Exact Solutions of Fractional Burgers and Cahn-Hilliard Equations Using Extended Fractional Riccati Expansion Method

Abstract

Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.

1. Introduction

During recent years, fractional differential equations (FDEs) have attracted much attention because of their potential applications in engineering and applied sciences such as signal processing, materials and mechanics, biology systems, anomalous diffusion, and medical [1–4]. Thus, it is very important to find some proper methods for solving fractional differential equations. Many effective methods have been established to obtain the solutions of FDEs, such as the variational iteration method [5, 6], the finite difference method [7], the fractional complex transform [8, 9], the exponential function method [10], the fractional subequation method [11], the (𝐺^′/𝐺)-expansion method [12, 13], and the first integral method [14]. However, the solutions of most fractional differential equations cannot be obtained explicitly in general. As a consequence, it is necessary to extend the existing methods for the fractional differential equations.

Recently, based on Jumarie’s modified Riemann-Liouville derivative and the fractional Riccati equation 𝐷^𝛼_𝜉𝜙(𝜉)=𝜎+𝜙²(𝜉), S. Zhang and H.-Q. Zhang [15] first proposed a new direct method called fractional subequation method in solving nonlinear time fractional biological population model and (4+1)-dimensional space-time fractional Fokas equation. Soon, the method was further improved by Guo et al. [16] and they obtained the analytical solutions of the space-time fractional Whitham-Broer-Kaup and generalized Hirota-Satsuma coupled KdV equations by introducing a new general ansatz. In a similar way, Lu [17] modified the method to derive the rational formal solutions of the space-time fractional Whitham-Broer-Kaup, the foam drainage equation with time and space-fractional derivatives, and nonlinear time fractional biological population model. Very lately, by extending the fractional Riccati equation in [15–17] to the more general form 𝐷^𝛼_𝜉𝐹(𝜉)=𝐴+𝐵𝐹²(𝜉), Abdel-Salam and Yousif [18] presented the fractional Riccati expansion method to obtain exact solutions of the space-time fractional Korteweg-de Vries equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the space-time fractional Klein-Gordon equation. In this paper, based on Jumarie’s modified Riemann-Liouville derivative and the general fractional Riccati equation 𝐷^𝛼_𝜉𝐹(𝜉)=𝐴+𝐵𝐹(𝜉)+𝐶𝐹²(𝜉), we will introduce an extended fractional Riccati expansion method to the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation [19] 𝜕^𝛼𝑢𝜕𝑡^𝛼+𝜀𝑢𝜕𝑢𝜕𝜕𝑥−𝜈²𝑢𝜕𝑥²=0,𝑡>0,0<𝛼≤1,𝐷^𝛼_𝑡𝑢−𝛾𝐷^𝛼_𝑥𝑢−6𝑢(𝐷^𝛼_𝑥𝑢)²−3𝑢²−1𝐷_𝑥^2𝛼𝑢+𝐷_𝑥^4𝛼𝑢=0,0<𝛼≤1, (1)(2) to construct new exact solutions. This paper is organized as follows. In Section 2, some basic properties of Jumarie’s modified Riemann-Liouville derivative are given. In Section 3, the main steps of the extended fractional Riccati expansion method are given. In Sections 4 and 5, we construct the exact solutions of (1) and (2) by the proposed method. Some conclusions are given in Section 6.

