Solving Famous Nonlinear Coupled Equations with Parameters Derivative by Homotopy Analysis Method

Abstract

The homotopy analysis method (HAM) is employed to obtain symbolic approximate solutions for nonlinear coupled equations with parameters derivative. These nonlinear coupled equations with parameters derivative contain many important mathematical physics equations and reaction diffusion equations. By choosing different values of the parameters in general formal numerical solutions, as a result, a very rapidly convergent series solution is obtained. The efficiency and accuracy of the method are verified by using two famous examples: coupled Burgers and mKdV equations. The obtained results show that the homotopy perturbation method is a special case of homotopy analysis method.

1. Introduction

Fractional differential equations have gained importance and popularity during the past three decades or so, mainly due to its demonstrated applications in numerous seemingly diverse fields of science and engineering. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives, and the fluid-dynamic traffic model with fractional derivatives can eliminate the deficiency arising from the assumption of continuum traffic flow. The differential equations with fractional order have recently proved to be valuable tools to the modeling of many physical phenomena [1, 2]. This is because of the fact that the realistic modeling of a physical phenomenon does not depend only on the instant time, but also on the history of the previous time which can also be successfully achieved by using fractional calculus. Most nonlinear fractional equations do not have exact analytic solutions, so approximation and numerical techniques must be used. The Adomain decomposition method [3], the homotopy perturbation method [4], the variational iteration method [5], and other methods have been used to provide analytical approximation to linear and nonlinear problems. However, the convergence region of the corresponding results is rather small. In 1992, Liao employed the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely, homotopy analysis method [6–10]. This method has been successfully applied to solve many types of nonlinear problems in science and engineering, such as the viscous flows of non-Newtonian fluids [11], the KdV-type equations [12], finance problems [13], fractional Lorenz system [14], and delay differential equation [15]. The HAM contains a certain auxiliary parameter ℎ which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution.

The HAM offers certain advantages over routine numerical methods. Numerical methods use discretization which gives rise to rounding off errors causing loss of accuracy and requires large computer memory and time. This computational method yields analytical solutions and has certain advantages over standard numerical methods. The HAM method is better since it does not involve discretization of the variables and hence is free from rounding off errors and does not require large computer memory or time.

In this paper, we extend the application of HAM to discuss the explicit numerical solutions of a type of nonlinear-coupled equations with parameters derivative in this form: 𝜕^𝛼𝑢𝜕𝑡^𝛼𝜕=𝐿₁(𝑢,𝑣)+𝑁₁(𝑢,𝑣),𝑡>0,_𝛽𝑣𝜕𝑡^𝛽=𝐿₂(𝑢,𝑣)+𝑁₂(𝑢,𝑣),𝑡>0,(1.1) where 𝐿_𝑖 and 𝑁_𝑖 (𝑖=1,2) are the linear and nonlinear functions of 𝑢 and 𝑣, respectively, 𝛼 and 𝛽 are the parameters that describe the order of the derivative. Different nonlinear coupled systems can be obtained when one of the parameters 𝛼 or 𝛽 varies. The study of (1.1) is very necessary and significant because when 𝛼 and 𝛽 are integers, it contains many important mathematical physics equations.

The paper has been organized as follows. Notations and basic definitions are given in Section 2. In Section 3 the homotopy analysis method is described. In Section 4 applying HAM for two famous coupled examples: Burgers and mKdV equations. Discussion and conclusions are presented in Section 5.

2. Description on the Fractional Calculus

Definition 2.1. A real function 𝑓(𝑡),𝑡>0 is said to be in the space 𝐶_𝜇, 𝜇∈𝑅 if there exists a real number 𝑝>𝜇, such that 𝑓(𝑡)=𝑡^𝑝𝑓₁(𝑡) where 𝑓₁∈(0,∞), and it is said to be in the space 𝐶^𝜇𝑛 l if and only if ℎ(𝑛)∈𝐶_𝜇, 𝑛∈𝑁. Clearly 𝐶_𝜇⊂𝐶_𝜈 if 𝜈≤𝜇.

