Exact Solutions for Some Fractional Partial Differential Equations by the (𝐺^′/𝐺) Method

Abstract

We apply the (𝐺^′/𝐺) method to seek exact solutions for several fractional partial differential equations including the space-time fractional (2 + 1)-dimensional dispersive long wave equations, the (2 + 1)-dimensional space-time fractional Nizhnik-Novikov-Veselov system, and the time fractional fifth-order Sawada-Kotera equation. The fractional derivative is defined in the sense of modified Riemann-liouville derivative. Based on a certain variable transformation, these fractional partial differential equations are transformed into ordinary differential equations of integer order. With the aid of mathematical software, a variety of exact solutions for them are obtained.

1. Introduction

In recent decades, fractional differential equations have gained much attention as they are widely used to describe various complex phenomena in many fields such as the fluid flow, signal processing, control theory, systems identification, biology, and other areas. Among the investigations for fractional differential equations, research for seeking exact solutions and approximate solutions of fractional differential equations is a hot topic. Many powerful and efficient methods have been proposed so far, for example, the fractional variational iteration method [1–5], the Adomian’s decomposition method [6–8], the homotopy perturbation method [9], the finite difference method [10], the finite element method [11], the fractional subequation method [12–14], and so on. Using these methods, solutions with various forms for some given fractional differential equations have been established.

In this paper, we will apply the (𝐺^′/𝐺) method [15–21] for solving some fractional partial differential equations in the sense of modified Riemann-Liouville derivative by Jumarie. The definition and some important properties for the Jumarie’s modified Riemann-Liouville derivative of order 𝛼 are listed as follows [22]: 𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑1−𝛼𝑑𝑡×^𝑡₀(𝑡−𝜉)^−𝛼×𝑓(𝜉)(0𝑓−𝑓)𝑑𝜉,0<𝛼<1,_(𝑛)(𝑡)^{(𝛼−𝑛)}𝐷,𝑛≤𝛼<𝑛+1,𝑛≥1,_𝛼𝑡𝑡^𝑟=Γ(1+𝑟)Γ()𝑡1+𝑟−𝛼^𝑟−𝛼,𝐷^𝛼_𝑡𝑓(𝑡)𝑔(𝑡)=𝑔(𝑡)𝐷^𝛼_𝑡𝑓(𝑡)+𝑓(𝑡)𝐷^𝛼_𝑡𝑔(𝑡),𝐷^𝛼_𝑡𝑓𝑔(𝑡)=𝑓^′_𝑔𝑔(𝑡)𝐷^𝛼_𝑡𝑔(𝑡)=𝐷^𝛼_𝑔𝑓𝑔(𝑡)𝑔^′(𝑡)^𝛼. (1)(2)(3)

The rest of this paper is organized as follows. In Section 2, we give the description of the (𝐺^′/𝐺) method for solving fractional partial differential equations. Then in Section 3 we apply this method to establish exact solutions for the space-time fractional (2 + 1)-dimensional dispersive long wave equations, the space-time fractional (2 + 1)-dimensional Nizhnik-Novikov-Veselov system, and the time fractional fifth-order Sawada-Kotera equation. Some conclusions are presented at the end of the paper.

2. Description of the (𝐺^′/𝐺) Method for Fractional Partial Differential Equations

In this section we give the description of the (𝐺^′/𝐺) method for solving fractional partial differential equations.

Suppose that a fractional partial differential equation, say in the independent variables 𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛, is given by 𝑃𝑢₁,…,𝑢_𝑘,𝐷^𝛼_𝑡𝑢₁,…,𝐷^𝛼_𝑡𝑢_𝑘,𝐷^𝛽_𝑥1𝑢₁𝐷,…,_𝛽𝑥1𝑢_𝑘,…,𝐷^𝛾_𝑥𝑛𝑢₁,…,𝐷^𝛾_𝑥𝑛𝑢_𝑘,…=0, (4) where 𝑢_𝑖=𝑢_𝑖(𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛),𝑖=1,…,𝑘, are unknown functions and 𝑃 is a polynomial in 𝑢_𝑖 and their various partial derivatives including fractional derivatives.

