Improved (𝐺^′/𝐺)-Expansion Method for the Space and Time Fractional Foam Drainage and KdV Equations

Abstract

The fractional complex transformation is used to transform nonlinear partial differential equations to nonlinear ordinary differential equations. The improved (𝐺^′/𝐺)-expansion method is suggested to solve the space and time fractional foam drainage and KdV equations. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.

1. Introduction

The soliton solutions of nonlinear evolution equations have made a major impact in the flesh. These solitons appear in various areas of physical and biological sciences. They show up in nonlinear optics, plasma physics, fluid dynamics, biochemistry, and mathematical chemistry. Fractional partial differential equations (FPDEs) have received considerable interest in recent years and have been extensively investigated. These equations were applied for many real problems which are modeled in various areas, for example, in mathematical physics [1], fluid and continuum mechanics [2], viscoplastic and viscoelastic flow [3], biology, chemistry, acoustics, and psychology [4, 5]. Some FPDEs do not have exact solutions, so approximation and numerical techniques must be used. There are several approximation and numerical methods. The most commonly used ones are the homotopy perturbation method [6, 7], the Adomian decomposition method [8, 9], the variational iteration method [10–12], the homotopy analysis method [13, 14], the generalized differential transform method [15], the finite difference method [16], and the finite element method [17]. In recent years, some authors have got exact solutions of FPDEs by using analytical methods. S. Zhang and H.-Q. Zhang [18] proposed to solve the nonlinear time fractional biological population model and (4+1)-dimensional space-time fractional Fokas equation by using the fractional subequation method. Guo et al. [19] presented the improved subequation method to solve the space-time fractional Whitham-Broer-Kaup and the generalized Hirota-Satsuma coupled KdV equations. Tang et al. [20] used the generalized fractional subequation method to obtain exact solutions of the space-time fractional Gardner equation with variable coefficients. Lu [21] investigated the exact solutions of the nonlinear fractional Klein-Gordon equation, the generalized time fractional Hirota-Satsuma coupled KdV system, and the nonlinear fractional Sharma-Tasso-Olver equation. Bin [22] solved the time-space fractional generalized Hirota-Satsuma coupled KdV equations and the time fractional fifth-order Sawada-Kotera equation by using the (𝐺^′/𝐺)-expansion method. Omran and Gepreel [23] used the improved (𝐺^′/𝐺)-expansion method to calculate the exact solutions to the time-space fractional foam drainage and KdV equations. In this paper, we will apply the improved (𝐺^′/𝐺)-expansion method to obtain the exact solutions for the time-space fractional foam drainage and KdV equations with the modified Riemann-Liouville derivative defined by Jumarie [24–27]: 𝜕^𝛼𝑢𝜕𝑡^𝛼=12𝑢𝜕^2𝛽𝑢𝜕𝑥^2𝛽+2𝑢²𝜕^𝛽𝑢𝜕𝑥^𝛽+𝜕^𝛽𝑢𝜕𝑥^𝛽2,𝑡>0,𝛼>0,𝛽≤1,𝜕^𝛼𝑢𝜕𝑡^𝛼𝜕+𝛼𝑢^𝛽𝑢𝜕𝑥^𝛽+𝜕^3𝛽𝑢𝜕𝑥^3𝛽=0,𝑡>0,𝛼>0,𝛽≤1, (1)(2) where 𝛼 is arbitrary constant. This paper is organized as follows. In Section 2, we introduce some basic definitions of Jumarie’s modified Riemann-Liouville derivative. In Section 3, the main steps of the improved (𝐺^′/𝐺)-expansion method are given. In Section 4, we construct the exact solutions of (1) and (2) by the proposed method. Some conclusions are given in Section 5.

2. Preliminaries

There are several definitions for fractional differential equations. These definitions include Riemann-Liouville, Weyl, Grünwald-Letnikov, Riesz, and Jumarie fractional derivatives. The Riemann-Liouville fractional derivative of a constant is not zero. So the fractional derivative is only defined for differentiable function. In order to deal with nondifferentiable functions, Jumarie [24–27] presented a modification of the Riemann-Liouville definition which appears to provide a framework for a fractional calculus. This modification was successfully applied in the probability calculus, fractional Laplace problem, exact solutions of the nonlinear fractional differential equations, and many other types of linear and nonlinear fractional differential equations [28–30].