Definition 2.2. The Riemann-Liouville fractional integral operator (𝐽^𝛼) of order 𝛼≥0, of a function 𝑓∈𝐶_𝜇, 𝜇≥−1, is defined as 𝐽^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥0𝐽(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡,𝑥>0.₀𝑓(𝑥)=𝑓(𝑥).(2.1)Γ(𝛼) is the well-known Gamma function. Some of the properties of the operator 𝐽^𝛼, which we will need here, are as follows.
For 𝑓∈𝐶_𝜇, 𝜇≥−1, 𝛼,𝛽≥0 and 𝛾≥−1𝐽𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛼+𝛽𝑓(𝑥),_𝛼𝐽𝐽^𝛽𝑓(𝑥)=𝐽^𝛽𝐽^𝛼𝑓(𝑥),_𝛼𝑡^𝛾=Γ(𝛾+1)Γ(𝛼+𝛾+1)𝑡^𝛼+𝛾.(2.2)

Definition 2.3. For the concept of fractional derivative, there exist many mathematical definitions [1, 16–19]. In this paper, the two most commonly used definitions: the Caputo derivative and its reverse operator Riemann-Liouville integral are adopted. That is because Caputo fractional derivative [1] allows the traditional assumption of initial and boundary conditions. The Caputo fractional derivative is defined as 𝐷^𝛼𝑡𝑢(𝑥,𝑡)=𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=⎧⎪⎨⎪⎩Γ(𝑛1−𝛼)^𝑡0(𝑡−𝜏)^{𝑛−𝛼−1}𝜕^𝑛𝑢(𝑥,𝑡)𝜕𝑡^𝑛𝜕𝑑𝜏,𝑛−1<𝛼<𝑛,_𝑛𝑢(𝑥,𝑡)𝜕𝑡^𝑛,𝛼=𝑛∈𝑁.(2.3) Here, we also need two basic properties about them: 𝐽𝐷^𝛼𝐽^𝛼𝑓(𝑥)=𝑓(𝑥),_𝛼𝐷^𝛼𝑓(𝑥)=𝑓(𝑥)−^∞_𝑘=0𝑓^(𝑘)0⁺𝑥^𝑘𝑘!,𝑥>0.(2.4)

Definition 2.4. The Mittag-Leffler function 𝐸_𝛼(𝑧) with 𝑎>0 is defined by the following series representation, valid in the whole complex plane: 𝐸_𝛼(𝑧)=^∞_𝑛=0𝑧^𝑛Γ(𝛼𝑛+1),𝛼>0,𝑧∈C.(2.5)

3. Basic Idea of HAM

To describe the basic ideas of the HAM, we consider the operator form of (1.1):𝑁𝑁𝐷𝐷^𝛼𝑡𝑢(𝑥,𝑡)=0,𝑡>0,_𝛽𝑡𝑣(𝑥,𝑡)=0,𝑡>0,(3.1) where 𝑁 is nonlinear operator, 𝐷^𝛼𝑡 and 𝐷^𝛽𝑡 stand for the fractional derivative and are defined as in (2.3), 𝑡 denotes an independent operator, and 𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡) are unknown functions.