Step 1. Execute a certain variable transformation: 𝑢_𝑖𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛=𝑈_𝑖(𝜉),𝜉=𝜉𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛, (5) such that (4) can be turned into the following ordinary differential equation of integer order with respect to the variable 𝜉: 𝑃𝑈₁,…,𝑈_𝑘,𝑈^′₁,…,𝑈^′_𝑘,𝑈₁^′′,…,𝑈_𝑘^′′,…=0. (6)

Step 2. Suppose that the solution of (6) can be expressed by a polynomial in (𝐺^′/𝐺) as follows: 𝑈_𝑗(𝜉)=^𝑚𝑗_𝑖=0𝑎_𝑗,𝑖𝐺^′𝐺^𝑖,𝑗=1,2,…,𝑘, (7) where 𝐺=𝐺(𝜉) satisfies the second-order ODE in the form 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0, (8) and 𝜆, 𝜇, 𝑎_𝑗,𝑖, 𝑖=0,1,…,𝑚_𝑗, 𝑗=1,2,…,𝑘, are constants to be determined later, 𝑎_𝑗,𝑚≠0. The positive integer 𝑚_𝑗 can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (6).
By the generalized solutions of (8), we have 𝐺^′𝐺=−𝜆2+𝜆²−4𝜇2×𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹,𝜆²−𝜆−4𝜇>0,2+4𝜇−𝜆²2×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹,𝜆²−𝜆−4𝜇<0,2+𝐶₂𝐶₁+𝐶₂𝜉,𝜆²−4𝜇=0, (9) where 𝐶₁ and 𝐶₂ are arbitrary constants.

Step 3. Substituting (7) into (6) and using (8), collecting all terms with the same order of (𝐺^′/𝐺) together, the left-hand side of (6) is converted into another polynomial in (𝐺^′/𝐺). Equating each coefficient of this polynomial to zero yields a set of algebraic equations for 𝜆, 𝜇, 𝑎_𝑗,𝑖, 𝑖=0,1,…,𝑚_𝑗, 𝑗=1,2,…,𝑘.

Step 4. Solving the equations system in Step 3 and using (9), we can construct a variety of exact solutions for (4).

3. Applications of the Method

3.1. Space-Time Fractional (2 + 1)-Dimensional Dispersive Long Wave Equations

We consider the following space-time fractional (2 + 1)-dimensional dispersive long wave equations: 𝜕^2𝛼𝑢𝜕𝑦^𝛼𝜕𝑡^𝛼+𝜕^2𝛼𝑣𝜕𝑥^2𝛼+𝜕^𝛼𝑢𝜕^𝛼𝑢/𝜕𝑥^𝛼𝜕𝑦^𝛼𝜕=0,_𝛼𝑣𝜕𝑡^𝛼+𝜕^𝛼𝑢𝜕𝑥^𝛼+𝜕^𝛼(𝑢𝑣)𝜕𝑥^𝛼+𝜕^3𝛼𝑢𝜕𝑥^2𝛼𝜕𝑦^𝛼=0,0<𝛼≤1, (10) which are a variation of the known (2 + 1)-dimensional dispersive long wave equations [23–36]: 𝑢_𝑦𝑡+𝑣_𝑥𝑥+𝑢𝑢_𝑥𝑦𝑣=0,_𝑡+𝑢_𝑥+(𝑢𝑣)_𝑥+𝑢_𝑥𝑥𝑦=0. (11) Some types of exact solutions for (11) have been obtained in [23–36] by the use of various methods including the Riccati subequation method [23, 24, 29], the nonlinear transformation method [25], Jacobi function method [27, 28, 36], (𝐺^′/𝐺)-expansion method [26], modified CK’s direct method [30], EXP-function method [31], Hopf-Cole transformation method [32], modified extended Fan’s sub-equation method [33, 34], and generalized algebraic method [35]. But we notice so far that no research has been paid for (10). In the following we will apply the described method in Section 2 to (10).