2. Preliminaries

There are several definitions for fractional differential equations. These definitions include Caputo’s fractional derivative [20], Kolwankar-Gangal derivative [21], and Jumarie’s modified Riemann-Liouville derivative [22, 23]. In this, we give the modified Riemann-Liouville derivative defined by Jumarie: 𝐷^𝛼_𝑥𝑓(𝑥)=1Γ()𝑑1−𝛼𝑑𝑥^𝑥₀(𝑥−𝜉)^−𝛼𝑓(𝜉)(0𝐷−𝑓)𝑑𝜉,0<𝛼<1,_𝛼𝑥𝑓(𝑥)=𝑓^(𝑛)(𝑥)^{(𝛼−𝑛)},𝑛≤𝛼≤𝑛+1,𝑛≥1, (3) where 𝑓∶𝑅→𝑅, 𝑥→𝑓(𝑥) denote a continuous (but not necessarily differentiable) function. Some useful formulas of Jumarie’s modified Riemann-Liouville derivative were summarized in [22, 23]; three of them are 𝐷^𝛼_𝑥𝑥^𝛾=Γ1+𝛾Γ𝑥1+𝛾−𝛼^𝛾−𝛼𝐷,𝛾>0,_𝛼𝑥(𝑢(𝑥)𝑣(𝑥))=𝑣(𝑥)𝐷^𝛼_𝑥𝑢(𝑥)+𝑢(𝑥)𝐷^𝛼_𝑥𝑣(𝑥),𝐷^𝛼_𝑥𝑓[𝑢(𝑥)]=𝑓^′_𝑢[𝑢(𝑥)]𝐷^𝛼_𝑥𝑢(𝑥)=𝐷^𝛼_𝑢𝑓[𝑢(𝑥)](𝑢^′(𝑥))^𝛼, (4) which will be used in the following sections.

3. Extended Fractional Riccati Expansion Method

In this section, we give the description of the extended fractional Riccati expansion method for solving the nonlinear FDE as 𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝐷^𝛼_𝑡𝑢,𝐷^𝛼_𝑥𝑢,…=0,0<𝛼≤1, (5) where 𝑢 is an unknown function and 𝑃 is a polynomial of 𝑢 and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved.

Step 1. Using the traveling wave transformation 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑥+𝜔𝑡, (6) where 𝜔 is a constant to be determined later, the nonlinear FDE (5) is reduced to a nonlinear fractional ordinary differential equation 𝑄𝑈(𝜉),𝑈^′(𝜉),𝑈^′′(𝜉),𝐷^𝛼_𝜉𝑈(𝜉),…=0, (7) where “′” = 𝑑/𝑑𝜉.

Step 2. Suppose that the solution of ODE (7) can be written as follows: 𝑈=𝑎₀+^𝑛_𝑗=1𝑎_𝑗𝐹^𝑗(𝜉)+𝑏_𝑗𝐹^−𝑗(𝜉), (8) where 𝑎₀, 𝑎_𝑗, 𝑏_𝑗 (𝑗=1,2,…,𝑛) are constants to be determined later and 𝑛 is a positive integer that is given by the homogeneous balance principle, and 𝐹=𝐹(𝜉) satisfies the Riccati equation 𝐷^𝛼_𝜉𝐹(𝜉)=𝐴+𝐵𝐹(𝜉)+𝐶𝐹²(𝜉),0<𝛼≤1, (9) where 𝐴, 𝐵, and 𝐶 are constants. Using the Mittag-Leffler function in one parameter 𝐸_𝛼(𝑥)=∑^∞_𝑘=0(𝑥^𝑘/Γ(1+𝑘𝛼)), (𝛼>0), we obtain the following solutions of (9).

Type 1. When 𝐵=0 and 𝐴/𝐶<0, the solutions of (9) are 𝐹(𝜉)=−−𝐴𝐶tanh_𝛼𝐶−𝐴𝐶𝜉,𝐹(𝜉)=−−𝐴𝐶coth_𝛼𝐶−𝐴𝐶𝜉,𝐹(𝜉)=−−𝐴𝐶×coth_𝛼2𝐶−𝐴𝐶𝜉±csch_𝛼2𝐶−𝐴𝐶𝜉. (10)(11)(12)

When 𝐵=0 and 𝐴/𝐶>0, the solutions of (9) are 𝐹(𝜉)=𝐴𝐶tan_𝛼𝐶𝐴𝐶𝜉,𝐹(𝜉)=−𝐴𝐶cot_𝛼𝐶𝐴𝐶𝜉,𝐹(𝜉)=𝐴𝐶tan_𝛼2𝐶𝐴𝐶𝜉±sec_𝛼2𝐶𝐴𝐶𝜉,𝐹(𝜉)=−𝐴𝐶cot_𝛼2𝐶𝐴𝐶𝜉±csc_𝛼2𝐶𝐴𝐶𝜉. (13)(14)(15)(16)

Type 2. When 𝐴=0 and 𝐵𝐶≠0, the solution of (9) is 𝐹(𝜉)=−𝐵𝑑𝐶𝑑+cosh_𝛼(𝐵𝜉)−sinh_𝛼(,𝐵𝑑) (17) where 𝑑 is an arbitrary constant.

