Numerical Solutions for the Time and Space Fractional Nonlinear Partial Differential Equations

Abstract

We implement relatively analytical techniques, the homotopy perturbation method, and variational iteration method to find the approximate solutions for time and space fractional Benjamin-Bona Mahony equation. The fractional derivatives are described in the Caputo sense. These methods are used in applied mathematics to obtain the analytic approximate solutions for the nonlinear Bejamin-Bona Mahoney (BBM) partial fractional differential equation. We compare between the approximate solutions obtained by these methods. Also, we present the figures to compare between the approximate solutions. Also, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved (𝐺^′/𝐺)-expansion function method to find exact solutions of nonlinear fractional BBM equation.

1. Introduction

In recent years, there has been a great deal of interest in fractional differential equations. First there were almost no practical applications of fractional calculus, and it was considered by many as an abstract area containing only mathematical manipulations of little or no use. Nearly 30 years ago, the paradigm began to shift from pure mathematical formulations to applications in various fields. During the last decade, fractional calculus has been applied to almost every field of science, engineering, and mathematics. Several fields of application of fractional differentiation and fractional integration are already well established, and some others have just started. Many applications of fractional calculus can be found in turbulence and fluid dynamics, stochastic dynamical system, plasma physics and controlled thermonuclear fusion, nonlinear control theory, image processing, nonlinear biological systems, and astrophysics [1–11]. There has been some attempt to solve linear problems with multiple fractional derivatives (the so-called multiterm equations) [1, 12]. Not much work has been done for nonlinear problems, and only a few numerical schemes have been proposed to solve nonlinear fractional differential equations. More recently, applications have included classes of nonlinear equation with multiorder fractional derivative, and this motivates us to develop a numerical scheme for their solutions [13]. Numerical and analytical methods have included Adomian decomposition method (ADM) [14–17], variational iteration method (VIM) [18–20], homotopy perturbation method [21–24], homotopy analysis method [25–27], and the fractional complex transformation [28, 29] to get some special exact solutions for nonlinear partial fractional differential equation.

Recently, Zhao [30] investigated the nature of protein dynamics and thermodynamics. Olszewski [31] studied the uncertainty relation between intervals of energy and time derived for the electromagnetic radiation of a harmonic oscillator. El-Naggar et al. [32] investigated the initial stress, magnetic field, voids, and rotation effects on plane waves in generalized thermoelasticity. Khrennikov [33] discussed “Einstein’s Dream” Quantum Mechanics as Theory of Classical Random Fields.

The main objective of this paper is to use two different methods such as homotopy perturbation method and variational iteration method for calculating the analytic approximate solutions of the Bejamin-Bona Mahoney equation. We compare between the approximate solutions which obtained from these methods. We make the figures to compare the approximate solutions. Also, we find some exact solutions to time and space fractional Bejamin-Bona Mahoney equation by using the improved (𝐺^′/𝐺)-expansion function method.

2. Preliminaries and Notations

We give some basic definitions and properties of the fractional calculus theory which are used further in this paper [1, 2]. For the finite derivative [a,b], we define the following fractional integral and derivatives.

Definition 1. If 𝑓(𝑡)∈𝐿₁(𝑎,𝑏), the set of all integrable functions, and 𝛼>0 then, the Riemann-Liouville fractional integral of order 𝛼, denoted by 𝐼^𝛼_𝑎+, is defined by 𝐼^𝛼_𝑎+𝑓(𝜏)=1Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏. (1)

Definition 2. For 𝛼>0, the Caputo fractional derivative of order 𝛼, denoted by ^𝑐𝐷^𝛼_𝑎+, is defined by ^𝑐𝐷^𝛼_𝑎+𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎(𝑡−𝜏)^{𝑛−𝛼−1}𝐷^𝑛𝑓(𝜏)𝑑𝜏, (2) where 𝑛 is such that 𝑛−1<𝛼<𝑛 and 𝐷=𝑑/𝑑𝜏.
If 𝛼 is an integer, then this derivative takes the ordinary derivative ^𝑐𝐷^𝛼_𝑎+=𝐷^𝛼,𝛼=1,2,3,…. (3)
Finally the Caputo fractional derivative on the whole space 𝑅 is defined by the following.

Definition 3. For 𝛼>0, the Caputo fractional derivative of order 𝛼 on the whole space, denoted by ^𝑐𝐷^𝛼₊, is defined by ^𝑐𝐷^𝛼₊𝑓(𝑥)=1Γ(𝑛−𝛼)^𝑥_−∞(𝑥−𝜉)^{𝑛−𝛼−1}𝐷^𝑛𝑓(𝜉)𝑑𝜉. (4)

In recent years, in order to investigate the local behavior of fractional models, several local versions of fractional derivatives have been proposed, that is, the Kolwankar-Gangal local fractional derivative [35], Chen’s fractal derivative [36], Cresson’s derivative [37], and Jumarie’s modified Riemann-Liouville derivative [38, 39]. Jumarie’s derivative is defined as 𝐷^𝛼_𝑥𝑓(𝑥)=1Γ()𝑑1−𝛼×𝑑𝑥^𝑥_−∞(𝑥−𝜉)^−𝛼𝑓(𝜉)−𝑓(0)𝑑𝜉,0<𝛼<1, (5) where 𝑓∶𝑅→𝑅, 𝑥→𝑓(𝑥) denotes a continuous (but not necessarily first-order-differentiable) function. We can obtain the following properties.

