On Approximate Solutions for Fractional Logistic Differential Equation

Abstract

A new approximate formula of the fractional derivatives is derived. The proposed formula is based on the generalized Laguerre polynomials. Global approximations to functions defined on a semi-infinite interval are constructed. The fractional derivatives are presented in terms of Caputo sense. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. The new spectral Laguerre collocation method is presented for solving fractional Logistic differential equation (FLDE). The properties of Laguerre polynomials approximation are used to reduce FLDE to solve a system of algebraic equations which is solved using a suitable numerical method. Numerical results are provided to confirm the theoretical results and the efficiency of the proposed method.

1. Introduction

Ordinary and partial fractional differential equations (FDEs) have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics, and engineering [1]. Fractional calculus is a generalization of ordinary differentiation and integration to an arbitrary noninteger order. Many physical processes appear to exhibit fractional order behavior that may vary with time or space. Most FDEs do not have exact solutions, so approximate and numerical techniques [2–8] must be used. Several numerical and approximate methods to solve FDEs have been given such as variational iteration method [9–12], homotopy perturbation method [13], Adomian’s decomposition method [14, 15], and collocation method [16, 17].

The fractional Logistic model can be obtained by applying the fractional derivative operator on the Logistic equation. The model is initially published by Pierre Verhulst in 1838 [18, 19]. The continuous Logistic model is described by first-order ordinary differential equation. The discrete Logistic model is simple iterative equation that reveals the chaotic property in certain regions [20]. There are many variations of the population modeling [19, 21]. The Verhulst model is the classic example to illustrate the periodic doubling and chaotic behavior in dynamical system [20]. The model which is described the population growth may be limited by certain factors like population density [18, 19, 21].

Applications of Logistic Equation. A typical application of the Logistic equation is a common model of population growth. Another application of Logistic curve is in medicine, where the Logistic differential equation is used to model the growth of tumors. This application can be considered an extension of the above-mentioned use in the framework of ecology. Denoting by 𝑢(𝑡) the size of the tumor at time 𝑡.

The solution of Logistic equation is explained the constant population growth rate which not includes the limitation on food supply or spread of diseases [19]. The solution curve of the model is increasing exponentially from the multiplication factor up to saturation limit which is maximum carrying capacity [19], 𝑑𝑁/𝑑𝑡=𝜌𝑁(1−(𝑁/𝐾)) where 𝑁 is the population size with respect to time, 𝜌 is the rate of maximum population growth, and 𝐾 is the carrying capacity. The solution of continuous Logistic equation is in the form of constant growth rate as in formula 𝑁(𝑡)=𝑁₀𝑒^𝜌𝑡 where 𝑁₀ is the initial population [22].

In this paper, we consider FLDE of the form 𝐷^𝜈𝑢(𝑡)=𝜌𝑢(𝑡)(1−𝑢(𝑡)),𝑡>0,𝜌>0, (1) here, the parameter 𝜈 refers to the fractional order of time derivative with 0<𝜈≤1.

We also assume an initial condition 𝑢(0)=𝑢₀,𝑢₀>0. (2) For 𝜈=1, (1) is the standard Logistic differential equation 𝑑𝑢(𝑡)(𝑡(𝑡.𝑑𝑡=𝜌𝑢)(1−𝑢)) (3) The exact solution to this problem is 𝑢(𝑡)=𝑢₀/((1−𝑢₀)𝑒^−𝜌𝑡+𝑢₀).

The existence and the uniqueness of the proposed problem (1) are introduced in details in [23, 24].

The main aim of the presented paper is concerned with an extension of the previous work on FDEs and derive an approximate formula of the fractional derivative of the Laguerre polynomials and then we apply this approach to obtain the numerical solution of FLDE. Also, we present study of the convergence analysis and estimate an error upper bound of the proposed formula.

The structure of this paper is arranged in the following way: in Section 2, we introduce some basic definitions about Caputo fractional derivatives and properties of the Laguerre polynomials. In Section 3, we give an approximate formula of the fractional derivative of Laguerre polynomials and its convergence analysis. In Section 4, we implement the proposed method for solving FLDE to show the accuracy of the presented method. Finally, in Section 5, the paper ends with a brief conclusion and some remarks.

2. Preliminaries and Notations

In this section, we present some necessary definitions and mathematical preliminaries of the fractional calculus theory that will be required in the present paper.

