Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system

Highlights

• In this analysis, an attempt was made to study the fractional logistic model with the help of fractional derivative with Mittag-Leffler non-singular kernel, where the fractional derivative is considered in the ABC sense.

• The RKM has been applied to obtain approximate solutions for the non-linear fractional logistic deferential equation (FLDE).

• Three applications of this class of FLDEs are considered in the sense of ABC to verify the effectiveness of the presented method.

• Numerical and graphical results are also provided and conferred quantitatively to clarify the required solutions, where the results obtained are like those in previous studies that used Caputo type of fractional derivatives. So, since the ABC definition has a non-singular kernel, the use of ABC in DE modeling can be an appropriate substitute for Caputo fractional derivative and other fractional derivatives.

• We observed that the RKM method is very suitable, easy and effective to solve such a class of DEs and can be used to solve other types of differential equations.

Abstract

In this article, a class of population growth model, the fractional nonlinear logistic system, is studied analytically and numerically. This model is investigated by means of Atangana-Baleanu fractional derivative with a non-local smooth kernel in Sobolev space. Existence and uniqueness theorem for the fractional logistic equation is provided based on the fixed-point theory. In this orientation, two numerical techniques are implemented to obtain the approximate solutions; the reproducing-kernel algorithm is based on the Schmidt orthogonalization process to construct a complete normal basis, while the successive substitution algorithm is based on an appropriate iterative scheme. Convergence analysis associated with the suggested approaches is provided to demonstrate the applicability theoretically. The impact of the fractional derivative on population growth is discussed by a class of nonlinear logistical models using the derivatives of Caputo, Caputo-Fabrizio, and Atangana-Baleanu. Using specific examples, numerical simulations are presented in tables and graphs to show the effect of the fractional operator on the population curve as. The present results confirm the theoretical predictions and depict that the suggested schemes are highly convenient, quite effective and practically simplify computational time.

Introduction

The fractional logistic equation is one of the well-known nonlinear fractional differential equations that appear in ecology, biology and social studies. The population growth model, in particular, can be considered as one of the famous typical applications of the fractional logistic equations where the rate of system change depends on previous memory of entire historical states [1], [2]. In point of fact, Malthus was the first economist who proposed a systematic theory of population [2]. He gathered experimental data and assumed that human population grows exponentially. After that, Verhulst presented a model to population dynamics in 1838 to illustrate the periodic doubling and chaotic behavior of the dynamical system [3]. This standard model was closely related to the exponential growth model suggested by Malthus. More specifically, Verhulst presented a nonlinear first-order ordinary differential equation of population growth which subsequently became known as the logistic equation as follows:

$$\frac{dM}{dt}=\rho M\left(1-\frac{M}{k}\right),t\ge 0,$$

where M(t) indicates the size of population growth at a time t, ρ > 0 is Malthusian parameter related with the maximum growth rate, and k describes the carrying capacity.

Reciprocally, if we put $N\left(t\right)=M/k$, then the standard logistic differential equation (LDE) can be expressed as:

$$\frac{dN}{dt}=\rho N\left(1-N\right),$$

in which the exact solution of such problem can be given by

$$N\left(t\right)=\frac{N_0}{N_0+\left(1-N_0\right)e^{-\rho t}},$$

along with initial population data $N_0=N\left(0\right)$. Indeed, this solution explains the rate of population growth which doesn't include the reducing food supplies or spreading diseases [4]. Further, the LDE has been introduced in the fractional sense through many applications on different fractional operators for standard logistic issues, including the growth of tumors and epidemic disease spreading in medicine [5], the social dynamics of replacement technologies by Fisher and Pry [6] and the adaptability of society to innovation.

In fact, fractional calculus is a broad field of mathematics that provides derivatives and integrals of non-integer orders. It has been investigated extensively in describing memory and hereditary for various physical and engineering applications similar to rheology, continuum mechanics, entropy, electromagnetic problems, thermodynamics and so forth [7], [8], [9], [10], [11]. Recent researches have proved that fractional models are much better than integer ones since the variety of choices in order lead to better results. In literature, there are many definitions of fractional calculus such as the concepts of Graunwald-Letnikov, Erdelyi-Kober, Riemann-Liouville, Riesz, Caputo, and Caputo-Fabrizio [12], [13], [14], [15], [16], [17], [18], [19]. In view of this, the most common are the derivatives of Riemann-Liouville (R-LD) and Caputo (CD) that enjoy some privacy. Anyhow, both derivatives have singular and not nonlocal kernels which impose singularity on mathematical models and fail to accurately describe the memory effect. Also, the integral associated with these fractional operators is just the average of the function and their integral but not fractional. Such weakness is reflected in the modeling of real-world problems. The Caputo-Fabrizio derivative (C-FD) proposed in [15] includes an exponential kernel function to describe heterogeneities and structures with various scales, which cannot be well formulated by standard local derivatives that possess the singular kernel. Moreover, the classic Caputo concept is more influenced by the past state than the Caputo-Fabrizio concept that provides rapid stabilization due to the presence of a regular kernel [16].