To begin with, we suppose 𝑢(𝑥,𝑦,𝑡)=𝑈(𝜉), 𝑣(𝑥,𝑦,𝑡)=𝑉(𝜉), where 𝜉=(𝑐/Γ(1+𝛼))𝑡^𝛼+(𝑘₁/Γ(1+𝛼))𝑥^𝛼+(𝑘₂/Γ(1+𝛼))𝑦^𝛼+𝜉₀, 𝑘₁, 𝑘₂, 𝑐, 𝜉₀ are all constants with 𝑘₁, 𝑘₂, 𝑐≠0. Then by the use of the first equality in (3), we obtain 𝐷^𝛼_𝑥𝑢=𝐷^𝛼_𝑥𝑈(𝜉)=𝑈^′(𝜉)𝐷^𝛼_𝑥𝜉=𝑘₁𝑈^′(𝜉),𝐷^𝛼_𝑦𝑢=𝐷^𝛼_𝑦𝑈(𝜉)=𝑈^′(𝜉)𝐷^𝛼_𝑦𝜉=𝑘₂𝑈^′(𝜉), 𝐷^𝛼_𝑡𝑢=𝐷^𝛼_𝑡𝑈(𝜉)=𝑈^′(𝜉)𝐷^𝛼_𝑡𝜉=𝑐𝑈^′(𝜉), and then (10) can be turned into 𝑐𝑘₂𝑈^′′+𝑘²₁𝑉^′′+𝑘₁𝑘₂𝑈𝑈^′′+𝑈^′2=0,𝑐𝑉^′+𝑘₁𝑈^′+𝑘₁(𝑈𝑉)^′+𝑘²₁𝑘₂𝑈^′′′=0. (12) Suppose that the solutions of (12), can be expressed by 𝑈(𝜉)=^𝑚1_𝑖=0𝑎_𝑖𝐺^′𝐺^𝑖,𝑉(𝜉)=^𝑚2_𝑖=0𝑏_𝑖𝐺^′𝐺^𝑖. (13) Balancing the order of 𝑉^′′ and 𝑈𝑈^′′, (𝑈𝑉)^′ and 𝑈^′′′ in (12) we can obtain 𝑚₁=1, 𝑚₂=2. So we have 𝑈(𝜉)=𝑎₀+𝑎₁𝐺^′𝐺,𝑉(𝜉)=𝑏₀+𝑏₁𝐺^′𝐺+𝑏₂𝐺^′𝐺². (14) Substituting (14) into (12), using (8), collecting all the terms with the same power of (𝐺^′/𝐺) together, and equating each coefficient to zero, yields a set of algebraic equations. Solving these equations yields 𝑎₀=±𝑘²₁𝜆−𝑐𝑘₁,𝑎₁=±2𝑘₁,𝑏₀=−2𝑘₁𝑘₂𝜇−1,𝑏₁=−2𝑘₁𝑘₂𝑏𝜆,₂=−2𝑘₁𝑘₂. (15) Substituting the result above into (14) and combining it with (9) we can obtain the following exact solutions to (10).

When 𝜆²−4𝜇>0, 𝑢₁=𝑥,𝑦,𝑡±𝑘²₁𝜆−𝑐𝑘₁±2𝑘₁−𝜆2+𝜆²−4𝜇2×𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹,−32𝑘²₁−𝜆2+𝜆²−4𝜇2×𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹²,𝑣₁𝑥,𝑦,𝑡=−2𝑘₁𝑘₂𝜇−1−2𝑘₁𝑘₂𝜆×−𝜆2+𝜆²−4𝜇2×𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹−2𝑘₁𝑘₂−𝜆2+𝜆²−4𝜇2×𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹², (16) where 𝜉=(𝑐/Γ(1+𝛼))𝑡^𝛼+(𝑘₁/Γ(1+𝛼))𝑥^𝛼+(𝑘₂/Γ(1+𝛼))𝑦^𝛼+𝜉₀.

In particular, if we take 𝐶₂=0, then we obtain the following solitary wave solutions, which are shown in Figures 1 and 2.

Figure 1 

The solution 𝑢1 with the positive symbol selected in ± and 𝜆=4, 𝜇=3, 𝑐=𝑘1=𝑘2=1, 𝐶2=𝜉0=0, 𝛼=1/2, and 𝑡=0.1.

Figure 2 

The solution 𝑣1 with 𝜆=4, 𝜇=3, 𝑐=𝑘1=𝑘2=1, 𝐶2=𝜉0=0, 𝛼=1/2, and 𝑡=0.1.

When 𝜆²−4𝜇<0, 𝑢₂=𝑥,𝑦,𝑡±𝑘²₁𝜆−𝑐𝑘₁±2𝑘₁−𝜆2+4𝜇−𝜆²2×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹−32𝑘²₁−𝜆2+4𝜇−𝜆²2×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉²,𝑣₂𝑥,𝑦,𝑡=−2𝑘₁𝑘₂𝜇−1−2𝑘₁𝑘₂𝜆−𝜆2+4𝜇−𝜆²2×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹−2𝑘₁𝑘₂−𝜆2+4𝜇−𝜆²2×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉², (17) where 𝜉=(𝑐/Γ(1+𝛼))𝑡^𝛼+(𝑘₁/Γ(1+𝛼))𝑥^𝛼+(𝑘₂/Γ(1+𝛼))𝑦^𝛼+𝜉₀.

In Figures 3 and 4, the periodic function solutions (17) are demonstrated with some certain parameters.

Figure 3 

The solution 𝑢2 with the positive symbol selected in ± and 𝜆=2, 𝜇=2, 𝑐=𝑘1=𝑘2=1, 𝐶2=𝜉0=0, 𝛼=1/2, and 𝑡=0.1.

Figure 4 

The solution 𝑣2 with 𝜆=2, 𝜇=2, 𝑐=𝑘1=𝑘2=1, 𝐶2=𝜉0=0, 𝛼=1/2, and 𝑡=0.1.

When 𝜆²−4𝜇=0, 𝑢₃𝑥,𝑦,𝑡=±𝑘²₁𝜆−𝑐𝑘₁±2𝑘₁−𝜆2+𝐶₂𝐶₁+𝐶₂𝜉−32𝑘²₁−𝜆2+𝐶₂𝐶₁+𝐶₂𝜉²,𝑣₃𝑥,𝑦,𝑡=−2𝑘₁𝑘₂𝜇−1−2𝑘₁𝑘₂𝜆−𝜆2+𝐶₂𝐶₁+𝐶₂𝜉−2𝑘₁𝑘₂−𝜆2+𝐶₂𝐶₁+𝐶₂𝜉², (18) where 𝜉=(𝑐/Γ(1+𝛼))𝑡^𝛼+(𝑘₁/Γ(1+𝛼))𝑥^𝛼+(𝑘₂/Γ(1+𝛼))𝑦^𝛼+𝜉₀.

Remark 1. The established solutions in (16)–(18) are new exact solutions for the space-time fractional (2 + 1)-dimensional dispersive long wave equations.

3.2. Space-Time Fractional (2 + 1)-Dimensional Nizhnik-Novikov-Veselov System

We will consider the space-time fractional (2 + 1)-dimensional Nizhnik-Novikov-Veselov (NNV) system: 𝐷^𝛼_𝑡𝑢+𝑎𝐷_𝑥^3𝛽𝑢+𝑏𝐷_𝑦^3𝛾𝑢+𝑐𝐷^𝛽_𝑥𝑢+𝑑𝐷^𝛾_𝑦𝑢=3𝑎𝐷^𝛽_𝑥(𝑢𝑣)+3𝑏𝐷^𝛾_𝑦(),𝐷𝑢𝑤_𝛽𝑥𝑢=𝐷^𝛾_𝑦𝐷𝑣,_𝛾𝑦𝑢=𝐷^𝛽_𝑥𝑤,0<𝛼,𝛽,𝛾≤1, (19) which is a variation of the following (2 + 1)-dimensional Nizhnik-Novikov-Veselov (NNV) system [21, 37, 38]: 𝑢_𝑡+𝑎𝑢_𝑥𝑥𝑥+𝑏𝑢_𝑦𝑦𝑦+𝑐𝑢_𝑥+𝑑𝑢_𝑦=3𝑎(𝑢𝑣)_𝑥+3𝑏(𝑢𝑤)_𝑦,𝑢_𝑥=𝑣_𝑦,𝑢_𝑦=𝑤_𝑥. (20)

Suppose 𝑢(𝑥,𝑦,𝑡)=𝑈(𝜉), 𝑣(𝑥,𝑦,𝑡)=𝑉(𝜉), 𝑤(𝑥,𝑦,𝑡)=𝑊(𝜉), where 𝜉=(𝑐/Γ(1+𝛼))𝑡^𝛼+(𝑘/Γ(1+𝛽))𝑥^𝛽+(𝑙/Γ(1+𝛾))𝑦^𝛾+𝜉₀, 𝑐, 𝑘, 𝑙, 𝜉₀ are all constants with 𝑘₁,𝑘₂, 𝑐≠0. Then by the use of the first equality in (3), (19) can be turned into the following forms: 𝑐𝑢^′+𝑎𝑘³𝑢^′′′+𝑏𝑙³𝑢^′′′+𝑐𝑘𝑢^′+𝑑𝑙𝑢^′=3𝑎𝑘(𝑢𝑣)^′+3𝑏𝑙(𝑢𝑤)^′,𝑘𝑢^′=𝑙𝑣^′,𝑙𝑢^′=𝑘𝑤^′. (21) Integrating (21) once, we have 𝑐𝑢+𝑎𝑘³𝑢^′′+𝑏𝑙³𝑢^′′+𝑐𝑘𝑢+𝑑𝑙𝑢=3𝑎𝑘𝑢𝑣+3𝑏𝑙𝑢𝑤+𝑔₁,𝑘𝑢=𝑙𝑣+𝑔₂,𝑙𝑢=𝑘𝑤+𝑔₃, (22) where 𝑔₁,𝑔₂, and 𝑔₃ are the integration constants.

Suppose that the solutions of (22) can be expressed by a polynomial in (𝐺^′/𝐺) as follows: 𝑈(𝜉)=^𝑚1_𝑖=0𝑎_𝑖𝐺^′𝐺^𝑖,𝑉(𝜉)=^𝑚2_𝑖=0𝑏_𝑖𝐺^′𝐺^𝑖,𝑊(𝜉)=^𝑚3_𝑖=0𝑐_𝑖𝐺^′𝐺^𝑖, (23) where 𝑎_𝑖, 𝑏_𝑖, and 𝑐_𝑖 are constants. Balancing the order of 𝑢^′′ and 𝑢𝑣, and the order of 𝑢 and 𝑣, the order of 𝑢 and 𝑤 in (22), we can obtain 𝑚₁=𝑚₂=𝑚₃=2. So we have 𝑈(𝜉)=𝑎₀+𝑎₁𝐺^′𝐺¹+𝑎₂𝐺^′𝐺²,𝑉(𝜉)=𝑏₀+𝑏₁𝐺^′𝐺¹+𝑏₂𝐺^′𝐺²,𝑊(𝜉)=𝑐₀+𝑐₁𝐺^′𝐺¹+𝑐₂𝐺^′𝐺². (24)

Substituting (24) into (22), using (8), collecting all the terms with the same power of (𝐺^′/𝐺) together, and equating each coefficient to zero, yields a set of algebraic equations. Solving these equations yields.

Case 1. Consider 𝑎₂=2𝑘𝑙,𝑎₁=2𝑘𝑙𝜆,𝑎₀𝑏=2𝑘𝑙𝜇,₂=2𝑘²,𝑏₁=2𝑘²𝜆,𝑏₀=2𝑘²𝑐𝜇,₂=2𝑙²,𝑐₁=2𝑙²𝜆,𝑐₀=2𝑙²𝜇,𝑐=4𝑎𝑘³𝜇−𝑎𝑘³𝜆²−𝑏𝑙³𝜆²−𝑑𝑙+4𝑏𝑙³𝑔𝜇−𝑐𝑘,₁=𝑔₂=𝑔₃=0, (25) where 𝑘 and 𝑙 are arbitrary constants.
Substituting the result above into (24) and combining it with (9) we can obtain the corresponding exact solutions for (19).
When 𝜆²−4𝜇>0, 𝑢₁𝑥,𝑦,𝑡=2𝑘𝑙𝜇−𝑘𝑙𝜆²2+𝑘𝑙𝜆²−4𝜇2×𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹²,𝑣₁𝑥,𝑦,𝑡=2𝑘²𝑘𝜇−²𝜆²2+𝑘²2𝜆²×𝐶−4𝜇₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹²,𝑤₁𝑥,𝑦,𝑡=2𝑙²𝑙𝜇−²𝜆²2+𝑙²2𝜆²×𝐶−4𝜇₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹², (26) where 𝜉=((4𝑎𝑘³𝜇−𝑎𝑘³𝜆²−𝑏𝑙³𝜆²−𝑑𝑙+4𝑏𝑙³𝜇−𝑐𝑘)/Γ(1+𝛼))𝑡^𝛼+(𝑘/Γ(1+𝛽))𝑥^𝛽+(𝑙/Γ(1+𝛾))𝑦^𝛾+𝜉₀.
When 𝜆²−4𝜇<0, 𝑢₂𝑥,𝑦,𝑡=2𝑘𝑙𝜇−𝑘𝑙𝜆²2+𝑘𝑙4𝜇−𝜆²2×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹²,𝑣₂𝑥,𝑦,𝑡=2𝑘²𝑘𝜇−²𝜆²2+𝑘²24𝜇−𝜆²×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹²,𝑤₂𝑥,𝑦,𝑡=2𝑙²𝑙𝜇−²𝜆²2+𝑙²24𝜇−𝜆²×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹², (27) where 𝜉=((4𝑎𝑘³𝜇−𝑎𝑘³𝜆²−𝑏𝑙³𝜆²−𝑑𝑙+4𝑏𝑙³𝜇−𝑐𝑘)/Γ(1+𝛼))𝑡^𝛼+(𝑘/Γ(1+𝛽))𝑥^𝛽+(𝑙/Γ(1+𝛾))𝑦^𝛾+𝜉₀.
When 𝜆²−4𝜇=0, 𝑢₃𝑥,𝑦,𝑡=−𝑘𝑙𝜆²2+2𝑘𝑙𝐶²₂𝐶₁+𝐶₂𝜉²𝑣+2𝑘𝑙𝜇,₃𝑘𝑥,𝑦,𝑡=−²𝜆²2+2𝑘²𝐶²₂𝐶₁+𝐶₂𝜉²+2𝑘²𝑤𝜇,₃𝑙𝑥,𝑦,𝑡=−²𝜆²2+2𝑙²𝐶²₂𝐶₁+𝐶₂𝜉²+2𝑙²𝜇, (28) where 𝜉=((4𝑎𝑘³𝜇−𝑎𝑘³𝜆²−𝑏𝑙³𝜆²−𝑑𝑙+4𝑏𝑙³𝜇−𝑐𝑘)/Γ(1+𝛼))𝑡^𝛼+(𝑘/Γ(1+𝛽))𝑥^𝛽+(𝑙/Γ(1+𝛾))𝑦^𝛾+𝜉₀.

Case 2. Consider 𝑎₂=2𝑘𝑙,𝑎₁=2𝑘𝑙𝜆,𝑎₀=13𝑘𝑙𝜆²,𝑏+2𝜇₂=2𝑘²,𝑏₁=2𝑘²𝜆,𝑏₀=13𝑘²𝜆²,𝑐+2𝜇₂=2𝑙²,𝑐₁=2𝑙²𝜆,𝑐₀=13𝑙²𝜆²+2𝜇,𝑐=−4𝑎𝑘³𝜇+𝑎𝑘³𝜆²+𝑏𝑙³𝜆²−𝑑𝑙−4𝑏𝑙³𝑔𝜇−𝑐𝑘,₁=𝑔₂=𝑔₃=0. (29)
Substituting the result above into (24) and combining it with (9) we can obtain the corresponding exact solutions for (19).
When 𝜆²−4𝜇>0, 𝑢₄=1𝑥,𝑦,𝑡3𝑘𝑙𝜆²+2𝜇−𝑘𝑙𝜆²2+𝑘𝑙𝜆²−4𝜇2×𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹²,𝑣₄=1𝑥,𝑦,𝑡3𝑘²𝜆²−𝑘+2𝜇²𝜆²2+𝑘²2𝜆²×𝐶−4𝜇₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹²,𝑤₄=1𝑥,𝑦,𝑡𝑥𝑖3𝑙²𝜆²−𝑙+2𝜇²𝜆²2+𝑙²2𝜆²×𝐶−4𝜇₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹², (30) where 𝜉=((−4𝑎𝑘³𝜇+𝑎𝑘³𝜆²+𝑏𝑙³𝜆²−𝑑𝑙−4𝑏𝑙³𝜇−𝑐𝑘)/Γ(1+𝛼))𝑡^𝛼+(𝑘/Γ(1+𝛽))𝑥^𝛽+(𝑙/Γ(1+𝛾))𝑦^𝛾+𝜉₀.
When 𝜆²−4𝜇<0, 𝑢₅=1𝑥,𝑦,𝑡3𝑘𝑙𝜆²+2𝜇−𝑘𝑙𝜆²2+𝑘𝑙4𝜇−𝜆²2×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹²,𝑣₅=1𝑥,𝑦,𝑡3𝑘²𝜆²−𝑘+2𝜇²𝜆²2+𝑘²24𝜇−𝜆²×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹²,𝑤₅=1𝑥,𝑦,𝑡3𝑙²𝜆²−𝑙+2𝜇²𝜆²2+𝑙²24𝜇−𝜆²×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹², (31) where 𝜉=((−4𝑎𝑘³𝜇+𝑎𝑘³𝜆²+𝑏𝑙³𝜆²−𝑑𝑙−4𝑏𝑙³𝜇−𝑐𝑘)/Γ(1+𝛼))𝑡^𝛼+(𝑘/Γ(1+𝛽))𝑥^𝛽+(𝑙/Γ(1+𝛾))𝑦^𝛾+𝜉₀.
When 𝜆²−4𝜇=0, 𝑢₆𝑥,𝑦,𝑡=−𝑘𝑙𝜆²2+2𝑘𝑙𝐶²₂𝐶₁+𝐶₂𝜉²+13𝑘𝑙𝜆²,𝑣+2𝜇₆𝑘𝑥,𝑦,𝑡=−²𝜆²2+2𝑘²𝐶²₂𝐶₁+𝐶₂𝜉²+13𝑘²𝜆²,𝑤+2𝜇₆𝑙𝑥,𝑦,𝑡=−²𝜆²2+2𝑙²𝐶²₂𝐶₁+𝐶₂𝜉²+13𝑙²𝜆²+2𝜇, (32) where 𝜉=((−4𝑎𝑘³𝜇+𝑎𝑘³𝜆²+𝑏𝑙³𝜆²−𝑑𝑙−4𝑏𝑙³𝜇−𝑐𝑘)/Γ(1+𝛼))𝑡^𝛼+(𝑘/Γ(1+𝛽))𝑥^𝛽+(𝑙/Γ(1+𝛾))𝑦^𝛾+𝜉₀.

Remark 2. The established solutions in (26)–(32) for the space-time fractional (2 + 1)-dimensional Nizhnik-Novikov-Veselov system have not been reported by other authors so far.

Remark 3. In [21], Zayed solved the (2 + 1)-dimensional Nizhnik-Novikov-Veselov (NNV) system (20) by the (𝐺^′/𝐺)-expansion method. We note that if we put 𝛼=1, 𝑘=𝑙=1 in (26)–(32), then our results reduce to the results in [21] despite the slight difference of the symbols. So the solutions presented here are of more general forms.

3.3. Time Fractional Fifth-Order Sawada-Kotera Equation

We consider the time fractional fifth-order Sawada-Kotera equation: 𝐷^𝛼_𝑡𝑢+𝑢_{𝑥𝑥𝑥𝑥𝑥}+45𝑢_𝑥𝑢²+15𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥=0,0<𝛼≤1, (33) which is the variation of the fifth-order Sawada-Kotera equation [39]: 𝑢_𝑡+𝑢_{𝑥𝑥𝑥𝑥𝑥}+45𝑢_𝑥𝑢²+15𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥=0. (34) Now we apply the proposed method to (33). To begin with, we suppose 𝑢(𝑥,𝑡)=𝑈(𝜉), where 𝜉=𝑘𝑥+(𝑐/Γ(1+𝛼))𝑡^𝛼+𝜉₀, 𝑘, 𝑐, 𝜉₀ are all constants with 𝑘, 𝑐≠0. Then by the use of the first equality in (3), (33) can be turned into 𝑐𝑈^′+𝑘⁵𝑈⁽⁵⁾+45𝑘𝑈𝑈^′+15𝑘³𝑈^′𝑈^′′+𝑈𝑈^′′′=0. (35)