Property 1. Let 𝑓(𝑥) satisfy the definition of the modified Riemann-Liouville derivative, and let 𝑓(𝑥) be a 𝑘-order differentiable function. The generalized Taylor series is given by [38, 40] 𝑓(𝑥+ℎ)=^∞_𝑘=0ℎ^𝛼𝑘()!𝑓𝛼𝑘^(𝛼𝑘)(𝑥),0<𝛼<1. (6)

Property 2. Assume that 𝑓(𝑥) denotes a continuous 𝑅→𝑅 function. We use the following equality for the integral with respect to (𝑑𝑥)^𝛼 [39, 40]: 𝐼^𝛼_𝑥𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝜉)^𝛼−1𝑓(𝜉)𝑑𝜉. (7)

Property 3. Some useful formulas include 𝑓^(𝛼)[𝑥(𝑡)]𝑑𝑓𝑥𝑑𝑥^(𝛼)(𝑡),𝐷^𝛼_𝑥𝑥^𝛽=Γ1+𝛽Γ𝑥1+𝛽−𝛼^𝛽−𝛼,(𝑑𝑥)^𝛽=𝑥^𝛽. (8)(9)(10)

Function 𝑓(𝑥) should be differentiable with respect to 𝑥(𝑡), and 𝑥(𝑡) is fractional differentiable in (8). The above results are employed in the following sections. The modified Riemann-Liouville derivative has been successfully applied in the probability calculus [41], the fractional Laplace problems [42], the fractional variational approach with several variables [43], the fractional variational iteration method [44], the fractional variational approach with natural boundary conditions [45], and the fractional Lie group method [46].

3. Basic Idea of the Homotopy Perturbation Method (HPM)

In this section, we illustrate some basic concepts of the homotopy perturbation method for the following nonlinear partial fractional differential equation [21–24]: 𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=𝑣(𝑥,𝑡)−ℓ𝑢(𝑥,𝑡)−𝑁̌𝑢(𝑥,𝑡),𝑚−1<𝛼<𝑚,𝑚∈𝑁,𝑡≥0,𝑥∈𝑅^𝑛, (11) subject to the initial and boundary conditions 𝑢^(𝑖)(0,0)=𝑐_𝑖,𝐵𝑢,𝜕𝑢,𝜕𝑡𝜕𝑢𝜕𝑥_𝑗=0,𝑖=0,1,…,𝑚−1,𝑗=0,1,…,𝑛, (12) where ℓ is a linear operator, while 𝑁̌ is a nonlinear operator, 𝑣 is a known analytical function, and 𝐷^𝛼_𝑡 denotes the fractional derivative in the Caputo sense. The solution 𝑢 is assumed to be a causal function of time, that is, vanishing for 𝑡<0. Also, 𝑢^(𝑖)(𝑥,𝑡) is the 𝑖th derivative of 𝑢,𝑐_𝑖,𝑖=0,1,…,𝑚−1, are the specified initial conditions, and 𝐵 is a boundary operator. Using He’s homotopy perturbation technique [47], we can construct the following simple homotopy: 1−𝑝𝐷^𝛼_𝑡𝑢(𝑥,𝑡)+𝑝𝐷^𝛼_𝑡𝑢(𝑥,𝑡)+ℓ𝑢(𝑥,𝑡)+𝑁̌𝑢()(𝑥,𝑡−𝑣𝑥,𝑡)=0,𝑝∈[0,1], (13) or 𝐷^𝛼_𝑡𝑢(𝑥,𝑡)+𝑝ℓ𝑢(𝑥,𝑡)+𝑁̌𝑢(𝑥,𝑡)−𝑣([].𝑥,𝑡)=0,𝑝∈0,1 (14)

The homotopy parameter 𝑝 always changes from zero to unity. In the case of 𝑝=0, (13) or (14) becomes 𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=0, (15) and when 𝑝=1, (13) or (14) turns out to be the original fractional differential equation.

Applying the homotopy perturbation method, we use the homotopy parameter 𝑝 to expand the solution into the following form: 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑝𝑢₁(𝑥,𝑡)+𝑝²𝑢₂(𝑥,𝑡)+𝑝³𝑢₃(𝑥,𝑡)+⋯. (16)