Type 3. When 𝐴𝐵𝐶≠0 and Δ=𝐵²−4𝐴𝐶>0, the solutions of (9) are 𝐹(𝜉)=2𝐴cosh_𝛼√Δ/2𝜉√Δsinh_𝛼√Δ/2𝜉−𝐵cosh_𝛼√𝜉,Δ/2𝐹(𝜉)=−2𝐴sinh_𝛼√Δ/2𝜉𝐵sinh_𝛼√Δ/2𝜉−√Δcosh_𝛼√𝜉.Δ/2 (18)(19)

When 𝐴𝐵𝐶≠0 and Δ=𝐵²−4𝐴𝐶<0, the solutions of (9) are 𝐹(𝜉)=−2𝐴cos_𝛼√−Δ/2𝜉√−Δsin_𝛼√−Δ/2𝜉+𝐵cos_𝛼√𝜉,−Δ/2𝐹(𝜉)=−2𝐴sin_𝛼√−Δ/2𝜉𝐵sin_𝛼√−Δ/2𝜉−√−Δcos_𝛼√𝜉.−Δ/2 (20)(21) The generalized hyperbolic and trigonometric functions in the above solutions (10)–(21) are defined as sinh_𝛼(𝜉)=𝐸_𝛼𝜉^𝛼−𝐸_𝛼−𝜉^𝛼2,cosh_𝛼(𝜉)=𝐸_𝛼𝜉^𝛼+𝐸_𝛼−𝜉^𝛼2,sin_𝛼(𝜉)=𝐸_𝛼𝑖𝜉^𝛼−𝐸_𝛼−𝑖𝜉^𝛼,2𝑖cos_𝛼(𝜉)=𝐸_𝛼𝑖𝜉^𝛼+𝐸_𝛼−𝑖𝜉^𝛼,2𝑖tanh_𝛼(𝜉)=sinh_𝛼(𝜉)cosh_𝛼(𝜉),coth_𝛼(𝜉)=cosh_𝛼(𝜉)sinh_𝛼(𝜉),sech_𝛼(𝜉)=1cosh_𝛼(𝜉),csch_𝛼(𝜉)=1sinh_𝛼(𝜉),tan_𝛼(𝜉)=sin_𝛼(𝜉)cos_𝛼(𝜉),cot_𝛼(𝜉)=cos_𝛼(𝜉)sin_𝛼(𝜉),sec_𝛼(𝜉)=1cos_𝛼(𝜉),csc_𝛼(𝜉)=1sin_𝛼(𝜉). (22)

Step 3. Substituting (8) into (7) and using (9), collecting all terms with the same order of 𝐹(𝜉) together, and then setting coefficients of the polynomial to zero yield a set of algebraic equations for 𝑎₀, 𝑎_𝑗, 𝑏_𝑗 (𝑗=1,2,…,𝑛), and 𝜔.

Step 4. Solving the algebraic equations obtained in Step 3, the constants 𝑎₀, 𝑎_𝑗, 𝑏_𝑗 (𝑗=1,2,…,𝑛), and 𝜔 can be expressed. Substituting these values into expression (8), we can obtain the general form of the exact solution of (7).

Step 5. Substituting the exact solutions of (9) into the general form of exact solution obtained in Step 4, then we can obtain the exact solutions of (5).

Remark 1. If we take 𝐴=𝜎, 𝐵=0, and 𝐶=1, this agrees with the results obtained by S. Zhang and H.-Q. Zhang [15].