By means of generalizing the traditional homotopy method, Liao [6] constructs the so-called zero-order deformation equations:((11−−𝑞𝑞))𝐿𝐿𝜙𝜙₁₂(𝑥,𝑡,𝑞)−𝑢₀(𝑥,𝑡)=𝑞ℎ𝐻(𝑡)𝑁𝐷^𝛼𝑡𝜙₁(𝑥,𝑡,𝑞),(3.2)(𝑥,𝑡,𝑞)−𝑣₀(𝑥,𝑡)=𝑞ℎ𝐻(𝑡)𝑁𝐷^𝛽𝑡𝜙₂(𝑥,𝑡,𝑞),(3.3) where 𝑞∈[0,1] is the embedding parameter, ℎ≠0 is a non-zero auxiliary parameter, 𝐻(𝑡)≠0 is an auxiliary function, 𝐿 is an auxiliary linear operator, 𝑢₀(𝑥,𝑡), 𝑣₀(𝑥,𝑡) are initial guesses of 𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡) and 𝜙₁(𝑥,𝑡,𝑞), 𝜙₂(𝑥,𝑡,𝑞) are two unknown functions, respectively. It is important that one has great freedom to choose auxiliary things in HAM. Obviously, when 𝑞=0 and 𝑞=1, the following holds: 𝜙𝜙₁(𝑥,𝑡,0)=𝑢₀(𝑥,𝑡),𝜙₁(𝑥,𝑡,1)=𝑢(𝑥,𝑡),₂(𝑥,𝑡,0)=𝑣₀(𝑥,𝑡),𝜙₂(𝑥,𝑡,1)=𝑣(𝑥,𝑡),(3.4) respectively. Thus, as 𝑞 increases from 0 to 1, the solution 𝜙₁(𝑥,𝑡,𝑞), 𝜙₂(𝑥,𝑡,𝑞) varies from the initial guess 𝑢₀(𝑥,𝑡), 𝑣₀(𝑥,𝑡) to the solution 𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡). Expanding 𝜙₁(𝑥,𝑡,𝑞), 𝜙₂(𝑥,𝑡,𝑞) in Taylor series with respect to 𝑞, we have𝜙₁(𝑥,𝑡,𝑞)=𝑢₀(𝑥,𝑡)+^+∞_𝑚₌₁𝜙₂𝑢_𝑚(𝑥,𝑡)𝑞^𝑚,(𝑥,𝑡,𝑞)=𝑣₀(𝑥,𝑡)+^+∞_𝑚₌₁𝑣_𝑚(𝑥,𝑡)𝑞^𝑚,(3.5) where𝑢_𝑚(𝑥,𝑡)=1𝑚!𝜕^𝑚𝜙₁(𝑥,𝑡,𝑞)𝜕𝑞^𝑚𝑣_𝑚||_𝑞=0,(𝑥,𝑡)=1𝑚!𝜕^𝑚𝜙₂(𝑥,𝑡,𝑞)𝜕𝑞^𝑚||_𝑞=0.(3.6) If the auxiliary linear operator, the initial guess, the auxiliary parameter ℎ, and the auxiliary function are so properly chosen, the series (3.5) converges at 𝑞=1, then we have𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+^+∞_𝑚₌₁𝑣(𝑥,𝑡)=𝑣₀𝑢_𝑚(𝑥,𝑡),(𝑥,𝑡)+^+∞_𝑚₌₁𝑣_𝑚(𝑥,𝑡),(3.7) which must be one of solutions of original nonlinear equation, as proved by Liao [8]. As ℎ=−1 and 𝐻(t)=1, (3.2) and (3.3) become((11−−𝑞𝑞))𝐿𝐿𝜙𝜙₁₂(𝑥,𝑡,𝑞)−𝑢₀(𝑥,𝑡)+𝑞𝑁𝜙₁(𝑥,𝑡,𝑞)=0,(𝑥,𝑡,𝑞)−𝑢₀(𝑥,𝑡)+𝑞𝑁𝜙₂(𝑥,𝑡,;𝑞)=0,(3.8) which is used mostly