frac{1}{n}\left(a_{n-1}-b_{n-1}+n{b}_n\frac{\alpha -1}{\alpha }\right)$$

which in the limit as $\alpha \to 1$ converges to (4). This relation can be obtained from [18] (Equation (8)).

We have included in Figure 6 and Figure 7 some plots of the logistic function, solution to (1) with $x\left(0\right)=1/2$, as well as some approximations of the solution to the Caputo–Fabrizio logistic differential Equation (29).

Moreover, in Figure 8, we show a comparison between the results in [18] in implicit form, and the results presented here in terms of recurrence relation for the coefficients in the power series expansion. For this last comparison, we have chosen $\alpha =0.9$.

4. Conclusions

The Prabhakar fractional calculus, based on the three-parameter generalization of the Mittag–Leffler function, provides some physical examples such as anomalous phenomena showing the need for an extension of ordinary calculus based on the Prabhakar function [15]. We contribute to the study of the fractional logistic differential equation in the setting of Prabhakar fractional calculus by solving that logistic equation in some cases.