4. Exact Solutions of (1)

In this section, we use the above extended fractional Riccati expansion method to explore the exact solutions of time fractional Burgers equation (1). The study of Burgers equation is important since it arises in the approximate theory of flow through a shock wave propagating in a viscous fluid [24] and in the modeling of turbulence [25]. In [26], Inc used the variational iteration method to obtain the numerical solutions of (1) for 𝜀=1. The approximate solution of (1) for 𝜀=1 based on Von-Neumann method is recently considered by El-Danaf and Hadhoud [27]. Bekir et al. [19] solved (1) using the exponential function method based on fractional complex transform to convert fractional differential equations into ordinary differential equations.

Let 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑥+𝜔𝑡, (23) where 𝜔 is nonzero constant, and substituting (23) into (1), we obtain 𝜔^𝛼𝐷^𝛼_𝜉𝑈+𝜀𝑈𝑈^′−𝜈𝑈^′′=0. (24) Thus, the solution of (24) has the form 𝑈=𝑎₀+𝑎₁𝐹(𝜉)+𝑏₁𝐹(𝜉). (25) Substituting (25) into (24) and using (9), collecting the coefficients of 𝐹(𝜉), and then setting the coefficients of 𝐹(𝜉) to zero, we can obtain a set of algebraic equations about 𝑎₀, 𝑎₁, 𝑏₁, 𝜔. Solving the algebraic equations obtained above, we can have the following solutions.

Type 1. When 𝐵=0, we have the following cases.

Case 1. We have 𝑎₀𝜔=−^𝛼𝜀,𝑎₁=2𝜈𝐶𝜀,𝑏₁=0, (26) where 𝜔 is nonzero constants.
Substituting (26) into (25), then according to (10)–(16), we obtain the following hyperbolic function and trigonometric function solutions of (1), respectively. Consider 𝑢₁()𝜔𝑥,𝑡=−^𝛼𝜀−2𝜈𝐶𝜀−𝐴𝐶tanh_𝛼𝐶−𝐴𝐶𝜉, (27) where 𝐴/𝐶<0, 𝜉=𝑥+𝜔𝑡.
The profile for exact solution 𝑢₁(𝑥,𝑡) of (27) is shown in Figure 1. Consider 𝑢₂()𝜔𝑥,𝑡=−^𝛼𝜀−2𝜈𝐶𝜀−𝐴𝐶coth_𝛼𝐶−𝐴𝐶𝜉, (28) where 𝐴/𝐶<0, 𝜉=𝑥+𝜔𝑡. Consider 𝑢₃()𝜔𝑥,𝑡=−^𝛼𝜀−2𝜈𝐶𝜀−𝐴𝐶coth_𝛼2𝐶−𝐴𝐶𝜉±csch_𝛼2𝐶−𝐴𝐶𝜉, (29) where 𝐴/𝐶<0, 𝜉=𝑥+𝜔𝑡. Consider 𝑢₄()𝜔𝑥,𝑡=−^𝛼𝜀+2𝜈𝐶𝜀𝐴𝐶tan_𝛼𝐶𝐴𝐶𝜉, (30) where 𝐴/𝐶>0, 𝜉=𝑥+𝜔𝑡. Consider 𝑢₅()𝜔𝑥,𝑡=−^𝛼𝜀−2𝜈𝐶𝜀𝐴𝐶cot_𝛼𝐶𝐴𝐶𝜉, (31) where 𝐴/𝐶>0, 𝜉=𝑥+𝜔𝑡. Consider 𝑢₆()𝜔𝑥,𝑡=−^𝛼𝜀+2𝜈𝐶𝜀𝐴𝐶tan_𝛼2𝐶𝐴𝐶𝜉±sec_𝛼2𝐶𝐴𝐶𝜉, (32) where 𝐴/𝐶>0, 𝜉=𝑥+𝜔𝑡. Consider 𝑢₇()𝜔𝑥,𝑡=−^𝛼𝜀−2𝜈𝐶𝜀𝐴𝐶cot_𝛼2𝐶𝐴𝐶𝜉±csc_𝛼2𝐶𝐴𝐶𝜉, (33) where 𝐴/𝐶>0, 𝜉=𝑥+𝜔𝑡.