For nonlinear problems, let us set 𝑁̌𝑢(𝑥,𝑡)=𝑆(𝑥,𝑡). Substituting (16) into (13) or (14) and equating the terms with identical powers of 𝑝, we can obtain a series of equations of the form 𝑝⁰∶𝐷^𝛼_𝑡𝑢₀()𝑝𝑥,𝑡=0,₁∶𝐷^𝛼_𝑡𝑢₁(𝑥,𝑡)=−ℓ𝑢₀(𝑥,𝑡)−𝑆₀𝑢₀(𝑥,𝑡)+𝑣(),𝑝𝑥,𝑡₂∶𝐷^𝛼_𝑡𝑢₂(𝑥,𝑡)=−ℓ𝑢₁(𝑥,𝑡)−𝑆₁𝑢₀(𝑥,𝑡),𝑢₁(),𝑝𝑥,𝑡₃∶𝐷^𝛼_𝑡𝑢₃(𝑥,𝑡)=−ℓ𝑢₂(𝑥,𝑡)−𝑆₂𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡), (17) and so on, where the functions 𝑆₀,𝑆₁,𝑆₂,… satisfy the following equations: 𝑆𝑢₀(𝑥,𝑡)+𝑝𝑢₁(𝑥,𝑡)+𝑝²𝑢₂(𝑥,𝑡)+⋯=𝑆₀𝑢₀(𝑥,𝑡)+𝑝𝑆₁𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡)+𝑝²𝑆₂𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡)+⋯. (18)

Applying the operator 𝐼^𝛼_𝑡 on both sides of (17) and considering the initial and boundary conditions, the terms of the series solution can be given by 𝑢₀(𝑥,𝑡)=^𝑛−1_𝑖=0𝑐_𝑖𝑡^𝑖,𝑢𝑖!₁(𝑥,𝑡)=−𝐼^𝛼_𝑡𝐿́𝑢₀(𝑥,𝑡)−𝐼^𝛼_𝑡𝑆₀𝑢₀(𝑥,𝑡)+𝐼^𝛼_𝑡[𝑣(,𝑢𝑥,𝑡)]_𝑗(𝑥,𝑡)=−𝐼^𝛼_𝑡𝐿́𝑢_𝑗−1(𝑥,𝑡)−𝐼^𝛼_𝑡𝑆_𝑗−1𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),…,𝑢_𝑗−1(𝑥,𝑡),𝑗=2,3,…. (19)(20)(21)

On setting 𝑝=1, we get an accurate approximate solution in the following form: 𝑢(𝑥,𝑡)=^∞_𝑖=0𝑢_𝑖(𝑥,𝑡). (22)

4. Basic Idea of Variational Iteration Method (VIM)

The basic concepts of the variational iteration method can be expressed as follows. Consider the differential equation of the form ℓ𝑢(𝑥,𝑡)+𝑁̌𝑢(𝑥,𝑡)=𝑓(𝑥,𝑡), (23) where ℓ is a linear operator, 𝑁̌ is a nonlinear operator, and 𝑓(𝑥,𝑡) is the inhomogeneous term. According to VIM, a correction functional for (23) can be constructed as follows: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛()+𝑥,𝑡_𝑡0𝜆(𝑠)ℓ𝑢_𝑛(𝑥,𝑠)+𝑁̌𝑢_𝑛()()𝑥,𝑠−𝑓𝑥,𝑠𝑑𝑠,𝑛=0,1,2,…, (24) where 𝜆 is a general Lagrange multiplier, which can be identified optimally via the variational theory, the subscript 𝑛 denotes the 𝑛th approximations, and 𝑢_𝑛 is considered as restricted variation; that is, 𝛿𝑢_𝑛=0. The successive approximations 𝑢_𝑛+1(𝑥,𝑡), 𝑛=0,1,2,…, of the solution 𝑢(𝑥,𝑡) can be obtained after finding the Lagrange multiplier and by using the selective function 𝑢₀(𝑥,𝑡) which is usually selected from initial conditions. Consequently, the solution is obtained as 𝑢(𝑥,𝑡)=lim_𝑛→∞𝑢_𝑛(𝑥,𝑡). (25)

5. Approximate Solutions for the Time and Space Fractional BBM Equation

In this section, we use the homotopy perturbation method and variational iteration method to calculate the approximate solutions for the time and space fractional Bejamin-Bona Mahoney equation. Also, we compare the approximate solutions obtained from these methods.

5.1. The HPM for the Time and Space Fractional Nonlinear BBM Equation

In this subsection, we use the HPM to construct the approximate solution of the time and space fractional nonlinear BBM equation in the form 𝜕^𝛼𝑢𝜕𝑡^𝛼𝜕+𝐶^𝛽𝑢𝜕𝑥^𝛽+𝐿𝑢²𝜕^𝛽𝑢𝜕𝑥^𝛽𝜕+𝑁^𝛼𝜕𝑡^𝛼𝜕^𝛽𝜕𝑥^𝛽𝜕^𝛽𝑢𝜕𝑥^𝛽=0,𝑡>0,0<∝,𝛽≤1, (26) where 𝐶, 𝐿, and 𝑁 are constants.