2.1. The Caputo Fractional Derivative

Definition 1. The Caputo fractional derivative operator 𝐷^𝜈 of order 𝜈 is defined in the following form: 𝐷^𝜈𝑓(𝑥)=1Γ(𝑚−𝜈)^𝑥₀𝑓^(𝑚)(𝑡)(𝑥−𝑡)^{𝜈−𝑚+1}𝑑𝑡,𝜈>0,𝑥>0, (4) where 𝑚−1<𝜈≤𝑚, 𝑚∈ℕ.
Similar to integer-order differentiation, Caputo fractional derivative operator is linear 𝐷^𝜈𝜆𝑓(𝑥)+𝜇𝑔(𝑥)=𝜆𝐷^𝜈𝑓(𝑥)+𝜇𝐷^𝜈𝑔(𝑥), (5) where 𝜆 and 𝜇 are constants. For the Caputo’s derivative we have 𝐷^𝜈𝐷𝐶=0,𝐶isaconstant,_𝜈𝑥^𝑛=0,for𝑛∈ℕ₀⌈𝜈⌉;Γ(),𝑛<𝑛+1Γ()𝑥𝑛+1−𝜈^𝑛−𝜈,for𝑛∈ℕ₀,𝑛≥⌈𝜈⌉. (6)(7) We use the ceiling function ⌈𝜈⌉ to denote the smallest integer greater than or equal to 𝜈, and ℕ₀={0,1,2,…}. Recall that for 𝜈∈ℕ, the Caputo differential operator coincides with the usual differential operator of integer order.

For more details on fractional derivatives definitions and their properties see [1, 25–28].

2.2. The Definition and Properties of the Generalized Laguerre Polynomials

Spectral collocation methods are efficient and highly accurate techniques for numerical solution of nonlinear differential equations. The basic idea of the spectral collocation method is to assume that the unknown solution 𝑢(𝑡) can be approximated by a linear combination of some basis functions, called the trial functions, such as orthogonal polynomials. The orthogonal polynomials can be chosen according to their special properties, which make them particularly suitable for a problem under consideration. In [16], Khader introduced an efficient numerical method for solving the fractional diffusion equation using the shifted Chebyshev polynomials. In [29] the generalized Laguerre polynomials were used to compute a spectral solution of a nonlinear boundary value problems. The generalized Laguerre polynomials constitute a complete orthogonal sets of functions on the semi-infinite interval [0,∞). Convolution structures of Laguerre polynomials were presented in [30]. Also, other spectral methods based on other orthogonal polynomials are used to obtain spectral solutions on unbounded intervals [31].

The generalized Laguerre polynomials [𝐿_𝑛^(𝛼)(𝑥)]^∞_𝑛=0, 𝛼>−1 are defined on the unbounded interval (0,∞) and can be determined with the aid of the following recurrence formula: (𝑛+1)𝐿^(𝛼)_𝑛+1(𝑥)+(𝑥−2𝑛−𝛼−1)𝐿_𝑛^(𝛼)(𝑥)+(𝑛+𝛼)𝐿^(𝛼)_𝑛−1(𝑥)=0,𝑛=1,2,…, (8) where 𝐿₀^(𝛼)(𝑥)=1 and 𝐿₁^(𝛼)(𝑥)=𝛼+1−𝑥.

The analytic form of these polynomials of degree 𝑛 is given by 𝐿_𝑛^(𝛼)(𝑥)=^𝑛_𝑘=0(−1)^𝑘𝑥𝑘!𝑛+𝛼𝑛−𝑘^𝑘=𝑛𝑛+𝛼^𝑛_𝑘=0(−𝑛)_𝑘(𝛼+1)_𝑘𝑥^𝑘,𝑘! (9)𝐿_𝑛^(𝛼)(0)=(_𝑛^𝑛+𝛼). These polynomials are orthogonal on the interval [0,∞) with respect to the weight function 𝑤(𝑥)=(1/Γ(1+𝛼))𝑥^𝛼𝑒^−𝑥. The orthogonality relation is 1Γ(1+𝛼)^∞₀𝑥^𝛼𝑒^−𝑥𝐿_𝑚^(𝛼)(𝑥)𝐿_𝑛^(𝛼)(𝑥)𝑛𝑑𝑥=𝛿𝑛+𝛼_𝑚𝑛. (10) Also, they satisfy the differentiation formula 𝐷^𝑘𝐿_𝑛^(𝛼)(𝑥)=(−1)^𝑘𝐿^(𝛼+𝑘)_𝑛−𝑘(𝑥),𝑘=0,1,…,𝑛. (11) Any function 𝑢(𝑥) belongs to the space 𝐿²_𝑤[0,∞) of all square integrable functions on [0,∞) with weight function 𝑤(𝑥) can be expanded in the following Laguerre series: 𝑢(𝑥)=^∞_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(𝑥), (12) where the coefficients 𝑐_𝑖 are given by 𝑐_𝑖=Γ(𝑖+1)Γ(𝑖+𝛼+1)^∞₀𝑥^𝛼𝑒^−𝑥𝐿_𝑖^(𝛼)(𝑥)𝑢(𝑥)𝑑𝑥,𝑖=0,1,…. (13) Consider only the first (𝑚+1) terms of generalized Laguerre polynomials, so we can write 𝑢_𝑚(𝑥)=^𝑚_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(𝑥). (14) For more details on Laguerre polynomials, its definitions, and properties, see [29, 31, 32].