(a) 𝛼=0.5

(b) 𝛼=0.25, 𝛼=0.5, 𝛼=0.75, and 𝛼=1

Figure 1 

The solution 𝑢1(𝑥,𝑡) in (27) shows the 3D and 2D graphs for the given parameters 𝐴=−1, 𝐶=1, 𝜈=0.6, 𝜀=0.1, and 𝜔=0.2.

Case 2. We have 𝑎₀𝜔=−^𝛼𝜀,𝑎₁=0,𝑏₁=−2𝜈𝐴𝜀, (34) where 𝜔 is nonzero constants.
Substituting (34) into (25), then according to (12), (15), and (16), we obtain the following hyperbolic function and trigonometric function solutions of (1), respectively: 𝑢₈(𝑥,𝑡)=2𝜈𝐴sinh_𝛼2𝐶−𝐴𝐶𝜉−𝜔^𝛼−𝐴𝐶cosh_𝛼2𝐶−𝐴𝐶𝜉∓𝜔^𝛼−𝐴𝐶×𝜀−𝐴𝐶cosh_𝛼2𝐶−𝐴𝐶𝜉±1⁻¹, (35) where 𝐴/𝐶<0, 𝜉=𝑥+𝜔𝑡. Consider 𝑢₉(𝑥,𝑡)=−𝜔^𝛼𝐴𝐶sin_𝛼2𝐶𝐴𝐶𝜉+2𝜈𝐴cos_𝛼2𝐶𝐴𝐶𝜉±𝜔^𝛼𝐴𝐶×𝜀𝐴𝐶sin_𝛼2𝐶𝐴𝐶𝜉±1⁻¹, (36) where 𝐴/𝐶>0, 𝜉=𝑥+𝜔𝑡. Consider 𝑢₁₀(𝑥,𝑡)=2𝜈𝐴sin_𝛼2𝐶𝐴𝐶𝜉−𝜔^𝛼𝐴𝐶cos_𝛼2𝐶𝐴𝐶𝜉∓𝜔^𝛼𝐴𝐶×𝜀𝐴𝐶cos_𝛼2𝐶𝐴𝐶𝜉±1⁻¹, (37) where 𝐴/𝐶>0, 𝜉=𝑥+𝜔𝑡.

Case 3. We have 𝑎₀𝜔=−^𝛼𝜀,𝑎₁=2𝜈𝐶𝜀,𝑏₁=−2𝜈𝐴𝜀, (38) where 𝜔 is nonzero constants.
Substituting (38) into (25), then according to (10) and (13), we obtain the following hyperbolic function and trigonometric function solutions of (1), respectively: 𝑢₁₁()𝜔𝑥,𝑡=−^𝛼𝜀−2𝜈𝐶𝜀−𝐴𝐶tanh_𝛼𝐶−𝐴𝐶𝜉+2𝜈𝐴𝜀−𝐴/𝐶coth_𝛼𝐶−𝐴𝐶𝜉, (39) where 𝐴/𝐶<0, 𝜉=𝑥+𝜔𝑡. Consider 𝑢₁₂()𝜔𝑥,𝑡=−^𝛼𝜀+2𝜈𝐶𝜀𝐴𝐶tan_𝛼𝐶𝐴𝐶𝜉−2𝜈𝐴𝜀𝐴/𝐶cot_𝛼𝐶𝐴𝐶𝜉, (40) where 𝐴/𝐶>0, 𝜉=𝑥+𝜔𝑡.

Type 2. When 𝐴=0 and 𝐵𝐶≠0, we have 𝑎₀=𝜈𝐵−𝜔^𝛼𝜀,𝑎₁=2𝜈𝐶𝜀,𝑏₁=0, (41) where 𝜔 is nonzero constants.

Substituting (41) into (25), then according to (17), we obtain the following hyperbolic function solution of (1): 𝑢₁₃(𝑥,𝑡)=𝜈𝐵−𝜔^𝛼𝜀−2𝜈𝐵𝑑𝜀𝑑+cosh_𝛼(𝐵𝜉)−sinh_𝛼(,𝐵𝑑) (42) where 𝜉=𝑥+𝜔𝑡.

