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 Open Access
On numerical techniques for solving the fractional logistic differential equation
 Yves Yannick Yameni Noupoue^{1}Email authorView ORCID ID profile,
 Yücel Tandoğdu^{1} and
 Muath Awadalla^{2}
https://doi.org/10.1186/s136620192055y
© The Author(s) 2019
 Received: 15 December 2018
 Accepted: 5 March 2019
 Published: 14 March 2019
Abstract
This paper studied the existence and uniqueness of the solution of the fractional logistic differential equation using Hadamard derivative and integral. Previous work has shown that there is not an exact solution to this fractional model. Hence several numerical approaches, such as generalized Euler’s method (GEM), power series expansion (PSE) method, and Caputo–Fabrizio (CF) method, were used to compute the solution. The classical solution obtained from the first order nonlinear differential equation was also considered to enable the comparison of error levels.
Keywords
 Grunwald–Letnikov fractional derivative
 Mathematical modeling
 Nonlinear systems
 Numerical simulation
 Fractional logistic equation
1 Introduction
The name fractional calculus stems from the fact that the order of derivatives and integrals are fractions rather than integers. Early work on fractional calculus dates back to the early nineteenth century [1]. Researchers initially concentrated on the proof of the existence and uniqueness of the solution to a fractional model [2–4]. Theory of fractional calculus has been discussed by many authors [5–7].
Recent research has concentrated on showing the advantages of fractional over classical calculus [4, 8–11]. In many cases, the fractional calculus has provided better results compared to those obtained via the classical approach. In general, the better performance of the fractional calculus becomes evident based on lower error levels produced during an estimation process [8–10, 12, 13]. Computational methods and numerical simulations successfully applied to several works, such as those found in [14–19], aimed to show the results proven theoretically or to iteratively compute the solution to problem whose analytic form is not explicit.
Evidence of fractional calculus being applicable to real life problems has gradually been proven by researchers in various branches of science. Such works are for instance found in biology [8, 11, 12, 20, 21], in economy [9], and in physics [8, 10].
It is common in research to define a fractional calculus problem referring to its ordinary calculus counterpart when it exists. Such practice is often used in fractional differential equations. However, the structure of the solution to a fractional differential equation does not always correspond to its classical counterpart when it exists. Hence, an ordinary differential equation might have an exact solution whereas its fractional counterpart does not. The logistic differential equation [22] has an exact solution. However, West [23] proposed a power series referring to it as the exact solution to fractional logistic differential equation (FLDE). Subsequently Area et al. [24] claimed that West’s proposal is valid only when the order of the derivative is one. However, the solution proposed by West was proven valid by D’Ovidio et al. [25], subject to changing the structure of the FLDE for what they called ‘modified fractional logistic equation’. In [26] a numerical solution was proposed for FLDE based on Euler’s method.
In this work the FLDE is investigated. Considering the fact that the FLDE is a nonlinear equation whose exact solution does not exist, several numerical methods are used to compute its solution. Finally we compute the solution of FLDE by means of three numerical methods: the power series expansion (PSE) method also known as Letnikov method (LM) [27], the generalized Euler method (GEM) [28], and the Caputo–Fabrizio (CF) method [29]. The error rate of each of the mentioned methods is also computed for performance evaluation purpose. The introduction of the present work is followed by a preliminaries section in which some useful definitions are given, followed by a discussion on various numerical techniques used in the simulations. The fourth section is dedicated to the construction of the FLDE. The two last sections are respectively the simulation section and the concluding remarks.
2 Preliminaries
Some fractional calculus definitions and notation needed in the course of this work are discussed in this section.
Definition 2.1
([30])
Definition 2.2
([30])
Definition 2.3
([30])
Definition 2.4
([5])
Definition 2.5
([31])
The Caputo–Fabrizio fractional derivative has an advantage over other fractional derivatives of being a fractional derivative with nonsingular kernel [31].
Definition 2.6
([32])
Definition 2.7
([32])
Lemma 2.8
([32])
3 Numerical techniques for solving nonlinear differential equations
We discuss in this section some general numerical methods often used to find the numerical solution to nonlinear fractional differential equations. These methods are later applied to the FLDE.
3.1 The generalized Euler’s method (GEM)
3.2 The Grünwald–Letnikov method (GL) or power series expansion (PSE)
Grünwald provided a numerical approach for solving nonlinear differential equation. This approach is discussed in detail in [27].
Definition 3.1
([27])
3.3 The Caputo–Fabrizio method (CF)
4 Fractional logistic differential equation
4.1 Existence and uniqueness of the solution of fractional logistic equation
In this section we prove the existence and uniqueness of the solution of the FLDE. Without loss of generality, the Hadamard fractional derivative and integral symbols are used.