Definition 1. The Riemann-Liouville fractional integral is defined [31] as ₀𝐼^𝛼_𝑥𝑓(𝑥)=𝐼^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥₀𝑓(𝜉)(𝑥−𝜉)^𝛼−1𝐼𝑑𝜉,𝛼>0,₀𝑓(𝑥)=𝑓(𝑥). (3)

Definition 2. Jumarie [24–27] defined the fractional derivative in the limit form by 𝑓^(𝛼)(𝑥)=lim_ℎ→0Δ^𝛼𝑓(𝑥)−𝑓(0)ℎ^𝛼, (4) where 𝑓(𝑥) should be a continuous (but not necessarily differentiable) function and ℎ>0 denotes a constant discretization span. So, the modified form of the Riemann-Liouville derivative is defined as ₀𝐷^𝛼_𝑥𝑓(𝑥)=1Γ()𝑑𝑛−𝛼𝑑𝑥^𝑛𝑛𝑥₀(𝑥−𝜉)^{(𝑛−𝛼)}𝑓(𝜉)−𝑓(0), (5) where 𝑥∈[0,1], 𝑛−1≤𝛼<𝑛 and 𝑛≥1.

Lemma 3. The integral with respect to (𝑑𝑥)^𝛼 is defined by Jumarie [24, 25] as follows: ^𝑥₀𝑓(𝜉)(𝑑𝜉)^𝛼=𝛼^𝑥₀()𝑓(𝜉)𝑑𝑥−𝜉𝑑𝜉,0<𝛼≤1,_𝛼𝑑𝑥^𝛼₀^𝑢(𝑥)𝑓(𝜉)(𝑑𝜉)^𝛼=Γ(𝛼+1)𝑓𝑢(𝜉)𝑢^′(𝜉)^𝛼,0<𝛼≤1. (6)

Theorem 4. Assume that the continuous function 𝑓(𝑥) has a fractional derivative of order 𝛼; then 𝑑^𝛼𝑑𝑥^𝛼𝐼^𝛼𝑓(𝑥)(𝑥),𝐼=𝑓_𝛼𝑑^𝛼𝑑𝑥^𝛼𝑓(𝑥)=𝑓(𝑥)−𝑓(0),0<𝛼≤1, (7) hold.

3. Description of the Improved (𝐺^′/𝐺)-Expansion Method

In this section, we give the description of the improved (𝐺^′/𝐺)-expansion method for solving the nonlinear FPDEs as 𝐹𝑢,𝐷^𝛼_𝑡𝑢,𝐷^𝛽_𝑥𝑢,𝐷^𝛾_𝑦𝑢,𝐷^𝛿_𝑧𝑢,𝐷^𝛼_𝑡𝐷^𝛼_𝑡𝐷𝑢,_𝛼𝑡𝐷^𝛽_𝑥𝑢,𝐷^𝛽_𝑥𝐷^𝛽_𝑥𝑢,…=0,0<𝛼,𝛽,𝛾,𝛿≤1, (8) 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 nonlinear terms are involved. We offer an improved (𝐺^′/𝐺)-expansion method [32]. The essential steps of this method are described as follows.

Step 1. Li and He [33] and He and Li [34] presented a fractional complex transform to transform fractional differential equations into ordinary differential equations. So, all analytical methods devoted to advanced calculus can be easily dedicated to fractional calculus. The traveling wave variable is given as 𝑢𝑥,𝑦,𝑧,𝑡=𝑢(𝜉),𝜉=𝐾𝑥^𝛽Γ+𝛽+1𝑁𝑦^𝛾Γ+𝛾+1𝐿𝑡^𝛼Γ(),𝛼+1 (9) where 𝐾, 𝑁, and 𝐿 are nonzero arbitrary constants. So, (9) is reduced to (10): 𝑝𝑢,𝑢^′,𝑢^′′,𝑢^′′′,…=0, (10) where 𝑢=𝑢(𝜉).