3. An Approximate Fractional Derivative of 𝐿_𝑛^(𝛼)(𝑥) and Its Convergence Analysis

The main goal of this section is to introduce the following theorems to derive an approximate formula of the fractional derivatives of the generalized Laguerre polynomials and study the truncating error and its convergence analysis.

Lemma 2. Let 𝐿_𝑛^(𝛼)(𝑥) be a generalized Laguerre polynomial then 𝐷^𝜈𝐿_𝑛^(𝛼)(𝑥)=0,𝑛=0,1,…,⌈𝜈⌉−1,𝜈>0. (15)

Proof. This lemma can be proved directly by applying (6)-(7) on (9).

The main approximate formula of the fractional derivative of 𝑢(𝑥) is given in the following theorem.

Theorem 3. Let 𝑢(𝑥) be approximated by the generalized Laguerre polynomials as (14) and also suppose 𝜈>0; then its approximated fractional derivative can be written in the following form: 𝐷^𝜈𝑢_𝑚(𝑥)≅^𝑚_⌈𝜈⌉^𝑖_{𝑖=⌈𝜈⌉𝑘=}𝑐_𝑖𝑤^(𝜈)_𝑖,𝑘𝑥^𝑘−𝜈, (16) where 𝑤^(𝜈)_𝑖,𝑘 is given by 𝑤^(𝜈)_𝑖,𝑘=(−1)^𝑘Γ(𝑘+1−𝜈)𝑖+𝛼𝑖−𝑘. (17)

Proof. Since the Caputo’s fractional differentiation is a linear operation, we obtain 𝐷^𝜈𝑢_𝑚(𝑥)=^𝑚_𝑖=0𝑐_𝑖𝐷^𝜈𝐿_𝑖^(𝛼)(𝑥). (18) Also, from (9) we can get 𝐷^𝜈𝐿_𝑖^(𝛼)(𝑥)=0,𝑖=0,1,…,⌈𝜈⌉−1,𝜈>0. (19) Therefore, for 𝑖=⌈𝜈⌉,⌈𝜈⌉+1,…,𝑚, and by using (6)-(7) in (9), we get 𝐷^𝜈𝐿_𝑖^(𝛼)(𝑥)=^𝑖_𝑘=0(−1)^𝑘𝐷𝑘!𝑖+𝛼𝑖−𝑘^𝜈𝑥^𝑘=^𝑖_{⌈𝜈⌉𝑘=}(−1)^𝑘Γ()𝑥𝑘+1−𝜈𝑖+𝛼𝑖−𝑘^𝑘−𝜈. (20) A combination of (18)–(20) leads to the desired result (16) and ends the proof of the theorem.

Test Example. Consider the function 𝑢(𝑥)=𝑥³ with 𝑚=3, 𝜈=1.5, and 𝛼=−0.5, the generalized Laguerre series of 𝑥³ is 𝑥³=1.875𝐿₀^(𝛼)(𝑥)−11.25𝐿₁^(𝛼)(𝑥)+15𝐿₂^(𝛼)(𝑥)−6𝐿₃^(𝛼)(𝑥). (21) Now, by using formula (16), we obtain 𝐷^1.5𝑥³=^3𝑖_𝑖=2𝑘=2𝑐_𝑖𝑤^(1.5)_𝑖,𝑘𝑥^𝑘−1.5, (22) where 𝑤^(1.5)_2,2=1.12838, 𝑤^(1.5)_3,2=2.82095, 𝑤^(1.5)_3,3=−0.752253, therefore, 𝐷^1.5𝑥³=𝑐₂𝑤^(1.5)_2,2𝑥^0.5+𝑐₃𝑤^(1.5)_3,2𝑥^0.5+𝑐₃𝑤^(1.5)_3,3𝑥^1.5=Γ(4)Γ()𝑥2.5^1.5, (23) which agrees with the exact derivative (7).