in the homotopy perturbation method [20], where as the solution obtained directly, without using Taylor series. According to the definition (3.6), the governing equation can be deduced from the zero-order deformation equation (3.2). Define the vector 𝑢_𝑛⃗=𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),…,𝑢_𝑛(𝑥,𝑡),𝑣_𝑛=𝑣₀(𝑥,𝑡),𝑣₁(𝑥,𝑡),…,𝑣_𝑛(𝑥,𝑡).(3.9) Differentiating equations (3.2) and (3.3) 𝑚 times with respect to the embedding parameter 𝑞 and then setting 𝑞=0 and finally dividing them by 𝑚!, we have the so-called 𝑚th-order deformation equation: 𝐿𝐿𝑣𝑢_𝑚𝑚(𝑥,𝑡)−𝜒_𝑚𝑢_𝑚−1(𝑥,𝑡)=ℎ𝐻(𝑡)𝑅_1,𝑚𝑢_𝑚−1,(𝑥,𝑡)−𝜒_𝑚𝑣_𝑚−1(𝑥,𝑡)=ℎ𝐻(𝑡)𝑅_2,𝑚𝑣_𝑚−1,(3.10) where 𝑅_1,𝑚𝑢_𝑚−1=1(𝑚−1)!𝜕^𝑚−1𝐷^𝛼𝑡𝜙₁(𝑥,𝑡,𝑞)𝜕𝑞^𝑚−1𝑅_2,𝑚||_𝑞=0,𝑣_𝑚−1=(𝑚−11)!𝜕^𝑚−1𝐷^𝛽𝑡𝜙₂(𝑥,𝑡,𝑞)𝜕𝑞^𝑚−1𝜒_𝑚||_𝑞=0,=⎧⎪⎨⎪⎩01,,𝑚𝑚⩽>11,.(3.11) Applying the Riemann-Liouville integral operator 𝐽^𝛼,𝐽^𝛽 on both side of (3.10), we have𝑢_𝑚(𝑥,𝑡)=𝜒_𝑚𝑢_𝑚−1(𝑥,𝑡)−𝜒_𝑚𝑛−1_𝑖=0𝑢^𝑖𝑚−10⁺𝑡^𝑖𝑖!+ℎ𝐻(𝑡)𝐽^𝛼𝑣_𝑚𝑅_1,𝑚𝑢_𝑚−1,(𝑥,𝑡)=𝜒_𝑚𝑣_𝑚−1(𝑥,𝑡)−𝜒_𝑚𝑛−1_𝑖=0𝑣^𝑖𝑚−10⁺𝑡^𝑖𝑖!+ℎ𝐻(𝑡)𝐽^𝛽𝑅_2,𝑚𝑣_𝑚−1.(3.12) It should be emphasized that 𝑢_𝑚(𝑥,𝑡), 𝑣_𝑚(𝑥,𝑡) for 𝑚⩾1 is governed by the linear equation (3.10), under the linear boundary conditions that come from original problem, which can be easily solved by symbolic computation software such as MATLAB. For the convergence of the above method we refer the reader to Liao's work. Liao [7] proved that, as long as a series solution given by the homotopy analysis method converges, it must be one of exact solutions. So, it is important to ensure that the solution series is convergent. Note that the solution series contain the auxiliary parameter ℎ, which we can choose properly by plotting the so-called ℎ-curves to ensure solution series converge.