Type 3. When 𝐴𝐵𝐶≠0, we have 𝑎₀=𝜈𝐵−𝜔^𝛼𝜀,𝑎₁=2𝜈𝐶𝜀,𝑏₁=0, (43) where 𝜔 is nonzero constants.

Substituting (43) into (25), then according to (18)–(21), we obtain the following hyperbolic function and trigonometric function solutions of (1), respectively: 𝑢₁₄(𝑥,𝑡)=𝜈𝐵−𝜔^𝛼𝜀+4𝜈𝐶𝐴cosh_𝛼√Δ2𝜉×𝜀√Δsinh_𝛼√Δ2𝜉−𝐵cosh_𝛼√Δ2𝜉⁻¹, (44) where Δ=𝐵²−4𝐴𝐶>0, 𝜉=𝑥+𝜔𝑡.

The profile for exact solution 𝑢₁₄(𝑥,𝑡) of (44) is shown in Figure 2. Consider 𝑢₁₅(𝑥,𝑡)=𝜈𝐵−𝜔^𝛼𝜀−4𝜈𝐶𝐴sinh_𝛼√Δ2𝜉×𝜀𝐵sinh_𝛼√Δ2𝜉−√Δcosh_𝛼√Δ2𝜉⁻¹, (45) where Δ=𝐵²−4𝐴𝐶>0, 𝜉=𝑥+𝜔𝑡. Consider 𝑢₁₆(𝑥,𝑡)=𝜈𝐵−𝜔^𝛼𝜀−4𝜈𝐶𝐴cos_𝛼√−Δ2𝜉×𝜀√−Δsin_𝛼√−Δ2𝜉+𝐵cos_𝛼√−Δ2𝜉⁻¹, (46) where Δ=𝐵²−4𝐴𝐶<0, 𝜉=𝑥+𝜔𝑡. Consider 𝑢₁₇(𝑥,𝑡)=𝜈𝐵−𝜔^𝛼𝜀−4𝜈𝐶𝐴sin_𝛼√−Δ2𝜉×𝜀𝐵sin_𝛼√−Δ2𝜉−√−Δcos_𝛼√−Δ2𝜉⁻¹, (47) where Δ=𝐵²−4𝐴𝐶<0, 𝜉=𝑥+𝜔𝑡.

(a) 𝛼=0.4

(b) 𝛼=0.01, 𝛼=0.4, and 𝛼=1

Figure 2 

The solution 𝑢14(𝑥,𝑡) in (44) shows the 3D and 2D graphs for the given parameters 𝐴=−1, 𝐵=2, 𝐶=1, 𝜈=0.6, 𝜀=0.1, and 𝜔=0.1.

5. Exact Solutions of (2)

In this section, we use the above extended fractional Riccati expansion method to explore the exact solutions of Cahn-Hilliard equation (2). In the case of 𝛼=1, this equation is related to a number of interesting physical phenomena like the spinodal decomposition, phase separation, and phase ordering dynamics [19]. The study of Cahn-Hilliard equation is important since it arises in material sciences [28, 29]. However we notice that this equation is very difficult to be solved and several articles investigated it [19, 30]. Jafari et al. [31] used the fractional subequation method to calculate the numerical solutions of (2). Bekir et al. [19] solved (2) using the exponential function method based on fractional complex transform to convert fractional differential equations into ordinary differential equations.

Let 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑥+𝜔𝑡, (48) where 𝜔 is nonzero constant, and substituting (48) into (2), we obtain 𝜔^𝛼−𝛾𝐷^𝛼_𝜉𝑈−6𝑈(𝐷^𝛼_𝜉𝑈)²−3𝑈²−1𝐷_𝜉^2𝛼𝑈+𝐷_𝜉^4𝛼𝑈=0. (49) Thus, the solution of (49) has the form 𝑈=𝑎₀+𝑎₁𝐹(𝜉)+𝑏₁𝐹(𝜉). (50)

Substituting (50) into (49) and using (9), collecting the coefficients of 𝐹(𝜉), and then setting the coefficients of 𝐹(𝜉) to zero, we can obtain a set of algebraic equations about 𝑎₀, 𝑎₁, 𝑏₁, 𝜔. Solving the algebraic equations obtained above, we can have the following solutions.