With the initial condition, 𝑢(𝑥,0)=𝑒^−𝑘𝑥. (27)

By the homotopy perturbation technique, we construct a homotopy 𝑉(𝑟,𝑝)∶Ω×[0,1]→𝑅 which satisfies 𝐻𝑉,𝑝=1−𝑝𝐷^𝛼_𝑡𝑉−𝐷^𝛼_𝑡𝑉₀+𝑝𝐷^𝛼_𝑡𝜕𝑉+𝐶^𝛽𝑉𝜕𝑥^𝛽+𝐿𝑉²𝜕^𝛽𝑉𝜕𝑥^𝛽𝜕+𝑁^𝛼𝜕𝑡^𝛼𝜕^𝛽𝜕𝑥^𝛽𝜕^𝛽𝑉𝜕𝑥^𝛽=0,𝑟∈Ω. (28)

According to the homotopy perturbation method, we can first use the embedding parameter 𝑝 as a small parameter and assume that the solution of (28) can be written as a power series in 𝑝 as follows: 𝑉(𝑥,𝑡)=𝑉₀(𝑥,𝑡)+𝑝𝑉₁(𝑥,𝑡)+𝑝²𝑉₂(𝑥,𝑡)+𝑝³𝑉₃(𝑥,𝑡)+⋯. (29)

Substituting (29) into (28) and arranging the coefficients of the powers of 𝑝, after some calculation we obtain 𝜕^𝛼𝑉₀𝜕𝑡^𝛼𝜕=0,_𝛼𝑉₁𝜕𝑡^𝛼𝜕+𝐶^𝛽𝑉₀𝜕𝑥^𝛽+𝐿𝑉²₀𝜕^𝛽𝑉₀𝜕𝑥^𝛽𝜕+𝑁^𝛼𝜕𝑡^𝛼𝜕^𝛽𝜕𝑥^𝛽𝜕^𝛽𝑉₀𝜕𝑥^𝛽𝜕=0,_𝛼𝑉₂𝜕𝑡^𝛼𝜕+𝐶^𝛽𝑉₁𝜕𝑥^𝛽+𝐿𝑉²₀𝜕^𝛽𝑉₁𝜕𝑥^𝛽+2𝐿𝑉₀𝑉₁𝜕^𝛽𝑉₀𝜕𝑥^𝛽𝜕+𝑁^𝛼𝜕𝑡^𝛼𝜕^𝛽𝜕𝑥^𝛽𝜕^𝛽𝑉₁𝜕𝑥^𝛽𝜕=0,_𝛼𝑉₃𝜕𝑡^𝛼𝜕+𝐶^𝛽𝑉₂𝜕𝑥^𝛽+𝐿𝑉²₀𝜕^𝛽𝑉₂𝜕𝑥^𝛽+2𝐿𝑉₀𝑉₁𝜕^𝛽𝑉₁𝜕𝑥^𝛽+2𝐿𝑉₀𝑉₂𝜕^𝛽𝑉₀𝜕𝑥^𝛽+𝐿𝑉²₁𝜕^𝛽𝑉₀𝜕𝑥^𝛽𝜕+𝑁^𝛼𝜕𝑡^𝛼𝜕^𝛽𝜕𝑥^𝛽𝜕^𝛽𝑉₂𝜕𝑥^𝛽=0, (30) and so on. We substitute the initial condition (27) into the system (30), then the approximate solutions of (26) take the following form: 𝑉₀(𝑥,𝑡)=𝑒^−𝑘𝑥,𝑉₁(𝑥,𝑡)=−𝑡^𝛼Γ(𝛼+1)𝐶(−𝑘)^𝛽𝑒^−𝑘𝑥+𝐿(−𝑘)^𝛽𝑒^−3𝑘𝑥,𝑉₂()=𝑡𝑥,𝑡^2𝛼Γ()𝑓2𝛼+1¹𝐶+𝐿𝑉²₀𝑡+2^2𝛼Γ(2𝛼+1)𝐿𝑒^−2𝑘𝑥(−𝑘)^𝛽𝑡𝑓+𝑁^𝛼Γ()𝑓𝛼+1²,𝑉₃(𝑥,𝑡)=−𝐶+𝐿𝑒^−2𝑘𝑥𝐶+𝐿(−2𝑘)^𝛽𝑒^−2𝑘𝑥𝑓₂𝑡^3𝛼Γ(3𝛼+1)+2𝐿𝑒^−2𝑘𝑥(−2𝑘)^𝛽(−𝑘)^𝛽𝑡^3𝛼Γ()𝑓3𝛼+1¹+𝑁𝑓₃𝑡^2𝛼Γ(2𝛼+1)−2𝐿𝑒^−𝑘𝑥𝑡^2𝛼Γ(2𝛼+1)²𝑓𝑓₁−2𝐿𝑒^−𝑘𝑥𝐶+𝐿𝑉²₀𝑡^3𝛼Γ()𝑓3𝛼+1¹+2𝐿𝑒^−2𝑘𝑥(−𝑘)^𝛽×𝑡^3𝛼Γ()𝑡3𝛼+1𝑓+𝑁^2𝛼Γ()𝑓2𝛼+1²(−𝑘)^𝛽𝑒^−𝑘𝑥𝑡−𝐿^2𝛼Γ(2𝛼+1)²𝑓²(−𝑘)^𝛽𝑒^−𝑘𝑥−𝑁𝐶+𝐿(−2𝑘)^𝛽𝑒^−2𝑘𝑥𝑓₃𝑡^2𝛼Γ(2𝛼+1)+2𝐿𝑒^−2𝑘𝑥×(−2𝑘)^2𝛽(−𝑘)^𝛽𝑡^2𝛼Γ()𝑓2𝛼+1²+𝑁𝑓₄𝑡^𝛼Γ(),𝛼+1 (31) and so on. Thus the approximate solution of (26) is given by 𝑢(𝑥,𝑡)=𝑒^−𝑘𝑥−𝑡^𝛼Γ(𝛼+1)−𝐶(−𝑘)^𝛽𝑒^−𝑘𝑥−𝐿(−𝑘)^𝛽𝑒^−3𝑘𝑥+𝑡^2𝛼Γ()𝑓2𝛼+1¹𝐶+𝐿𝑒^−2𝑘𝑥𝑡+2^2𝛼Γ(2𝛼+1)𝐿𝑒^−2𝑘𝑥(−𝑘)^𝛽𝑡𝑓+𝑁^𝛼Γ()𝑓𝛼+1²−𝐶+𝐿𝑒^−2𝑘𝑥𝐶+𝐿(−2𝑘)^𝛽𝑒^−2𝑘𝑥×𝑓₂𝑡^3𝛼Γ(3𝛼+1)+2𝐿𝑒^−2𝑘𝑥(−2𝑘)^𝛽×(−𝑘)^𝛽𝑡^3𝛼Γ()𝑓3𝛼+1¹+𝑁𝑓₃𝑡^2𝛼Γ(2𝛼+1)−2𝐿𝑒^−𝑘𝑥𝑡^2𝛼Γ(2𝛼+1)²𝑓𝑓₁−2𝐿𝑒^−𝑘𝑥𝐶+𝐿𝑉²₀𝑡^3𝛼Γ()𝑓3𝛼+1¹+2𝐿𝑒^−2𝑘𝑥×(−𝑘)^𝛽𝑡^3𝛼Γ()𝑡3𝛼+1𝑓+𝑁^2𝛼Γ()𝑓2𝛼+1²×(−𝑘)^𝛽𝑒^−𝑘𝑥𝑡−𝐿^2𝛼Γ(2𝛼+1)²𝑓²(−𝑘)^𝛽𝑒^−𝑘𝑥−𝑁𝐶+𝐿(−2𝑘)^2𝛽𝑒^−2𝑘𝑥𝑓₃𝑡^2𝛼Γ(2𝛼+1)+2𝐿𝑒^−2𝑘𝑥(−2𝑘)^2𝛽(−𝑘)^𝛽𝑡^2𝛼Γ()𝑓2𝛼+1²+𝑁𝑓₄𝑡^𝛼Γ(𝛼+1)+⋯, (32) where 𝑓=𝐶(−𝑘)^𝛽𝑒^−𝑘𝑥+𝐿(−𝑘)^𝛽𝑒^−3𝑘𝑥,𝑓₁=𝐶(−𝑘)^2𝛽𝑒^−𝑘𝑥+𝐿(−𝑘)^𝛽(−3𝑘)^𝛽𝑒^−3𝑘𝑥,𝑓₂=𝐶(−𝑘)^3𝛽𝑒^−𝑘𝑥+𝐿(−𝑘)^𝛽(−3𝑘)^2𝛽𝑒^−3𝑘𝑥,𝑓₃=𝐶(−𝑘)^4𝛽𝑒^−𝑘𝑥+𝐿(−𝑘)^𝛽(−3𝑘)^3𝛽𝑒^−3𝑘𝑥,𝑓₄=𝐶(−𝑘)^5𝛽𝑒^−𝑘𝑥+𝐿(−𝑘)^𝛽(−3𝑘)^4𝛽𝑒^−3𝑘𝑥. (33)