Theorem 4. The Caputo fractional derivative of order 𝜈 for the generalized Laguerre polynomials can be expressed in terms of the generalized Laguerre polynomials themselves in the following form: 𝐷^𝜈𝐿_𝑖^(𝛼)(𝑥)=^𝑖_{⌈𝜈⌉𝑘=}^{𝑘−⌈𝜈⌉}_𝑗=0Ω_𝑖𝑗𝑘𝐿_𝑗^(𝛼)(𝑥),𝑖=⌈𝜈⌉,⌈𝜈⌉+1,…,𝑚, (24) where Ω_𝑖𝑗𝑘=^𝑗_𝑟=0(−1)^𝑟+𝑘(𝛼+𝑖)!𝑗!(𝑘+𝛼−𝜈+𝑟)!()!()!()!()!.𝑘−𝜈𝑖−𝑘𝛼+𝑘!𝑟!𝑗−𝑟𝛼+𝑟 (25)

Proof. From the properties of the generalized Laguerre polynomials [33] and expanding 𝑥^𝑘−𝜈 in (20) in the following form: 𝑥^𝑘−𝜈=^{𝑘−⌈𝜈⌉}_𝑗=0𝑐_𝑘𝑗𝐿_𝑗^(𝛼)(𝑥), (26) where 𝑐_𝑘𝑗 can be obtained using (13), where 𝑢(𝑥)=𝑥^𝑘−𝜈, then 𝑐_𝑘𝑗=Γ𝑗+1Γ𝑗+1+𝛼^∞₀𝑥^{𝑘+𝛼−𝜈}𝑒^−𝑥𝐿_𝑗^(𝛼)(𝑥)=𝑑𝑥_𝑗𝑟=0(−1)^𝑟𝑗!(𝑘−𝜈+𝛼+𝑟)!!()!,𝑟!𝑗−𝑟𝛼+𝑟𝑗=0,1,…, (27) this by substituting from (9) and using the definition of Gamma function. Now, we can write (26) in the following form: 𝑥^𝑘−𝜈=^{𝑘−⌈𝜈⌉𝑗}_𝑗=0𝑟=0(−1)^𝑟𝑗!(𝑘−𝜈+𝛼+𝑟)!!()!𝐿𝑟!𝑗−𝑟𝛼+𝑟^𝑗(𝛼)(𝑥). (28) Therefore, the Caputo fractional derivative 𝐷^𝜈𝐿_𝑖^(𝛼)(𝑥) in (20) can be rewritten in the following form: 𝐷^𝜈𝐿_𝑖^(𝛼)(𝑥)=^𝑖_{⌈𝜈⌉𝑘=}^{𝑘−⌈𝜈⌉𝑗}_𝑗=0𝑟=0(−1)^𝑟+𝑘()!𝑗!()!×()!()!()!×()!𝛼+𝑖𝑘−𝜈+𝛼+𝑟𝑘−𝜈𝑖−𝑘𝛼+𝑘!𝑟!𝑗−𝑟𝛼+𝑟⁻¹𝐿_𝑗^(𝛼)(𝑥), (29) for 𝑖=⌈𝜈⌉,⌈𝜈⌉+1,…,𝑚. Equation (29) leads to the desired result (24) and this completes the proof of the theorem.

Theorem 5. For the Laguerre polynomials 𝐿_𝑛^(𝛼)(𝑥), one has the following global uniform bounds estimates: 𝐿_𝑛^(𝛼)(𝑥≤())𝛼+1_𝑛𝑒𝑛!^𝑥/2(),𝑓𝑜𝑟𝛼≥0,𝑥≥0,𝑛=0,1,…;2−𝛼+1_𝑛𝑒𝑛!^𝑥/2,𝑓𝑜𝑟−1<𝛼≤0,𝑥≥0,𝑛=0,1,…. (30)

Proof. These estimates were presented in [33–35].

Theorem 6. The error in approximating 𝐷^𝜈𝑢(𝑥) by 𝐷^𝜈𝑢_𝑚(𝑥) is bounded by 𝐸_𝑇(𝑚)≤^∞_𝑖=𝑚+1𝑐_𝑖Π_𝜈()𝑖,𝑗𝛼+1_𝑗𝑒𝑗!^𝑥/2,𝐸𝛼≥0,𝑥≥0,𝑗=0,1,…,_𝑇(𝑚)≤^∞_𝑖=𝑚+1𝑐_𝑖Π_𝜈()𝑖,𝑗2−𝛼+1_𝑗𝑒𝑗!^𝑥/2,−1<𝛼≤0,𝑥≥0,𝑗=0,1,…, (31) where |𝐸_𝑇(𝑚)|=|𝐷^𝜈𝑢(𝑥)−𝐷^𝜈𝑢_𝑚(𝑥)| and Π_𝜈(𝑖,𝑗)=∑^𝑖_{𝑘=⌈𝜈⌉}∑^{𝑘−⌈𝜈⌉}_𝑗=0Ω_𝑖𝑗𝑘.