Step 2. Suppose that (10) has the solution (11): 𝑢(𝜉)=^𝑛_𝑖=0𝑎_𝑖𝐹^𝑖(𝜉), (11) where 𝑎_𝑖  (𝑖=0,1,…,𝑛) are real constants to be determined, the balancing number 𝑛 is a positive integer which can be determined by balancing the highest derivative terms with the highest power nonlinear terms in (10). More precisely, we define the degree of 𝑢(𝜉) as 𝐷[𝑢(𝜉)]=𝑚, which gives rise to the degrees of other expressions, as follows: 𝐷𝑑^𝑞𝑢𝑑𝜉^𝑞𝐷𝑢=𝑚+𝑞,_𝑝𝑑^𝑞𝑢𝑑𝜉^𝑞𝑠=𝑚𝑝+𝑠𝑞+𝑚. (12) Therefore, we can obtain the value of 𝑚 in (11).

Step 3. 𝐹(𝜉) is 𝐹(𝜉)=𝐺^′(𝜉)𝐺(𝜉), (13) where 𝐺(𝜉) expresses the solution of the following auxiliary ordinary differential equation 𝐺(𝜉)𝐺^′′(𝜉)=𝐴𝐺²(𝜉)+𝐵𝐺(𝜉)𝐺^′(𝜉)+𝐶𝐺(𝜉)², (14) where the prime denotes derivative with respect to 𝜉. 𝐴, 𝐵, and 𝐶 are real parameters.

Step 4. Substituting (13) into (10), using (14), collecting all terms with the same order of (𝐺^′(𝜉)/𝐺(𝜉)) together, and then equating each coefficient of the resulting polynomial to zero, we obtain a set of algebraic equations for 𝑎_𝑖  (𝑖=0,1,…,𝑛), 𝐴, 𝐵, 𝐶, 𝐾, 𝑁, and 𝐿.

Step 5. Using the general solutions of (14), with the aid of Mathematica, we have the following four solutions of (13).

Case 1. If 𝐵≠0 and Δ=𝐵²+4𝐴−4𝐴𝐶≥0, then 𝐹(𝜉)=𝐵2()+𝐵√1−𝐶Δ2()×𝑐1−𝐶₁exp√Δ/2𝜉+𝑐₂exp−√Δ/2𝜉𝑐₁exp√Δ/2𝜉−𝑐₂exp−√𝜉.Δ/2 (15)

Case 2. If 𝐵≠0 and Δ=𝐵²+4𝐴−4𝐴𝐶<0, then 𝐹(𝜉)=𝐵2()+𝐵√1−𝐶−Δ2()×1−𝐶𝑖𝑐₁cos√−Δ/2𝜉−𝑐₂sin√−Δ/2𝜉𝑖𝑐₁sin√−Δ/2𝜉+𝑐₂cos√𝜉.−Δ/2 (16)

Case 3. If 𝐵=0 and Δ=𝐴(𝐶−1)≥0, then 𝐹(𝜉)=√Δ()𝑐1−𝐶¹cos√Δ𝜉+𝑐₂sin√Δ𝜉𝑐₁sin√Δ𝜉−𝑐₂cos√.Δ𝜉 (17)

Case 4. If 𝐵=0 and Δ=𝐴(𝐶−1)<0, then 𝐹(𝜉)=√−Δ(1−𝐶)𝑖𝑐₁cosh√−Δ𝜉−𝑐₂sinh√−Δ𝜉𝑖𝑐₁sinh√−Δ𝜉−𝑐₂cosh√,−Δ𝜉 (18) where 𝜉=𝐾𝑥^𝛽Γ+𝛽+1𝑁𝑦^𝛾Γ+𝛾+1𝐿𝑡^𝛼Γ(𝛼+1) (19) and 𝐴, 𝐵, 𝐶, 𝑐₁, and 𝑐₂ are real parameters.

4. Applications

We use the improved (𝐺^′/𝐺)-expansion method on the time-space fractional nonlinear foam drainage equation and the time-space fractional nonlinear KdV equation in this section.

