Abstract
Abstract: This paper considers the Freedman model using the Liouville-Caputo fractional-order derivative and the fractional-order derivative with Mittag-Leffler kernel in the Liouville-Caputo sense. Alternative solutions via Laplace transform, Sumudu-Picard and Adams-Moulton rules were obtained. We prove the uniqueness and existence of the solutions for the alternative model. Numerical simulations for the prediction and interaction between a unilingual and a bilingual population were obtained for different values of the fractional order.
Introduction
Interactions between groups that speak different languages are occurring continuously in several countries in the world due to globalization and cultural openness. Multilingualism is the use of more than one language, either by an individual speaker or by a community of speakers. A mathematical model portraying the interaction dynamics of a population considering bilingual components and a monolingual component was proposed in [1, 2] . Baggs in [2] studied the condition under which the bilingual component could persist and conditions under which it could become extinct. The weakness of these models is that they do not take into account the degree of interest in time and also the memory of the interaction, meaning the recall of the original meeting or interaction or contact up to a particular period of time in the present. Fractional calculus is one of the most powerful mathematical tools used in recent decades to model real-world problems in many fields, such as science, technology and engineering. The Liouville-Caputo fractional derivative involves the power-law function. The Liouville-Caputo fractional-order derivative allows usual initial conditions when playing with the integral transform, for instance the Laplace transform [3] [4] [5] . Recently, Abdon Atangana and Dumitru Baleanu proposed two fractional-order operators involving the generalized Mittag-Leffler function. The generalized Mittag-Leffler function was introduced in the literature to improve the limitations posed by the power-law [6] [7] [8] [9] [10] [11] [12] . The two-parametric, three-parametric, four-parametric and multiple Mittag-Leffler functions were presented by Wiman, Prabhakar, Shukla and Srivastava in [13] [14] [15] [16] [17] [18] . The kernel used in Atangana-Baleanu fractional differentiation appears naturally in several physical problems as generalized exponential decay and as a power-law asymptotic for a very large time [19] [20] [21] [22] [23] [24] . The choice of this derivative is motivated by the fact that the interaction is not local, but global, and also, the trend observed in the field does not follow the power-law. The generalized
Introduction
where 0 < A i , M i , |F i | ≤ 1 are the birth, death and emigration parameters for i ∈ [1, 2]. 0 < G 1 ≤ 1 is the infant language acquisition parameter, that proportion of births in the x 2 population raised unilingually. 0 < α ≤ 1 is the non-infant language acquisition rate, the proportion of x 1 learning the x 2 language per unit time after infancy. The term α ·
Introduction
describes that part of population x 1 lost to x 2 due to virtual predation on the part of x 2 [26] .
Introduction
The aim of this work is to obtain alternative representations of the Freedman model considering Liouville-Caputo and Atangana-Baleanu-Caputo fractional derivatives. The paper is organized as follows: Section 2 introduces the fractional operators. Alternative representations of the Freedman model are shown in Section 3. Finally, in Section 4, we conclude the manuscript.
Fractional Operators
The Liouville-Caputo fractional-order derivative of order γ is defined by [27] :
Fractional Operators
The Laplace transform to Liouville-Caputo fractional-order derivative gives [27] :
Fractional Operators
The Atangana-Baleanu-Caputo fractional-order derivative is defined as follows [19] [20] [21] [22] 24] :
Fractional Operators
where B(β) = B(0) = B(1) = 1 is a normalization function and E γ is the Mittag-Leffler function [6] [7] [8] [9] [10] [11] [12] . The Mittag-Leffler kernel is a combination of both the exponential-law and power-law. For this fractional derivative, we have at the same time the power-law and the stretched exponential as the waiting time distribution. The Laplace transform of Equation (4) is defined as follows:
Fractional Operators
The Sumudu transform is derived from the classical Fourier integral [28] . The Sumudu transform of Equation (4) is defined as:
Fractional Operators
The Atangana-Baleanu fractional integral of order γ of a function f (t) is defined as:
Freedman Model
In this section, we obtain alternative representations of the Freedman model considering the Liouville-Caputo fractional derivative, and the special solution is obtained using a Laplace transform method.