Proof. A combination of (12), (14), and (24) leads to 𝐸_𝑇(𝑚)=𝐷^𝜈𝑢(𝑥)−𝐷^𝜈𝑢_𝑚(𝑥≤_{∞𝑖=𝑚+1})𝑐_𝑖Π_𝜈𝑖,𝑗𝐿_𝑗^(𝛼)(𝑥), (32) using (30) and subtracting the truncated series from the infinite series, bounding each term in the difference, and summing the bounds completes the proof of the theorem.

4. Implementation of Laguerre Spectral Method for Solving FLDE

In this section, we introduce a numerical algorithm using Laguerre spectral method for solving the fractional Logistic differential equation of the form (1).

The procedure of the implementation is given by the following steps. (1) Approximate the function 𝑢(𝑡) using the formula (14) and its Caputo fractional derivative 𝐷^𝜈𝑢(𝑡) using the presented formula (16) with 𝑚=5, then FLDE (1) is transformed to the following approximated form: ^5𝑖_𝑖=1𝑘=1𝑐_𝑖𝑤^(𝜈)_𝑖,𝑘𝑡^𝑘−𝜈−𝜌⁵_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(𝑡×)1−⁵_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(𝑡)=0, (33) where 𝑤^(𝜈)_𝑖,𝑘 is defined in (17). We now collocate (33) at (𝑚+1−⌈𝜈⌉) points 𝑡_𝑝, 𝑝=0,1,…,𝑚−⌈𝜈⌉ as ^5𝑖_𝑖=1𝑘=1𝑐_𝑖𝑤^(𝜈)_𝑖,𝑘𝑡_𝑝^𝑘−𝜈−𝜌⁵_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)𝑡_𝑝×1−⁵_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)𝑡_𝑝=0. (34)(2) From the initial condition (2) we obtain the following equation: ⁵_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(0)=𝑢₀. (35) Equations (34)-(35) represent a system of nonlinear algebraic equations which contains six equations for the unknowns 𝑐_𝑖, 𝑖=0,1,…,5. (3) Solve the resulting system using the Newton iteration method to obtain the unknowns 𝑐_𝑖, 𝑖=0,1,…,5. Therefore, the approximate solution will take the form 𝑢(𝑡)=𝑐₀𝐿₀^(𝛼)(𝑡)+𝑐₁𝐿₁^(𝛼)(𝑡)+𝑐₂𝐿₂^(𝛼)(𝑡)+𝑐₃𝐿₃^(𝛼)(𝑡)+𝑐₄𝐿₄^(𝛼)(𝑡)+𝑐₅𝐿₅^(𝛼)(𝑡). (36)The numerical results of the proposed problem (1) are given in Figures 1 and 2 with different values of 𝜈 in the interval [0,1] with 𝜌=0.5 and 𝑢₀=0.25. Where in Figure 1, we presented a comparison between the behavior of the exact solution and the approximate solution using the introduced technique at 𝜈=1 (Figure 1(a)), and the behavior of the approximate solution using the proposed method at 𝜈=0.85 (Figure 1(b)). But, in Figure 2, we presented the behavior of the approximate solution with different values of 𝜈 (𝜈=0.65 (Figure 2(a)) and 𝜈=0.45 (Figure 2(b))).

(a)

(b)

Figure 1 

A comparison between the approximate solution and the exact solution at 𝜈=1 (a). The behavior of the approximate solution using the proposed method at 𝜈=0.85 (b).

(a)

(b)

Figure 2 

The behavior of the approximate solution using the proposed method at 𝜈=0.65 (a) and at 𝜈=0.45 (b).

5. Conclusion and Remarks

In this paper, we introduced a new spectral collocation method based on Laguerre polynomials for solving FLDE. We have introduced an approximate formula for the Caputo fractional derivative of the generalized Laguerre polynomials in terms of generalized Laguerre polynomials themselves. In the proposed method we used the properties of the Laguerre polynomials to reduce FLDE to solve a system of algebraic equations. The error upper bound of the proposed approximate formula is stated and proved. The obtained numerical results show that the proposed algorithm converges as the number of 𝑚 terms is increased. The solution is expressed as a truncated Laguerre series and so it can be easily evaluated for arbitrary values of time using any computer program without any computational effort. From illustrative examples, we can conclude that this approach can obtain very accurate and satisfactory results. Comparisons are made between the approximate solution and the exact solution to illustrate the validity and the great potential of the technique. All computations are done using Matlab.