Remark 3.1. The parameters 𝛼 and 𝛽 can be arbitrarily chosen as, integer or fraction, bigger or smaller than 1. When the parameters are bigger than 1, we will need more initial and boundary conditions such as 𝑢^0(𝑥,0), 𝑢^0(𝑥,0),… and the calculations will become more complicated correspondingly. In order to illustrate the problem and make it convenient for the readers, we only confine the parameters to [0,1] to discuss.

4. Application

4.1. The Nonlinear Coupled Burgers Equations with Parameters Derivative

In order to illustrate the method discussed above, we consider the nonlinear coupled Burgers equations with parameters derivative in an operator form:𝐷𝐷^𝛼𝑡𝑢−𝐿_𝑥𝑥𝑢−2𝑢𝐿_𝑥𝑢+𝐿_𝑥𝑢𝑣=0,(0<𝛼≤1),_𝛽𝑡𝑣−𝐿_𝑥𝑥𝑣−2𝑣𝐿_𝑥𝑣+𝐿_𝑥𝑢𝑣=0,(0<𝛽≤1),(4.1) where 𝑡>0, 𝐿_𝑥=𝜕/𝜕𝑥 and the fractional operators 𝐷^𝛼𝑡 and 𝐷^𝛽𝑡 are defined as in (2.3). Assuming the initial value as𝑢(𝑥,0)=sin𝑥,𝑣(𝑥,0)=sin𝑥.(4.2) The exact solutions of (4.1) for the special case: 𝛼=𝛽=1 are𝑢(𝑥,𝑡)=𝑒^−𝑡sin𝑥,𝑣(𝑥,𝑡)=𝑒^−𝑡sin𝑥.(4.3) For application of homotopy analysis method, in view of (4.1) and the initial condition given in (4.2), it is convenient to choose𝑢₀(𝑥,𝑡)=sin𝑥,𝑣₀(𝑥,𝑡)=sin𝑥,(4.4) as the initial approximate of (4.1). We choose the linear operators𝐿𝐿₁𝜙₁(𝑥,𝑡,𝑞)=𝐷^𝛼𝑡𝜙₁(𝑥,𝑡,𝑞)₂,𝜙₂(𝑥,𝑡,𝑞)=𝐷^𝛽𝑡𝜙₂(𝑥,𝑡,𝑞),(4.5) with the property 𝐿(𝑐)=0 where 𝑐 is constant of integration. Furthermore, we define a system of nonlinear operators as 𝑁₁𝜙_𝑖(𝑥,𝑡,𝑞)=𝐷^𝛼𝑡𝜙₁(𝑥,𝑡,𝑞)−𝜕²𝜙₁(𝑥,𝑡,𝑞)𝜕𝑥²−2𝜙₁(𝑥,𝑡,𝑞)𝜕𝜙¹(𝑥,𝑡,𝑞)+𝜕𝜙𝜕𝑥₁(𝑥,𝑡,𝑞)𝜙₂(𝑥,𝑡,𝑞)𝑁𝜕𝑥₂𝜙_𝑖(𝑥,𝑡,𝑞)=𝐷^𝛽𝑡𝜙₂(𝑥,𝑡,𝑞)−𝜕²𝜙₂(𝑥,𝑡,𝑞)𝜕𝑥²−2𝜙₂(𝑥,𝑡,𝑞)𝜕𝜙²(𝑥,𝑡,𝑞)+𝜕𝜙𝜕𝑥₁(𝑥,𝑡,𝑞)𝜙₂(𝑥,𝑡,𝑞)𝜕𝑥.(4.6) We construct the zeroth-order and the 𝑚th-order deformation equations where 𝑅_1,𝑚𝑢_𝑚−1=𝐷^𝛼𝑡𝑢_𝑚−1−𝑢_𝑚−1_𝑥𝑥−2^𝑚−1_𝑘=0𝑢_𝑘𝑢_{𝑚−1−𝑘}_𝑥+^𝑚−1_𝑘=0𝑢_𝑘𝑣_{𝑚−𝑘−1}_𝑥𝑅_2,𝑚,𝑣_𝑚−1=𝐷^𝛽𝑡𝑣_𝑚−1−𝑣_𝑚−1_𝑥𝑥−2^𝑚−1_𝑘=0𝑣_𝑘𝑣_{𝑚−1−𝑘}_𝑥+^𝑚−1_𝑘=0𝑢_𝑘𝑣_{𝑚−𝑘−1}_𝑥.(4.7) We start with an initial approximation 𝑢(𝑥,0)=sin(𝑥), 𝑣(𝑥,0)=sin(𝑥), thus we can obtain directly the other components as 𝑢₁=ℎ𝑡^𝑎sin(𝑥)𝑢Γ(𝑎+1)₂,=−sin𝑥×𝑎²Γ𝑎(𝑏+1)Γ(𝑎)²Γ(𝑎+(1/2))₂Γ(𝑏+1)Γ(𝑎)²Γ𝑎+12−𝑎²Γ(𝑏+1)Γ(𝑎)²Γ𝑎+12ℎ+ℎ𝑎Γ(𝑎)Γ(2𝑎+1)𝑡^𝑎Γ(𝑏+1)+ℎ²𝑎Γ(𝑎)Γ𝑎+12𝑡^𝑎Γ𝑎+1Γ(𝑏+1)+2ℎ²𝑎Γ(𝑎)2𝑡^(𝑏+𝑎)cos(𝑥)−2ℎ²𝑡^(2𝑎)cos(𝑥)Γ(𝑏+1)Γ𝑎+1+ℎ₂2𝑣₁⋮𝑡^(2𝑎)𝑎²Γ(𝑏+1)Γ(𝑎)²=ℎ𝑡^𝑏sin(𝑥)𝑣Γ(𝑏+1)₂,=−sin𝑥×𝑏²𝑏Γ(𝑎+1)Γ(𝑏)²Γ(𝑏+(1/2))₂Γ(Γ2𝑏(𝑎++11))𝑡_𝑏Γ(𝑏)²Γ(2𝑏+1)ℎ+ℎ𝑏Γ(𝑏),𝑡Γ(𝑎+1)+ℎ²𝑏Γ(𝑏)Γ(2𝑏+1)𝑡^𝑏Γ(𝑏+1)+2ℎ²𝑏Γ(𝑏)Γ(2𝑏+1),_(𝑎+𝑏)cos(𝑥)−2ℎ²𝑡^(2𝑏)cos(𝑥)Γ(𝑎+1)Γ(2𝑏+1)+ℎ²1𝑡^(−𝑏)Γ(𝑏)₂𝑡^𝑏𝑏²Γ(𝑎+1),⋮(4.8) The absolute error of the 6th-order HAM and exact solution with ℎ=−1 as shown in Figure 1. Also the absolute errors |𝑢(𝑡)−𝜙₆(𝑡)| have been calculated in Table 1. Figure 2 shows the numerical solutions of the nonlinear coupled Burgers equations with parameters derivative with ℎ=−1, 𝛼=𝛽=1. Figure 3 shows the explicit numerical solutions with ℎ=−1, 𝛼=1/4, and 𝛽=1/3 at 𝑡=0.02.