Suppose that the solution of (35) can be expressed by 𝑈(𝜉)=^𝑚_𝑖=0𝑎_𝑖𝐺^′𝐺^𝑖. (36) Balancing the order of 𝑈⁽⁵⁾ and 𝑈𝑈^′′′ in (36) we have 𝑚=2. So 𝑈(𝜉)=𝑎₀+𝑎₁𝐺^′𝐺+𝑎₂𝐺^′𝐺². (37)

Substituting (37) into (35), using (8), collecting all the terms with the same power of (𝐺^′/𝐺) together, and equating each coefficient to zero, yields a set of algebraic equations. Solving these equations yields 𝑎₀=−40𝑘³𝜇+5𝑘³𝜆²±80𝑘⁶𝜇²−40𝑘⁶𝜇𝜆²+5𝑘⁶𝜆⁴×()−20𝑘𝑐30𝑘⁻¹,𝑎₁=−2𝑘²𝜆,𝑎₂=−2𝑘². (38)

Substituting the result above into (37) and combining it with (9) we can obtain the corresponding exact solutions to (33).

When 𝜆²−4𝜇>0, 𝑢₁(𝑥,𝑡)=−40𝑘³𝜇+5𝑘³𝜆²±80𝑘⁶𝜇²−40𝑘⁶𝜇𝜆²+5𝑘⁶𝜆⁴×()−20𝑘𝑐30𝑘⁻¹−2𝑘²𝜆−𝜆2+𝜆²−4𝜇2×𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹−2𝑘²−𝜆2+𝜆²−4𝜇2×𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹², (39) where 𝜉=𝑘𝑥+(𝑐/Γ(1+𝛼))𝑡^𝛼+𝜉₀.

When 𝜆²−4𝜇<0, 𝑢₂(𝑥,𝑡)=−40𝑘³𝜇+5𝑘³𝜆²±80𝑘⁶𝜇²−40𝑘⁶𝜇𝜆²+5𝑘⁶𝜆⁴×(−20𝑘𝑐30𝑘)⁻¹−2𝑘²𝜆−𝜆2+4𝜇−𝜆²2×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹−2𝑘²−𝜆2+4𝜇−𝜆²2×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹², (40) where 𝜉=𝑘𝑥+(𝑐/Γ(1+𝛼))𝑡^𝛼+𝜉₀.

When 𝜆²−4𝜇=0, 𝑢₃(𝑥,𝑡)=−40𝑘³𝜇+5𝑘³𝜆²±80𝑘⁶𝜇²−40𝑘⁶𝜇𝜆²+5𝑘⁶𝜆⁴×()−20𝑘𝑐30𝑘⁻¹−2𝑘²𝜆−𝜆2+𝐶₂𝐶₁+𝐶₂𝜉−2𝑘²−𝜆2+𝐶₂𝐶₁+𝐶₂𝜉², (41) where 𝜉=𝑘𝑥+(𝑐/Γ(1+𝛼))𝑡^𝛼+𝜉₀.

Remark 4. The established solutions in (39)–(41) are new exact solutions for the time fractional fifth-order Sawada-Kotera equation.

4. Conclusions

We have solved the space-time fractional (2 + 1)-dimensional dispersive long wave equations, the space-time fractional (2 + 1)-dimensional Nizhnik-Novikov-Veselov system, and the time fractional fifth-order Sawada-Kotera equation by the known (𝐺^′/𝐺) method successfully. In the process, the transformation for 𝜉 is very important, which ensures that certain fractional partial differential equations can be turned into ordinary differential equations of integer order. With the aid of mathematical software, a variety of exact solutions for these equations are obtained, which may help to understand better physical phenomenon related to these equations. Finally, we note that this approach can also be applied to solve other fractional partial differential equations.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.