Freedman Model with the Power-Law Kernel
Considering Equation (2) , the modified Freedman model with the power-law kernel is given as:
Freedman Model with the Power-Law Kernel
where 0 < γ ≤ 1 is the fractional order, x 1 (t) represents the interaction of the majority unilingual population and x 2 (t) is a bilingual population. Applying the Laplace transform operator (3) and the inverse Laplace transform on both sides of Equation (8), we obtain:
Freedman Model with the Power-Law Kernel
The following iterative formula is then proposed:
Freedman Model with the Power-Law Kernel
where,
Freedman Model with the Power-Law Kernel
where the approximate solution is assumed to be obtained as a limit when "n" tend to infinity:
Stability Analysis of the Iteration Method
Theorem 1. We demonstrate that the recursive method given by Equation (10) is stable.
Stability Analysis of the Iteration Method
Proof. It is possible to find two positive constants Y and Z such that, for all:
Stability Analysis of the Iteration Method
Now, we consider a subset of C 2 ((a, b)(0, T)) defined by:
Stability Analysis of the Iteration Method
we now consider the operator φ defined as:
Stability Analysis of the Iteration Method
Then:
Stability Analysis of the Iteration Method
where,
Stability Analysis of the Iteration Method
, and
Stability Analysis of the Iteration Method
Applying the absolute value on both sides, we have:
Stability Analysis of the Iteration Method
Then,
Stability Analysis of the Iteration Method
where,
Stability Analysis of the Iteration Method
with:
Stability Analysis of the Iteration Method
, and:
Stability Analysis of the Iteration Method
Furthermore, if we consider a given non-null vector (x 1 , x 2 ), then using the some routine as the above case, we obtain:
Stability Analysis of the Iteration Method
From the results obtained in Equations (18) and (20), we conclude that the iterative method used is stable. This complete the proof. Now, we consider the Adams method [29, 30] to solve the system given by Equation (8) . The basic idea of the n-step Adams-Bashforth method is to use a polynomial interpolation for f (t, y(t)) passing through n points: (t i , f i ), (t i−1 , f i−1 ), ..., (t i−n+1 , f i−n+1 ). Correspondingly, the n-step Adams-Moulton method uses a polynomial interpolation for f (t, y(t)) passing through n + 1 points:
Stability Analysis of the Iteration Method
The fractional Adams method is derived as follows [30] :
Stability Analysis of the Iteration Method
where,
Stability Analysis of the Iteration Method
Following this procedure, we can propose a numerical solution for System (8) using the Adams method (21) as follows:
Freedman Model with the Mittag-Leffler Kernel
Considering Equation (4), the modified Freedman model is given as:
Freedman Model with the Mittag-Leffler Kernel
where 0 < γ ≤ 1 is the fractional order, x 1 (t) represents the interaction of the majority unilingual population and x 2 (t) is a bilingual population. Now, we obtain an alternative solution using an iterative scheme. The technique involves coupling the Sumudu transform and its inverse. The Sumudu transform is an integral transform similar to the Laplace transform, introduced by Watugala to solve differential equations [28] [29] [30] [31] [32] . Applying the Sumudu transform (6) and the inverse Sumudu transform on both sides of the system (24) yields:
Freedman Model with the Mittag-Leffler Kernel
The following recursive formula for Equation (25) is obtained:
Freedman Model with the Mittag-Leffler Kernel
and the solution of Equation (26) is provided by:
Stability Analysis of the Iteration Method
Now, we provide in detail the stability analysis of this method and show the uniqueness of the special solutions using the fixed point theory and properties of the inner product and the Hilbert space, respectively.
Stability Analysis of the Iteration Method
Let (X, | · |) be a Banach space and Ha self-map of X. Let z n+1 = g(H, z n ) be a particular recursive procedure. The following conditions must be satisfied for z n+1 = Hz n .
1.
The fixed point set of H has at least one element.
2.
z n converges to a point P ∈ F(H).
3.
lim n→∞ x n (t) = P. 