Table 1 

The comparison of the results of the HAM (ℎ=−1) and exact solution for the 𝑢(𝑥,𝑡), 𝛼=𝛽=1.

(a)

(b)

Figure 1 

The comparison of the 6th-order HAM and exact solution with ℎ=−1,𝛼=𝛽=1.

(a)

(b)

Figure 2 

Explicit numerical solutions withℎ=−1, 𝛼=𝛽=1.

(a)

(b)

Figure 3 

Explicit numerical solutions with ℎ=−1, 𝛼=1/4, and𝛽=1/3.

As suggested by Liao [7], the appropriate region for ℎ is a horizontal line segment. We can investigate the influence of ℎ on the convergence of the solution series gevin by the HAM, by plotting its curve versus h, as shown in Figure 4.

Figure 4 

The ℎ-curves obtained from the 5-order HAM approximate solution.

Remark 4.1. This example has been solved using homotopy perturbation method [21]. The graphs drawn and tables by ℎ=−1 are in excellent agreement with that graphs drawn with HPM.

4.2. The Nonlinear Coupled mKdV Equations with Parameters Derivative

In order to illustrate the method discussed above, we consider the nonlinear coupled mKdV equations with parameters derivative in an operator form:𝐷^𝛼𝑡𝑢−12𝑢^𝑥𝑥𝑥+3𝑢²𝑢_𝑥−32𝑣^𝑥𝑥𝐷−3(𝑢𝑣)_𝑥+3𝜆𝑢_𝑥=0,_𝛽𝑡𝑣+𝑣_𝑥𝑥𝑥+3𝑣𝑣_𝑥+3𝑢_𝑥𝑣_𝑥−3𝑢²𝑣_𝑥−3𝜆𝑣_𝑥=0,(4.9) with the initial conditions,𝑢(𝑥,0)=𝑏2𝑘+𝑘tanh(𝑘𝑥),𝑣(𝑥,0)=𝜆21+𝑘𝑏+𝑏tanh(𝑘𝑥).(4.10) As we know, when 𝛼=𝛽=1 (4.9) has the kink-type soliton solutions𝑢(𝑥,𝑡)=𝑏𝑣(𝑥,𝑡)=𝜆2𝑘+𝑘tanh(𝑘𝜉),21+𝑘𝑏+𝑏tanh(𝑘𝜉),(4.11) constructed by Fan [22], where 𝜉=𝑥+(1/4)(−4𝑘²−6𝜆+6𝑘𝜆/𝑏+3𝑏²/𝑘²)𝑡, 𝑘≠0, and 𝑏≠0.