Note that the initial value condition implies \(c_{1} = N_{a}\).
4.1.1 Existence of solution
Using the setting above, the existence of the solution is stated and proven by Theorem 4.1 as follows.
Theorem 4.1
 (\(A_{1}\)):

\(\exists N_{Q} > 0\) such that \(\vert Q ( t,N_{1} )  Q ( t,N_{2} ) \vert \le N_{Q} \vert N_{1}  N_{2} \vert \), \(\forall t \in [ a,T ]\), \(\forall N_{1},N_{2} \in \mathbb{{R}}\).
 (\(A_{2}\)):

\(\vert Q ( t,N ) \vert \le y ( t )\), \(\forall ( t,N ) \in [ 0,T ] \times \mathbb{{R}}\), where, \(y \in C ( [ a,T ], \mathbb{{R}}^{ +} )\) with \(\sup_{a \le t \le T} \vert y ( t ) \vert = \Vert y \Vert \).
In addition, it is assumed that \(\frac{N_{Q}}{\varGamma ( q + 1 )} ( \ln \frac{T}{a} ) ^{q  1} < 1\), then there is at least one solution for the initial value problem given by Eq. (4.3).
Proof of Theorem 4.1
For \(N_{1},N_{2} \in B_{\lambda }\), then \(\Vert E_{1}N_{1} + E_{2}N _{2} \Vert \le N_{a} ( \ln \frac{T}{a} )^{q  1} + \frac{1}{ \varGamma ( q + 1 )} ( \ln \frac{T}{a} )^{q} \Vert y \Vert \le \lambda \), thus \(E_{1}N_{1} + E_{2}N_{2} \in B_{\lambda }\).
The righthand side of the above inequality approaches zero as \(t_{1} \to t_{2}\). Note that \(\Vert ( E_{1}N ) ( t_{2} )  ( E_{1}N ) ( t_{1} ) \Vert \) is independent of N implies that \(E_{1}\) is relatively compact, by Arzela–Ascoli theorem we conclude that \(E_{1}\) is compact on \(B_{\lambda } \). Hence, the existence of the solution of the initial value problem given by Eq. (4.3) holds by Krasnoselskii’s fixed point theorem. □
4.1.2 Uniqueness of solution
Theorem 4.2
Let \(Q: [ a,T ] \times \mathbb{R} \to \mathbb{R}\) be a continuous function satisfying \(( A_{1} )\) and assume that \(( \frac{1}{\varGamma ( q + 1 )} ( \ln \frac{T}{a} )^{q} )N_{Q} < 1\), then the initial value problem given by Eq. (4.3) has a unique solution.
Proof Theorem 4.2
So E is a contraction. By Banach’s contraction mapping theorem, the initial value problem given by Eq. (4.3) has a unique solution on \([ a,T ]\). □
5 Simulation studies
In the literature review it has become evident that no one has so far proposed a valid exact form of solution to FLDE. In Sect. 4 the existence and uniqueness of the solution of FLDE is proven. Since there is no exact solution to the FLDE problem, approximate solutions are found using the GEM, PSE, and CF methods.
Experimental data comes from the study of the annual growth rate of helianthus plant. The data is retrieved from [36]. The height of the plants measured at a constant spacing time of 7 days is given in centimeters. Twelve measurements were considered, ranging from the seventh day to the eightyfourth day of the plant life. The mean values of the heights are considered.
6 Conclusion
The FLDE is studied, the existence and uniqueness of a solution are proposed and proved by means of the Hadamard fractional derivative and integral formula. The proposed solution is shown to be applicable in practice by using a data set and employing three numerical approaches, namely the GEM, PSE, and CF.
The values of the fractional derivative of order q, for which each of the method produces a minimum error rate ER, were found iteratively. It appeared that the GEM and PSE methods produce a common minimum error rate of \(ER=3.2\)% for a value of \(q=1\). The CF method produces the same minimum error rate of \(ER=3.2\)% for a value of \(q=1.005\).
It is worth mentioning that the fractional derivative of order \(q=1\) coincides with the classical approach of the solution. Through the application problem it has become evident that the CF method has performed better in solving the FLDE, by producing a minimum error rate for q value different from 1.
Declarations
Acknowledgements
The authors thank the reviewers for their useful comments, which led to the improvement of the original manuscript.
Funding
This research received no specific grant from any funding agency in the public, commercial, or notforprofit sectors. The work was not performed as part of the employment of the authors.
Authors’ contributions
Supervision, YT; writing – original draft, MA and YYYN. All authors read and approved the final manuscript.
Competing interests
The authors declare no conflict of interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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