4.1. The Time and Space-Fractional Nonlinear Foam Drainage Equation

We apply the improved (𝐺^′/𝐺)-expansion method to construct the exact solutions for the time-space fractional nonlinear foam drainage equation in this subsection. Foams are of great importance in many technological processes and applications. Their properties are subject of intensive studies from practical and scientific points of view [27, 35–37]. Liquid foam is an example of soft matter with a very well-defined structure, described by Joseph Plateau in the 19th century. Foams are common in foods and personal care products such as lotions and creams. They have important applications in food and chemical industries, mineral processing, fire fighting, and structural material sciences [27, 35–37]. This equation is numerically and analytically taken into account by different authors [38–40]. The space-time fractional nonlinear foam drainage equation is solved analytically only by Omran and Gepreel [23]. We can see the fractional complex transform as 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝐾𝑥^𝛽Γ+𝛽+1𝐿𝑡^𝛼Γ(),𝛼+1 (20) where 𝐾 and 𝐿 are constants. So, (20) reduces to (21): −𝐿𝑢^′+12𝐾²𝑢𝑢^′′+2𝐾𝑢²𝑢^′+𝐾²𝑢^′2=0. (21)

Balancing the highest order nonlinear term and the highest order linear term, we get 𝑛=1. Thus, we obtain 𝑢(𝜉)=𝑎₀+𝑎₁𝐹(𝜉)(𝜉)=𝐺,𝐹^′(𝜉)𝐺(𝜉), (22) where 𝑎₀ and 𝑎₁ will be determined constants. Substituting (22) into (21), using (14), collecting all the terms of powers of (𝐺^′/𝐺), and setting each coefficient to zero, we have the following system of algebraic equations: 𝐺^′𝐺⁰∶2𝐴𝑎²₀𝑎₁𝐾+𝐴²𝑎²₁𝐾²+12𝐴𝑎₀𝑎₁𝐵𝐾²−𝐴𝑎₁𝐺𝐿=0,^′𝐺¹∶2𝑎²₀𝑎₁𝐵𝐾−𝐴𝑎₀𝑎₁𝐾²+52𝐴𝑎²₁𝐵𝐾²+12𝑎₀𝑎₁𝐵²𝐾²+𝐴𝑎₀𝑎₁𝐶𝐾²−𝑎₁𝐺𝐵𝐿=0,^′𝐺²∶−2𝑎²₀𝑎₁𝐾+2𝐴𝑎³₁𝐾+4𝑎₀𝑎²₁𝐵𝐾+2𝑎²₀𝑎₁𝐶𝐾−3𝐴𝑎²₁𝐾²+32𝑎₀𝑎₁𝐵𝐾²+32𝑎²₁𝐵²𝐾²+3𝐴𝑎²₁𝐶𝐾²+32𝑎₀𝑎₁𝐵𝐶𝐾²+32𝑎²₁𝐵²𝐾²+3𝐴𝑎²₁𝐶𝐾²+32𝑎₀𝑎₁𝐵𝐶𝐾²+𝑎₁𝐿−𝑎₁𝐺𝐶𝐿=0,^′𝐺³∶−4𝑎₀𝑎²₁𝐾+2𝑎³₁𝐵𝐾+4𝑎₀𝑎²₁𝐶𝐾+𝑎₀𝑎₁𝐾²−72𝑎²₁𝐵𝐾²+2𝑎₀𝑎₁𝐶𝐾²+72𝑎²₁𝐵𝐶𝐾²+𝑎₀𝑎₁𝐶²𝐾²𝐺=0,^′𝐺⁴∶−2𝑎³₁𝐾+2𝑎³₁𝐶𝐾+2𝑎²₁𝐾²−4𝑎²₁𝐶𝐾²+2𝑎²₁𝐶²𝐾²=0. (23) Solving the set of the above algebraic equations, we get the following result: 𝑎₀=8𝐴𝐵𝐾(𝐶−1)𝐵²+6𝐴−6𝐴𝐶,𝑎₁(),1=𝐾𝐶−1𝐿=24𝑎²₀𝐾+𝑎₀𝐵𝐾²+2𝐴𝐾³−2𝐴𝐶𝐾³,𝐾𝐵(𝐶−1)≠0. (24)