For application of homotopy analysis method, in view of (4.9) and the initial condition given in (4.10), it in convenient to choose𝑢(𝑥,0)=𝑏2𝑘+𝑘tanh(𝑘𝑥),𝑣(𝑥,0)=𝜆21+𝑘𝑏+𝑏tanh(𝑘𝑥),(4.12) as the initial approximate of (4.10). We choose the linear operators𝐿𝐿₁𝜙₁(𝑥,𝑡,𝑞)=𝐷^𝛼𝑡𝜙₁(𝑥,𝑡,𝑞)₂,𝜙₂(𝑥,𝑡,𝑞)=𝐷^𝛽𝑡𝜙₂(𝑥,𝑡,𝑞),(4.13) with the property 𝐿(𝑐)=0 where 𝑐 is constant of integration. Furthermore, we define a system of nonlinear operators as 𝑁₁𝜙_𝑖(𝑥,𝑡,𝑞)=𝐷^𝛼𝑡𝜙₁(𝑥,𝑡,𝑞)−12𝜕³𝜙₁(𝑥,𝑡,𝑞)𝜕𝑥³+3𝜙₁(𝑥,𝑡,𝑞)²𝜕𝜙₁(𝑥,𝑡,𝑞)−3𝜕𝑥,2𝜕²𝜙₂(𝑥,𝑡,𝑞)𝜕𝑥²−3𝜕𝜙¹(𝑥,𝑡,𝑞)𝜙₂(𝑥,𝑡,𝑞)𝜕𝑥+3𝜆𝜕𝜙¹(𝑥,𝑡,𝑞)𝑁𝜕𝑥₂,𝜙_𝑖(𝑥,𝑡,𝑞)=𝐷^𝛽𝑡𝜙₂(𝑥,𝑡,𝑞)+𝜕³𝜙₂(𝑥,𝑡,𝑞)𝜕𝑥³+3𝜙₂(𝑥,𝑡,𝑞)𝜕𝜙²(𝑥,𝑡,𝑞)+3𝜕𝜙₁𝜕𝑥,(𝑥,𝑡,𝑞)𝜕𝑥𝜕𝜙²(𝑥,𝑡,𝑞)𝜕𝑥−3𝜙¹(𝑥,𝑡,𝑞)²𝜕𝜙₂(𝑥,𝑡,𝑞)𝜕𝑥−3𝜆𝜕𝜙²(𝑥,𝑡,𝑞)𝜕𝑥.(4.14) We construct the zeroth-order and the 𝑚th-order deformation equations where 𝑅_1,𝑚𝑢_𝑚−1=𝐷^𝛼𝑡𝑢_𝑚−1−12𝑢^𝑚−1_𝑥𝑥𝑥+3^𝑚−1_𝑖=0𝑢_{𝑖𝑚−1−𝑖}_𝑘=0−3𝑢_𝑘𝑣_{𝑚−1−𝑖−𝑘}_𝑥,2𝑣^𝑚−1_𝑥𝑥−3^𝑚−1_𝑘=0𝑢_𝑘𝑣_{𝑚−𝑘−1}_𝑥𝑅_2,𝑚+3𝜆𝑢_𝑚−1_𝑥,𝑣_𝑚−1=−𝐷3^𝛽𝑡𝑣_{𝑚−1𝑚−1}+𝑣_𝑚−1_𝑥𝑥𝑥+3^𝑚−1_𝑘=0𝑣_𝑘𝑣_{𝑚−1−𝑘}_𝑥+3^𝑚−1_𝑘=0𝑢_𝑘_𝑥𝑣_{𝑚−𝑘−1}_𝑥,_𝑖=0𝑢_{𝑖𝑚−1−𝑖}_𝑘=0𝑢_𝑘𝑣_{𝑚−1−𝑖−𝑘}_𝑥−3𝜆𝑣_𝑚−1_𝑥.(4.15) We start with an initial approximation 𝑢(𝑥,0)=(𝑏/2𝑘)+𝑘tanh(𝑘𝑥), 𝑣(𝑥,0)=(𝜆/2)(1+(𝑘/𝑏))+𝑏tanh(𝑘𝑥), with 𝑘=0.1, 𝑏=1,𝑘=1/3, thus we can obtain directly the other components as follows: 𝑢₁=11771620ℎ𝑡^𝑎−1+tanh((1/3)𝑥)²𝑢Γ(𝑎+1)₂,=1437400−656100Γ(𝑎+𝑏+1)𝑎²Γ(𝑎+𝑏+1)𝑎₂Γ(𝑏+1)Γ(𝑎)²Γ(2𝑎+1)−145800,Γ(𝑏+1)Γ(𝑎)²Γ(2a+1)tanh13𝑥−656100Γ(𝑎+𝑏+1)𝑎²Γ(𝑏+1)Γ(𝑎)₂,ℎtanh1Γ(2𝑎+1)ℎ−145800Γ(𝑎+𝑏+1)𝑎²Γ(𝑏+1)Γ(𝑎)²Γ(2𝑎+1),3𝑥−327510ℎ²𝑎₂𝑡^(𝑏+𝑎)𝑎²Γ(𝑏+1)Γ(𝑎)²Γ(2𝑎+1)+1310040ℎ²𝑡^(𝑏+𝑎),Γ(𝑏+1)Γ(𝑎)²Γ(2𝑎+1)tanh13𝑥²Γ(2𝑎+1)tanh1−982530ℎ²𝑡^(𝑏+𝑎)𝑎²Γ(𝑏+1)Γ(𝑎)²,3𝑥⁴𝑣⋮+⋯₁−317790ℎΓ(𝑎+𝑏+1)𝑎Γ(𝑎)Γ(2𝑎+1)𝑡^𝑎Γ(𝑏+1),=1213540ℎ𝑡^𝑏−1+tanh((1/3)𝑥)²𝑣Γ(𝑏+1)₂,=1145800−145800𝑏²Γ(𝑎+1)Γ(𝑏)²Γ(2𝑏+1)ℎtanh13𝑥−145800𝑏²Γ(𝑎+1)Γ(𝑏)₂,Γ(2𝑏+1)tanh13𝑥−9720𝑏²−327510ℎ𝑏Γ(𝑏)Γ(2𝑏+1)𝑡_𝑏Γ(𝑎+1)Γ(𝑏)²Γ(2𝑏+1)ℎ,+317790ℎ₂Γ(𝑎+1)−327510ℎ²𝑏Γ(𝑏)Γ(2𝑏+1)𝑡^𝑏Γ(𝑎+1),tanh1𝑏Γ(𝑏)Γ(2𝑏+1)𝑡^(𝑏+𝑎)−635580ℎ²𝑏Γ(𝑏)Γ(2𝑏+1)𝑡^(𝑏+𝑎),3𝑥²−282480ℎ²𝑏Γ(𝑏)Γ(2𝑏+1)𝑡^(𝑏+𝑎)tanh13𝑥³tanh1+141240ℎ²𝑏,3𝑥⁴⋮+⋯(4.16)