5.2. The VIM for the Time-Space Fractional Nonlinear BBM Equation

In this subsection, we use the VIM to construct the approximate solution of the time and space fractional nonlinear BBM equation (26).

The correction functional for (26) can be approximately expressed as follows: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡₀𝜆(𝜏)×𝜕^𝛼𝑢_𝑛(𝑥,𝜏)𝜕𝑡^𝛼𝜕+𝐶^𝛽𝑢_𝑛(𝑥,𝜏)𝜕𝑥^𝛽+𝐿𝑢_𝑛(𝑥,𝜏)²𝜕^𝛽𝑢_𝑛(𝑥,𝜏)𝜕𝑥^𝛽𝜕+𝑁^𝛼𝜕𝑡^𝛼𝜕^𝛽𝜕𝑥^𝛽𝜕^𝛽𝑢_𝑛(𝑥,𝜏)𝜕𝑥^𝛽𝑑𝜏, (34) where 𝜆 is a general Lagrange multiplier, 𝑢_𝑛(𝑥,𝜏) is considered as restricted variations, and 𝛿𝑢_𝑛 is considered as a restricted variation. Making the above correction functional stationary and noticing that 𝛿𝑢_𝑛=0, we obtain 𝛿𝑢_𝑛+1(𝑥,𝑡)=𝛿𝑢_𝑛(𝑥,𝑡)+^𝑡₀(𝜏𝜕𝛿𝜆)^𝛼𝑢_𝑛(𝑥,𝜏)𝜕𝑡^𝛼𝑑𝜏,𝛿𝑢_𝑛+1(𝑥,𝑡)=𝛿𝑢_𝑛(𝑥,𝑡)+𝜆(𝜏)𝛿𝑢_𝑛()+𝑥,𝜏_𝑡0𝛿𝑢_𝑛(𝑥,𝜏)𝜆(𝜏̀)𝑑𝜏=0. (35) Therefore, it can be identified as 𝜆=−1, and the following variational iteration formula can be obtained: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)−^𝑡₀𝜕^𝛼𝑢_𝑛𝜕𝑡^𝛼𝜕+𝐶^𝛽𝑢_𝑛𝜕𝑥^𝛽+𝐿(𝑢_𝑛)²𝜕^𝛽𝑢_𝑛𝜕𝑥^𝛽𝜕+𝑁^𝛼𝜕𝑡^𝛼𝜕^𝛽𝜕𝑥^𝛽𝜕^𝛽𝑢_𝑛𝜕𝑥^𝛽𝑑𝜏. (36)