Type 1. When 𝐵=0, we have the following cases.

Case 1. We have 𝑎₀=0,𝑎₁=√22𝐴,𝑏₁=0,𝛾=𝜔^𝛼1,𝐶=−.2𝐴 (51)
Substituting (51) into (50), then according to (10)–(12), we obtain the following hyperbolic function solutions of (2), respectively: 𝑢₁(𝑥,𝑡)=tanh_𝛼√22𝜉, (52) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼. Consider 𝑢₂(𝑥,𝑡)=coth_𝛼√22𝜉, (53) where 𝜉=𝑥+𝜔𝑡 and 𝛾=𝜔^𝛼. Consider 𝑢₃(𝑥,𝑡)=coth_𝛼√2𝜉±𝑐𝑠𝑐ℎ_𝛼√2𝜉, (54) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼.

Case 2. We have 𝑎₀=0,𝑎₁=0,𝑏₁=√2𝐴,𝛾=𝜔^𝛼1,𝐶=−.2𝐴 (55)
Substituting (55) into (50), then according to (12), we obtain the following hyperbolic function solutions of (2): 𝑢₄(𝑥,𝑡)=sinh_𝛼√2𝜉cosh_𝛼√,2𝜉±1 (56) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼.
The profile for exact solution 𝑢₄(𝑥,𝑡) of (56) is shown in Figure 3.

(a) 𝛼=0.5

(b) 𝛼=0.35, 𝛼=0.5, 𝛼=0.75, and 𝛼=1

Figure 3 

The solution 𝑢4(𝑥,𝑡) in (56) shows the 3D and 2D graphs for the given parameter 𝛾=2.3.

Case 3. We have 𝑎₀=0,𝑎₁=√28𝐴,𝑏₁=√2𝐴,𝛾=𝜔^𝛼1,𝐶=−.8𝐴 (57)
Substituting (57) into (50), then according to (10), we obtain the following hyperbolic function solution of (2): 𝑢₅()=1𝑥,𝑡2tanh_𝛼√24𝜉+12coth_𝛼√24𝜉, (58) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼.
The profile for exact solution 𝑢₅(𝑥,𝑡) of (58) is shown in Figure 4.

(a) 𝛼=0.4

(b) 𝛼=0.01, 𝛼=0.4, and 𝛼=1

Figure 4 

The solution 𝑢5(𝑥,𝑡) in (58) shows the 3D and 2D graphs for the given parameter 𝛾=2.3.

Case 4. We have 𝑎₀=0,𝑎₁=√24𝐴,𝑏₁=√2𝐴,𝛾=𝜔^𝛼1,𝐶=.4𝐴 (59)
Substituting (59) into (50), then according to (13), (15), and (16), we obtain the following trigonometric function solutions of (2), respectively: 𝑢₆()=√𝑥,𝑡22tan_𝛼12𝜉+√22cot_𝛼12𝜉, (60) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼. Consider 𝑢₇(𝑥,𝑡)=±√2sec_𝛼(𝜉), (61) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼. Consider 𝑢₈(𝑥,𝑡)=∓√2csc_𝛼(𝜉), (62) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼.

Type 2. When 𝐴=0 and 𝐵𝐶≠0, we have 𝑎₀=−1,𝑎₁=√2𝐶,𝑏₁=0,𝛾=𝜔^𝛼,𝐵=−√2. (63)

Substituting (63) into (50), then according to (17), we obtain the following hyperbolic function solution of (2): 𝑢₉(𝑥,𝑡)=𝑑−cosh_𝛼√2𝜉−sinh_𝛼√2𝜉𝑑+cosh_𝛼√2𝜉+sinh_𝛼√,2𝜉 (64) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼.

Type 3. When 𝐴𝐵𝐶≠0, we have the following cases.