3.
for all x, z ∈ X, where 0 ≤ η, 0 ≤ η < 1. Suppose that H is Picard H-stable.
3.
Considering the following recursive formula, we have:
3.
where:
3.
correspond to the fractional Lagrange multiplier.
Theorem 2.
Let K be a self-map defined as:
Theorem 2.
is K-stable in L 1 (a, b) if:
Theorem 2.
where f (γ), d(γ), ω(γ), k(γ), i(γ), r(γ), s(γ) and o(γ) are functions from:
Theorem 2.
Proof. The proof consists of showing that K has a fixed point. To achieve this, we consider:
Theorem 2.
and:
Theorem 2.
Using the properties of the norm and considering the triangular inequality, we get:
Theorem 2.
we consider that the solutions play the some role, i.e., ||x 2(n) (t) − x 2(m) (t)|| ∼ = ||x 1(n) (t) − x 1(m) (t)||.
Theorem 2.
Using the linearity of the inverse Sumudu transform, we get:
Theorem 2.
Since x 1(n) (t) and x 1(m) (t) are bounded, we can find the following positive constants, , β, θ, η and ρ such that for all t:
Theorem 2.
Considering Equations (36) and (37), we obtain:
Theorem 2.
and:
Theorem 2.
where
Theorem 2.
, r(γ), s(γ) and o(γ) are functions from (33). We next show that K satisfies Property 1. Consider Equations (38) and (39), yielding:
Theorem 2.
We conclude that K is Picard K-stable.
Uniqueness of the Special Solution
Theorem 3. We consider the Hilbert space H = L 2 ((a, b) × (0, k)) that can be defined as the set of those functions:
Uniqueness of the Special Solution
We now consider the following operator:
Uniqueness of the Special Solution
Proof. We prove that the inner product of:
Uniqueness of the Special Solution
where (x 11 (t) − x 12 (t)), x 21 (t) − x 22 (t) are special solutions of the system. We can assume that
Uniqueness of the Special Solution
Using the relationship between the norm and the inner function, we get:
Uniqueness of the Special Solution
and:
Uniqueness of the Special Solution
(43) for large number m and n, both solutions converge to the exact solution; if η = ||X 1 − X 11 ||, ||X 1 − X 12 || and ν = ||x 2 − x 21 ||, ||x 2 − x 22 ||, we have:
Uniqueness of the Special Solution
and:
Uniqueness of the Special Solution
where λ n and λ m are two very small positive parameters.
Uniqueness of the Special Solution
Using the topology concept, we conclude that λ n < 0 and λ m < 0, where:
Uniqueness of the Special Solution
This completes the proof.
Uniqueness of the Special Solution
Involving the Atangana-Baleanu fractional integral, we can propose a numerical solution using the Adams-Moulton rule:
Uniqueness of the Special Solution
where:
Uniqueness of the Special Solution
Considering the above numerical scheme, we have:
Uniqueness of the Special Solution
and:
Uniqueness of the Special Solution
The proof of existence is described in detail by Alkahtani in [20] .
Numerical Results
Example 1. We present numerical simulations of the special solution of our model using the Adams-Moulton rule given by Equation ( 
Conclusions
A Freedman model was considered using the fractional derivatives of the Liouville-Caputo and Atangana-Baleanu-Caputo types. The solutions of the alternative models were obtained using an iterative scheme based on the Laplace transform and the Sumudu transform. Furthermore, we employed the fixed point theorem to study the stability analysis of the iterative methods, and using properties of the inner product and the Hilbert space, the uniqueness of the special solution was presented in detail. Additionally, special solutions via the Adams-Moulton rule were obtained for both fractional derivatives. The results obtained using the Liouville-Caputo and Atangana-Baleanu-Caputo derivatives are exactly the same as the ordinary case. However, as γ takes values smaller than one, the results obtained become a little different, having a remarkable difference when γ < 0.9. This is due to the kernel involved in the definitions of the fractional derivative. The computer used for obtaining the results in this paper is an Intel Core i7, 2.6-GHz processor, 16.0 GB RAM (MATLAB R.2013a).

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