By the iteration formula (36), we can obtain the other components as 𝑢₀=𝑒^−𝑘𝑥,𝑢₁=𝑒^−𝑘𝑥𝑢−𝑓𝑡,₂=3𝑒^−𝑘𝑥𝑡−3𝑓𝑡+^2−𝛼(2−𝛼)Γ(2−𝛼)𝑓+𝑁𝑓₂+𝑡⁴4𝐿𝑓²𝑓₁−𝑡³32𝐿𝑒^−𝑘𝑥𝑓𝑓₁+𝐿𝑓²(−𝑘)^𝛽𝑒^−𝑘𝑥+𝑡²2𝐶𝑓₁+𝐿𝑒^−2𝑘𝑥𝑓₁+2𝐿𝑒^−2𝑘𝑥(−𝑘)^𝛽𝑓, (37)(38) and so on, and in the same manner, the rest of components of the iteration formula (36) can be obtained using Mathematica package where 𝑓=𝐶(−𝑘)^𝛽𝑒^−𝑘𝑥+𝐿(−𝑘)^𝛽𝑒^−3𝑘𝑥,𝑓₁=𝐶(−𝑘)^2𝛽𝑒^−𝑘𝑥+𝐿(−𝑘)^𝛽(−3𝑘)^𝛽𝑒^−3𝑘𝑥,𝑓₂=𝐶(−𝑘)^3𝛽𝑒^−𝑘𝑥+𝐿(−𝑘)^𝛽(−3𝑘)^2𝛽𝑒^−3𝑘𝑥. (39)

The comparison between the approximate solutions which obtained, by using HPM (32) and VIM (38) is shown in Table 1.

Table 1 

This table shows the absolute error between the approximate solutions (32), (38) and the approximate solution in [34] when 𝑡=0.01 and 0<𝑥<1.

Now we make the comparison between the approximate solutions obtained by the variation iteration method and the approximate solution obtained by the homotopy perturbation method as shown in Figures 1, 2, 3, 4, and 5.

(a)

(b)

Figure 1 

The approximate solution obtained by the variational iteration method (38) shown in (a) in comparison with that of the approximate solutions obtained by homotopy perturbation method (32) shown in (b) when 𝛼=1, 𝛽=1, −5<𝑥<5, and 0<𝑡<1.

(a)

(b)

Figure 2 

The real part of the approximate solution obtained by the variational itearation method (38) shown in (a) in comparison with the of that approximate solutions obtained by homotopy perturbation method (32) shown in (b) when 𝛼=0.2, 𝛽=0.4, −5<𝑥<5, and 0<𝑡<1.

(a)

(b)

Figure 3 

The imaginary part of the approximate solution obtained by the variational itearation method (38) shown in (a) in comparison with that of the approximate solutions obtained by homotopy perturbation method (32) shown in (b) when 𝛼=0.2, 𝛽=0.4, −5<𝑥<5, and 0<𝑡<1.

(a)

(b)

Figure 4 

The real part of the approximate solution obtained by the variational itearation method (38) shown in (a) in comparison with that of the approximate solutions obtained by homotopy perturbation method (32) shown in (b) when 𝛼=0.99, 𝛽=0.99, −5<𝑥<5, and 0<𝑡<1.

(a)

(b)

Figure 5 

The imaginary part of the approximate solution which obtained by the variational itearation method (38) shown in (a) in comparison with the of the approximate solutions which obtained by homotopy perturbation method (32) shown in (b) when 𝛼=0.99, 𝛽=0.99, −5<𝑥<5, and 0<𝑡<1.

Remark 4. From Figures 2–5, the approximate solution obtained by the homotopy perturbation method is different from the approximate solutions obtained by variational iteration when 𝛼 and 𝛽 are not positive integers.

6. The Improved (𝐺^′/𝐺)-Expansion Function Method

Consider the following nonlinear partial fractional differential equation: 𝐹𝑢,𝐷^𝛼_𝑡𝑢,𝐷^𝛽_𝑥𝑢,𝐷^𝛾_𝑦𝑢,𝐷^𝛿_𝑧𝑢,𝐷^𝛼_𝑡𝐷^𝛼_𝑡𝑢,𝐷^𝛼_𝑡𝐷^𝛽_𝑥𝐷𝑢,_𝛽𝑥𝐷^𝛽_𝑥𝑢,𝐷^𝛽_𝑥𝐷^𝛾_𝑦𝑢,…=0,0<𝛼,𝛽,𝛿,𝛾<1, (40) where 𝑢 is an unknown function and 𝐹 is a polynomial of 𝑢 and its partial fractional derivatives, in which the highest order derivatives and the nonlinear terms are involved.

In the following, we give the main steps of the improved (𝐺^′/𝐺)-expansion method.

Step 1. Li and He [48] proposed a fractional complex transformation to convert fractional differential equations into ordinary differential equations (ODE), so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus. The traveling wave variable 𝑢𝑥,𝑦,𝑧,𝑡=𝑢(𝜉),𝜉=𝑅𝑥^𝛽Γ+𝛽+1𝑄𝑦^𝛾Γ+𝛾+1𝑇𝑧^𝛿Γ()+𝛿+1𝑆𝑡^𝛼Γ(),𝛼+1 (41)(42) where 𝑅, 𝑄, 𝑇, and 𝑆 are nonzero arbitrary constants, permits us to reduce (40) to an ODE of 𝑢=𝑢(𝜉) in the form 𝑃𝑢,𝑢^′,𝑢^′′,𝑢^′′′,…=0. (43)
If possible, we should integrate (43) term by term one or more times.

Step 2. Suppose that the solution of (43) can be expressed as a polynomial of (𝐺^′/𝐺) in the form [49–51] 𝑢(𝜉)=^𝑚_𝑖=−𝑚𝑎_𝑖𝐺^′𝐺^𝑖,𝑎_𝑖≠0, (44) where 𝑎_𝑖(𝑖=0,±1,…,±𝑚) are constants, while 𝐺(𝜉) satisfies the following second-order linear ODE: 𝐺^′′(𝜉)+𝜆𝐺^′(𝜉)+𝜇𝐺(𝜉)=0, (45) with 𝜆 and 𝜇 being constants.

Step 3. The positive integer 𝑚 can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in (43). More precisely, we define the degree of 𝑢(𝜉) as 𝐷[𝑢(𝜉)]=𝑚, which gives rise to the degrees of the other expressions as follows: 𝐷𝑑^𝑞𝑢𝑞𝜉^𝑞𝐷𝑢=𝑚+𝑞,_𝑝𝑑^𝑞𝑢𝑞𝜉^𝑞𝑠=𝑚𝑝+𝑠𝑞+𝑚. (46)
Therefore, we can obtain the value of 𝑚 in (44).

Step 4. Substituting (44) into (43), using (45), collecting all terms with the same order of (𝐺^′/𝐺) together and equating each coefficient of the resulting polynomial to zero, we obtain a set of algebraic equations for 𝑎_𝑖(𝑖=0,±1,…,±𝑚), 𝜆, 𝜇, 𝑅, 𝑄, 𝑇, and 𝑆.

Step 5. Since the general solutions to (45) are well known, then substituting 𝑎_𝑖(𝑖=0,±1,…,±𝑚), 𝜆, 𝜇, 𝑅, 𝑄, 𝑇, 𝑆, and the general solutions of (45) into (44), we get more traveling wave solutions of the nonlinear partial fractional derivatives (40).

7. Applications

In this section, we use the improved (𝐺^′/𝐺)-expansion method to construct the exact solutions for some nonlinear partial fractional differential equations, via the time-space fractional nonlinear BBM equation.

We can see that the fractional complex transformation 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑅𝑥^𝛽Γ+𝛽+1𝑆𝑡^𝛼Γ(),𝛼+1 (47) where 𝑅 and 𝑆 are constants, permits us to reduce (26) to an ordinary differential equation. After that, we integrate to obtain (𝑆+𝐶𝑅)𝑈+𝐿𝑅3𝑈³+𝑁𝑅²𝑆𝑈^′′+𝑐₁=0, (48) where 𝑐₁ is the integration constant.

Considering the homogeneous balance between the highest order derivative and the nonlinear term in (48), we deduce that 𝑛=1. Thus, we get 𝑢(𝜉)=𝑎₁𝐺^′𝐺+𝑎₀+𝑏₁𝐺^′𝐺⁻¹, (49) where 𝑎₁, 𝑎₀, 𝑏₁, 𝑅, and 𝑆 are arbitrary constants to be determined later. Substituting (49) into (48), collecting all the terms of powers of (𝐺^′/𝐺), and setting each coefficient to zero, we get a system of algebraic equations. With the aid of Maple or Mathematica, we can solve this system of algebraic equations to obtain the following sets of solutions.

Case 1𝑎₀=𝜆𝑎₁2,𝑆=−𝐿𝑎²₁,6𝑁𝑅𝜇=12𝐶𝑅²𝑁−2𝐿𝑎²₁+𝐿𝑎²₁𝑅²𝜆²𝑁4𝐿𝑎²₁𝑅²𝑁,𝑏₁=𝑐₁=0, (50) where 𝜆, 𝑎₁, 𝑅, 𝐿, 𝑁, and 𝐶 are arbitrary constants. Substituting (50) into (49) yields 𝑢(𝜉)=𝜆𝑎₁2+𝑎₁𝐺^′𝐺. (51)

From the solutions of (45) and (51), we deduce the following types of traveling wave solutions for (26).

Family 1. If 𝜆²−4𝜇>0, then we have the hyperbolic solution 𝑢(𝜉)=𝑎₁2𝜆²×()−4𝜇𝐴cosℎ1/2𝜆²𝜉()−4𝜇+𝐵sinℎ1/2𝜆²𝜉×()−4𝜇𝐴sinℎ1/2𝜆²𝜉()−4𝜇+𝐵cosℎ1/2𝜆²−4𝜇𝜉⁻¹. (52)

Family 2. If 𝜆²−4𝜇<0, then we have the trigonometric solution 𝑢(𝜉)=𝑎₁24𝜇−𝜆²×−𝐴sin(1/2)4𝜇−𝜆²𝜉+𝐵cos(1/2)4𝜇−𝜆²𝜉×𝐴cos(1/2)4𝜇−𝜆²𝜉+𝐵sin(1/2)4𝜇−𝜆²𝜉⁻¹. (53)

Family 3. If 𝜆²−4𝜇=0, then we have 𝑢(𝜉)=𝑎₁𝐵,𝐴+𝐵𝜉 (54) where 𝜉=𝑅𝑥^𝛽Γ−𝛽+1𝐿𝑎²₁𝑡^𝛼().6𝑁𝑅Γ𝛼+1 (55)

Case 2𝑏₁=−6𝐶𝑅²𝑁−𝐿𝑎²₁8𝐿𝑎²₁𝑅²𝑁,𝑆=−𝐿𝑎²₁,6𝑁𝑅𝜇=6𝐶𝑅²𝑁−𝐿𝑎²₁8𝐿𝑎²₁𝑅²𝑁,𝑎₀=𝑐₁=𝜆=0, (56) where 𝑎₁, 𝑅, 𝐿, 𝑁, and 𝐶 are arbitrary constants. Substituting (56) into (49) yields 𝑢(𝜉)=𝑎₁𝐺^′𝐺−6𝐶𝑅²𝑁−𝐿𝑎²₁8𝐿𝑎²₁𝑅²𝑁𝐺^′𝐺⁻¹. (57)

From the solutions of (45) and (57), we deduce the following types of traveling wave solutions for (26).

Family 1. If 𝜇>0, then we have the hyperbolic solution 𝑢(𝜉)=𝑎₁12√𝜇𝐴cosℎ(1/2)𝜇𝜉+𝐵sinℎ(1/2)𝜇𝜉×𝐴sinℎ(1/2)𝜇𝜉+𝐵cosℎ(1/2)𝜇𝜉⁻¹−6𝐶𝑅²𝑁−𝐿𝑎²₁8𝐿𝑎²₁𝑅²𝑁×12√𝜇𝐴cosℎ(1/2)𝜇𝜉+𝐵sinℎ(1/2)𝜇𝜉×𝐴sinℎ(1/2)𝜇𝜉+𝐵cosℎ(1/2)𝜇𝜉⁻¹⁻¹. (58)

Family 2. If 𝜇<0, then we have the trigonometric solution 𝑢(𝜉)=𝑎₁12√−𝜇−𝐴sin(1/2)𝜉()−𝜇+𝐵cos1/2𝜉×()−𝜇𝐴cos1/2𝜉()−𝜇+𝐵sin1/2−𝜇𝜉⁻¹−6𝐶𝑅²𝑁−𝐿𝑎²₁8𝐿𝑎²₁𝑅²𝑁×12√−𝜇−𝐴sin(1/2)𝜉()−𝜇+𝐵cos1/2𝜉×()−𝜇𝐴cos1/2𝜉()−𝜇+𝐵sin1/2−𝜇𝜉⁻¹⁻¹. (59)

Family 3. If 𝜇=0, then we have 𝑢(𝜉)=𝑎₁𝐵−𝐴+𝐵𝜉6𝐶𝑅²𝑁−𝐿𝑎²₁8𝐿𝑎²₁𝑅²𝑁𝐵𝐴+𝐵𝜉⁻¹, (60) where 𝜉=𝑅𝑥^𝛽Γ−𝛽+1𝐿𝑎²₁𝑡^𝛼().6𝑁𝑅Γ𝛼+1 (61)

8. Conclusion

In this paper, we use the homotopy perturbation method and variational iteration method to obtain the analytic approximate solutions for linear and nonlinear partial fractional differential equations. Section 5 shows the comparative study between the factional variational itearation methods and the homotopy perturbation method. The two methods are powerful and effective tools for the solutions of fractional partial differential equations, and both give two different approximate solutions of higher accuracy and closed form solutions if existing. Also, we use the improved (𝐺^′/𝐺)-expansion function method to calculate the exact solutions for the time and space fractional nonlinear BBM equation. This method is reliable and simple and gives many new exact solutions for the nonlinear fractional differential equations. When the parameters take certain special values, the solitary waves are derived from the traveling waves. This method is very efficient and powerful in finding the exact solutions for the nonlinear fractional differential equations.