Case 2. We have 𝑎₀=√22𝐵,𝑎₁=√2𝐵²−24𝐴,𝑏₁=0,𝛾=𝜔^𝛼𝐵,𝐶=²−2.4𝐴 (65)
Substituting (65) into (50), then according to (18) and (19), we obtain the following hyperbolic function solutions of (2), respectively: 𝑢₁₀(𝑥,𝑡)=𝐵sinh_𝛼√2/2𝜉−√2cosh_𝛼√2/2𝜉√2sinh_𝛼√2/2𝜉−𝐵cosh_𝛼√𝜉,2/2 (66) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼, and Δ=𝐵²−4𝐴𝐶=2>0. Consider 𝑢₁₁()=√𝑥,𝑡2sinh_𝛼√2/2𝜉−𝐵cosh_𝛼√2/2𝜉𝐵sinh_𝛼√2/2𝜉−√2cosh_𝛼√𝜉,2/2 (67) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼, and Δ=𝐵²−4𝐴𝐶=2>0.

Case 5. We have 𝑎₀=√22𝐵,𝑎₁√=−2𝐵²−28𝐴,𝑏₁=√2𝐴,𝛾=𝜔^𝛼𝐵,𝐶=−²−2.8𝐴 (68)
Substituting (68) into (50), then according to (18) and (19), we obtain the following hyperbolic function solutions of (2), respectively: 𝑢₁₂()=√𝑥,𝑡22𝐵𝐵−²−2√2/4cosh_𝛼√Δ/2𝜉√Δsinh_𝛼√Δ/2𝜉−𝐵cosh_𝛼√𝜉+√Δ/22/2√Δsinh_𝛼√Δ/2𝜉−𝐵cosh_𝛼√Δ/2𝜉cosh_𝛼√𝜉,Δ/2 (69) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼, and Δ=𝐵²−4𝐴𝐶=(3𝐵²/2)−1>0. Consider 𝑢₁₃()=√𝑥,𝑡22𝐵𝐵+²−2√2/4sinh_𝛼√Δ/2𝜉𝐵sinh_𝛼√Δ/2𝜉−√Δcosh_𝛼√𝜉−√Δ/22/2𝐵sinh_𝛼√Δ/2𝜉−√Δcosh_𝛼√Δ/2𝜉sinh_𝛼√𝜉,Δ/2 (70) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼, and Δ=𝐵²−4𝐴𝐶=(3𝐵²/2)−1>0.
Substituting (68) into (50), then according to (20) and (21), we obtain the following trigonometric function solutions of (2), respectively: 𝑢₁₄()=√𝑥,𝑡22𝐵𝐵+²−2√2/4cos_𝛼√−Δ/2𝜉√−Δsin_𝛼√−Δ/2𝜉+𝐵cos_𝛼√𝜉−√−Δ/22/2√−Δsin_𝛼√−Δ/2𝜉+𝐵cos_𝛼√−Δ/2𝜉cos_𝛼√𝜉,−Δ/2 (71) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼, and Δ=𝐵²−4𝐴𝐶=(3𝐵²/2)−1<0. Consider 𝑢₁₅()=√𝑥,𝑡22𝐵𝐵+²−2√2/4sin_𝛼√−Δ/2𝜉𝐵sin_𝛼√−Δ/2𝜉−√−Δcos_𝛼√𝜉−√−Δ/22/2𝐵sin_𝛼√−Δ/2𝜉−√−Δcos_𝛼√−Δ/2𝜉sin_𝛼√𝜉,−Δ/2 (72) where 𝜉=𝑥+𝜔𝑡, 𝛾=𝜔^𝛼, and Δ=𝐵²−4𝐴𝐶=(3𝐵²/2)−1<0.

6. Conclusions

In this paper, we introduced an extended fractional Riccati expansion method and carried it out to obtain more new exact solutions of the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation. These exact solutions include hyperbolic function solutions and trigonometric function solutions. To the best of our knowledge, the solutions obtained in this Letter have not been reported in literature. Though some exact solutions of the aforementioned two equations have been obtained via the variational iteration method [26], the Von-Neumann method [27], the exponential function method [19], and fractional subequation method [31], they are different from the solutions obtained in this paper. The paper shows that this method is reliable and simple. We will investigate other types of nonlinear fractional differential equations and so on.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported by the Natural Science Foundations of Yunnan Province, China (nos. 2011FZ193 and 2013FZ117).