Substituting this value in (22) and by Cases 1–4, we obtain the following exponential, hyperbolic and triangular function solutions of (1).(1) If we choose 𝐵≠0 and Δ=𝐵²+4𝐴−4𝐴𝐶≥0, then the exponential function solutions can be found as 𝑢()=()[(𝑥,𝑡16𝐴𝐵𝐾𝐶−1−𝐾𝐵Δ+2𝐴𝐶−1)]()−√2Δ+2𝐴𝐶−1𝐾𝐵Δ2×𝑐₁exp√Δ/2𝜉+𝑐₂exp−√Δ/2𝜉𝑐₁exp√Δ/2𝜉−𝑐₂exp−√𝜉,Δ/2 (25) where 𝜉=𝐾𝑥^𝛽Γ+1𝛽+124𝑎²₀𝐾+𝑎₀𝐵𝐾²+2𝐴𝐾³−2𝐴𝐶𝐾³×𝑡^𝛼Γ().𝛼+1 (26)(2) If we choose 𝐵≠0 and Δ=𝐵²+4𝐴−4𝐴𝐶<0, then the triangular function solution will be 𝑢()=()[(𝑥,𝑡16𝐴𝐵𝐾𝐶−1−𝐾𝐵Δ+2𝐴𝐶−1)]()−√2Δ+2𝐴𝐶−1𝐾𝐵−Δ2×𝑖𝑐₁cos√−Δ/2𝜉−𝑐₂sin√−Δ/2𝜉𝑖𝑐₁sin√−Δ/2𝜉+𝑐₂cos√𝜉,−Δ/2 (27) where 𝜉=𝐾𝑥^𝛽Γ+1𝛽+124𝑎²₀𝐾+𝑎₀𝐵𝐾²+2𝐴𝐾³−2𝐴𝐶𝐾³×𝑡^𝛼Γ().𝛼+1 (28)(3) If we choose 𝐵=0 and Δ₁=𝐴(𝐶−1)≥0, then we get another triangular function solution 𝑢(𝑥,𝑡)=−𝐾Δ₁𝑐₁cosΔ₁𝜉+𝑐₂sinΔ₁𝜉𝑐₁sinΔ₁𝜉−𝑐₂cosΔ₁𝜉, (29) where 𝜉=𝐾𝑥^𝛽/Γ(𝛽+1)−Δ₁(𝐾³𝑡^𝛼/Γ(𝛼+1)).(4) If we choose 𝐵=0 and Δ₁=𝐴(𝐶−1)<0, then we obtain the hyperbolic function solution 𝑢(𝑥,𝑡)=−𝐾Δ₁𝑖𝑐₁cosh−Δ₁𝜉−𝑐₂sinh−Δ₁𝜉𝑖𝑐₁sinh−Δ₁𝜉−𝑐₂cosh−Δ₁𝜉, (30) where 𝜉=𝐾𝑥^𝛽/Γ(𝛽+1)−Δ₁(𝐾³𝑡^𝛼/Γ(𝛼+1)).

If we take 𝑐₁=−𝑐₂ and 𝑐₁=𝑐₂ in (25), respectively, then we get 𝑢()=()[(𝑥,𝑡16𝐴𝐵𝐾𝐶−1−𝐾𝐵Δ+2𝐴𝐶−1)]()−√2Δ+2𝐴𝐶−1𝐾𝐵Δ2√tanhΔ2𝜉,𝑢()=()[(𝑥,𝑡16𝐴𝐵𝐾𝐶−1−𝐾𝐵Δ+2𝐴𝐶−1)]()−√2Δ+2𝐴𝐶−1𝐾𝐵Δ2√cothΔ2𝜉. (31)

4.2. The Nonlinear Space-Time Fractional KdV Equation

The KdV equation is the most popular soliton equation, and it has been largely investigated. In addition, the space and time fractional KdV equations with initial conditions were widely worked by [27, 38, 39]. Integrating (2) with respect to 𝑢 and ignoring the integral constants leads to 12𝐿𝑢²+16𝑎𝐾𝑢³+12𝐾³𝑢^′2=0. (32)