The absolute error of the 6th-order HAM and exact solution with ℎ=−1 as shown in Figure 5. Also the absolute errors |𝑢(𝑡)−𝜙₆(𝑡)| have been calculated for in Table 2. Figure 6 shows the numerical solutions of the nonlinear coupled Burgers equations with parameters derivative with ℎ=−1, 𝛼=𝛽=1. Figure 7 shows the explicit numerical solutions with ℎ=−1, 𝛼=1/2, and 𝛽=2/3 at 𝑡=0.002.

Table 2 

The comparison of the results of the HAM (ℎ=−1) and exact solution for the 𝑢(𝑥,𝑡), 𝛼=𝛽=1.

(a)

(b)

Figure 5 

The comparison of the 6th-order HAM and exact solution withℎ =−1, 𝛼=𝛽=1, 𝜆=0.1, 𝑏=1, and 𝑘=1/3.

(a)

(b)

Figure 6 

Explicit numerical solutions withℎ=−1, 𝛼=𝛽=1, 𝜆=0.1, 𝑏=1, and 𝑘=1/3.

(a)

(b)

Figure 7 

Explicit numerical solutions with ℎ=−1, 𝛼=1/2, 𝛽=2/3, 𝜆=0.1, 𝑏=1, and 𝑘=1/3.

As suggested by Liao [7], the appropriate region for ℎ is a horizontal line segment. We can investigate the influence of ℎ on the convergence of the solution series gevin by the HAM, by plotting its curve versus h, as shown in Figure 8.

Figure 8 

The ℎ-curves obtained from the 5-order HAM approximate solution.

Remark 4.2. This example has been solved using homotopy perturbation method [21]. The graphs drawn and tables by ℎ=−1 are in excellent agreement with those graphs drawn with HPM.

5. Conclusion

In this paper, based on the symbolic computation MATLAB, the HAM is directly extended to derive explicit and numerical solutions of the nonlinear coupled equations with parameters derivative. HAM provides us with a convenient way to control the convergence of approximation series by adapting ℎ, which is a fundamental qualitative difference in analysis between HAM and other methods. The obtained results demonstrate the reliability of the HAM and its wider applicability to fractional differential equation. It, therefore, provides more realistic series solutions that generally converge very rapidly in real physical problems. MATLAB has been used for computations in this paper.