In [17], Atangana and Baleanu suggested a novel derivative as generalization of the Caputo-Fabrizio derivative. They used the generalized Mittag-Leffler function $E_{\alpha }\left(-z^{\alpha }\right)={\sum }_{n=0}^{\infty }\frac{{\left(-t\right)}^{\alpha n}}{\Gamma \left(n\alpha +1\right)}$ to build the non-local and non-singular kernel. Their fractional operator has all advantages of C-FD, R-LD, and CD. Some of the advantages of Atangana-Baleanu derivative (A-BD) appear in the differences between fractional operators [18]: The R-LD and CD are Markovian, C-FD is non-Markovian, while the A-BD has both Markovian and non-Markovian aspects; The R-LD and CD have power low kernel, and C-FD has exponential decay kernel, while the A-BD has a Mittag-Leffler function as a kernel which is power low and stretched exponential kernel; The R-LD and CD are power low and scale invariant, so they fail in describing phenomena beyond their boundaries or the full history of physical problems. However, the A-BD successes in describing real-world problems with different scales or problems with changeable properties with respect to time and space; The R-LD and CD are deterministic while the A-BD is deterministic and stochastic. In fact, if the order of fractional A-BD is 1, then we get the exponential distribution, and when it is between 0 & 1, we get a heavy-tailed distribution, and a nice observation on the A-BD is that the fractional integral associated with it is the fractional average of Riemann-Liouville of a function and the function itself. Because of these important and other characteristics of A-BD, many researchers have relied on their research to deal with realistic models. Some of these models are a fractional model of bank data [19], the fractional groundwater flow model [20], and the Keller-Segel Model [21]. In [22], the authors obtained with the help of A-BD a chaotic behavior that is not obtained by local derivatives.

Consider the following nonlinear fractional logistic differential equation (FLDE):

$${}^{ABC}{D}_0^{\alpha }N\left(t\right)=\rho N\left(t\right)\left(1-N\left(t\right)\right),0〈\alpha \le 1,t\ge 0,\rho 〉0$$

associated with the initial condition

$$N\left(0\right)=N_0,N_0>0,$$

where α is a parameter describing the order of the fractional derivatives of N(t) and N(t) is a smooth function to be obtained.The fractional derivative is considered by Atangana-Baleanu in Caputo sense, denoted by ${}^{ABC}{D}_a^{\alpha }$. In the case of $\alpha =1$, the FLDE (3) reduces to the classical logistic equation.

Unfortunately, most fractional models have no exact solutions. So, numeric and approximate methods are needed. The variational iteration method [23], the Bernstein operational matrix method [24], and the finite difference method [25] are some of the advanced numerical and approximate techniques that have been implemented in solving the fractional logistic problems in the classical Caputo sense. On the other aspect as well, the reproducing-kernel method (RKM) received great attention from mathematicians and became a powerful tool for solving large-scale problems in applied sciences [26], [27], [28]. Interesting applications in acoustics, finance, stochastic, statistics and economics are well investigated using the RKM. Recently, there are many fundamental works of RKM including fuzzy differential equations, fractional differential equations, partial differential equations and integral equation; the reader can refer to [29], [30], [31], [32] for more details of results, modifications and applications of the RK method.

This paper is dedicated to analyzing a class of FLDE with Atangana-Baleanu fractional derivative in Caputo sense (ABC), in which the IVP in (3) and (4) has a unique solution in the Sobolev space. Moreover, analytical and approximate solutions for (3) and (4) are built using two different schemes, the successive substitution (SS) algorithm and reproducing kernel (RK) method. Using these techniques, the effect of the ABC derivative is shown by graphical and tabulated results for different values of α. The paper is organized as follows. Definitions and preliminaries of fractional calculus and reproducing kernel theory are given in Section 2. The analysis of the FLDE in the ABC sense is studied in Section 3. Short description of the proposed techniques, including the SS method and the RK method with convergence analysis to handle the nonlinear FLDEs with ${}^{ABC}{D}_a^{\alpha }$ is explained in Section 4. In Section 5, illustrative examples are provided to show the reliability and simplicity of the suggested methods and the effect of various fractional operators on the curves of FLDE solutions. Finally, a brief conclusion is outlined in Section 6.