Considering the homogeneous balance between the highest order derivatives and the nonlinear term in (32), we get 𝑛=2. So, we can suppose that (32) has the following ansatz: 𝑢(𝜉)=𝑎₀+𝑎₁𝐹(𝜉)+𝑎₂𝐹²(𝜉), (33) where 𝑎₀, 𝑎₁, 𝑎₂, 𝐿, and 𝐾 are arbitrary constants to be determined later. Substituting (33) and (14), along with (13), into (32) and using Mathematica yields a system of Equations of (𝐺^′/𝐺): 𝐺^′𝐺⁰∶13𝑎𝑎³₀𝐾+𝐴²𝑎²₁𝐾³+𝑎²₀𝐺𝐿=0,^′𝐺¹∶12𝑎𝑎²₀𝑎₁𝐾+2𝐴²𝑎₁𝑎₂𝐾³+𝐴𝑎²₁𝐵𝐾³+𝑎₀𝑎₁𝐺𝐿=0,^′𝐺²∶12𝑎𝑎₀𝑎²₁1𝐾+2𝑎𝑎²₀𝑎₂𝐾−𝐴𝑎²₁𝐾³+2𝐴²𝑎²₂𝐾³+4𝐴𝑎₁𝑎₂𝐵𝐾³+12𝑎²₁𝐵²𝐾³+𝐴𝑎²₁𝐶𝐾³+12𝑎²₁𝐿+𝑎₀𝑎₂𝐺𝐿=0,^′𝐺³∶16𝑎𝑎³₁𝐾+𝑎𝑎₀𝑎₁𝑎₂𝐾−4𝐴𝑎₁𝑎₂𝐾³−𝑎²₁𝐵𝐾³+4𝐴𝑎²₂𝐵𝐾³+2𝑎₁𝑎₂𝐵²𝐾³+4𝐴𝑎₁𝑎₂𝐶𝐾³+𝑎²₁𝐵𝐶𝐾³+𝑎₁𝑎₂𝐺𝐿=0,^′𝐺⁴∶12𝑎²₁𝑎₂1𝐾+2𝑎𝑎₀𝑎²₂1𝐾+2𝑎²₁𝐾³−4𝐴𝑎²₂𝐾³−4𝑎₁𝑎₂𝐵𝐾³+2𝑎²₂𝐵²𝐾³−𝑎²₁𝐶𝐾³+4𝑎₁𝑎₂𝐵𝐶𝐾³+12𝑎²₁𝐶²𝐾³+12𝑎²₂𝐺𝐿=0,^′𝐺⁵∶12𝑎𝑎₁𝑎²₂𝐾+2𝑎₁𝑎₂𝐾³−4𝑎²₂𝐵𝐾³−4𝑎₁𝑎₂𝐶𝐾³+4𝑎²₂𝐵𝐶𝐾³+2𝑎₁𝑎₂𝐶²𝐾³𝐺=0,^′𝐺⁶∶16𝑎₁𝑎²₂𝐾+2𝑎²₂𝐾³−4𝑎²₂𝐶𝐾³+2𝑎²₂𝐶²𝐾³=0. (34) Solving the set of the above algebraic equations by use of Mathematica, we get the following results: 𝑎≠0,𝑎₀=−12𝐴𝐾²𝑎(),𝑎𝐶−1₁=12𝐵𝐾²𝑎(),𝑎𝐶−1₂=−12𝐾²𝑎(𝐶−1),𝐿=−𝐾³𝐵²−4𝐴𝐶+4𝐴. (35) Substituting (35) into (33) and according to (15)–(18), we obtain the following exponential function solutions, hyperbolic function solutions, and triangular function solutions of (2), respectively.(1)  If we choose 𝐵≠0 and Δ=𝐵²+4𝐴−4𝐴𝐶≥0, then the exponential function solution can be found as 𝑢(𝑥,𝑡)=3𝐾²(Δ−2𝐶)𝑎()−𝐶−16𝐶𝐾²√Δ𝑎()×𝑐𝐶−1₁exp√Δ𝜉/2+𝑐₂exp−√Δ/2𝜉𝑐₁exp√Δ/2𝜉−𝑐₂exp−√𝜉−Δ/26𝐶𝐾²Δ𝑎()×𝑐𝐶−1₁exp√Δ/2𝜉+𝑐₂exp−√Δ/2𝜉𝑐₁exp√Δ/2𝜉−𝑐₂exp−√Δ/2𝜉², (36) where 𝜉=𝐾𝑥^𝛽/Γ(𝛽+1)−𝐾³Δ(𝑡^𝛼/Γ(𝛼+1)).(2)  If we choose 𝐵≠0 and Δ=𝐵²+4𝐴−4𝐴𝐶<0, then the triangular function solution will be 𝑢(𝑥,𝑡)=3𝐾²(Δ−2𝐶)𝑎()+𝐶−16𝐶𝐾²√Δ𝑎()×𝐶−1𝑖𝑐₁cos√−Δ𝜉/2−𝑐₂sin√−Δ/2𝜉𝑖𝑐₁sin√−Δ/2𝜉+𝑐₂cos√𝜉+−Δ/26𝐶𝐾²Δ𝑎()×𝐶−1𝑖𝑐₁cos√−Δ/2𝜉−𝑐₂sin√−Δ/2𝜉𝑖𝑐₁sin√−Δ/2𝜉+𝑐₂cos√−Δ/2𝜉², (37) where 𝜉=𝐾𝑥^𝛽/Γ(𝛽+1)−𝐾³Δ(𝑡^𝛼/Γ(𝛼+1)).(3)  If we choose 𝐵=0 and Δ₁=𝐴(𝐶−1)≥0, then the triangular function solution is given as 𝑢(𝑥,𝑡)=−12𝐾²Δ₁𝑎−12𝐾²Δ₁𝑎()×𝑐𝐶−1₁cosΔ₁𝜉+𝑐₂sinΔ₁𝜉𝑐₁sinΔ₁𝜉−𝑐₂cosΔ₁𝜉², (38) where 𝜉=𝐾𝑥^𝛽/Γ(𝛽+1)+4𝐾³Δ₁(𝑡^𝛼/Γ(𝛼+1)).(4)   If we choose 𝐵=0 and Δ₁=𝐴(𝐶−1)<0, then the hyperbolic function solution is given as 𝑢(𝑥,𝑡)=−12𝐾²Δ₁𝑎+12𝐾²Δ₁𝑎()×𝐶−1𝑖𝑐₁cosh−Δ₁𝜉−𝑐₂sinh−Δ₁𝜉𝑖𝑐₁sinh−Δ₁𝜉−𝑐₂cosh−Δ₁𝜉², (39) where 𝜉=𝐾𝑥^𝛽/Γ(𝛽+1)−4𝐾³Δ₁(𝑡^𝛼/Γ(𝛼+1)). Equation (36) can be rewritten at 𝑐₁=−𝑐₂; so we get the other hyperbolic function solution of (2): 𝑢(𝑥,𝑡)=3𝐾²(Δ−2𝐶)𝑎()−𝐶−16𝐶𝐾²√Δ𝑎()√𝐶−1×tanhΔ2𝐾𝑥^𝛽Γ𝛽+1−𝐾³Δ𝑡^𝛼Γ()+𝛼+16𝐶𝐾²√Δ𝑎(𝐶−1)×tanh²√Δ2𝐾𝑥^𝛽Γ𝛽+1−𝐾³Δ𝑡^𝛼Γ().𝛼+1 (40)Equation (36) becomes 𝑢(𝑥,𝑡)=3𝐾²(Δ−2𝐶)𝑎()−𝐶−16𝐶𝐾²√Δ𝑎()√𝐶−1×cothΔ2𝐾𝑥^𝛽Γ𝛽+1−𝐾³Δ𝑡^𝛼Γ()+𝛼+16𝐶𝐾²√Δ𝑎(𝐶−1)×coth²√Δ2𝐾𝑥^𝛽Γ𝛽+1−𝐾³Δ𝑡^𝛼Γ(𝛼+1) (41) at 𝑐₁=𝑐₂.

Remark 5. Kudryashov et al. [41–44] have showed that every solution, which was obtained when soliton solutions have been found by some analytic methods, is not a new solution. They also showed that these methods are very similar. Furthermore, they mentioned that authors who used these methods should check the obtained results very carefully. The reason for using improved (𝐺^′/𝐺)-expansion method in this work is to use nonlinear equation (14) instead of linear equation 𝐺^′′−𝜆𝐺^′−𝜇𝐺=0, (42) which was used in standard (𝐺^′/𝐺) method and to obtain lots of different solutions.

5. Conclusion

In this paper, we introduced an improved (𝐺^′/𝐺)-expansion method and carried it out to obtain new travelling wave solutions of the space-time fractional foam drainage equation and the space-time fractional KdV equation. This method gives new exact solutions for nonlinear FPDEs. These solutions include the hyperbolic function solution, the exponential function solution, the triangular function solution, and the trigonometric function solution. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena.