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Exact solutions for fractional partial differential equations by a new fractional sub-equation method

Advances in Difference Equations20132013:199

https://doi.org/10.1186/1687-1847-2013-199

  • Received: 10 November 2012
  • Accepted: 21 June 2013
  • Published:

Abstract

In this paper, we propose a new fractional sub-equation method for finding exact solutions of fractional partial differential equations (FPDEs) in the sense of modified Riemann-Liouville derivative, which is the fractional version of the known (G′/G) method. To illustrate the validity of this method, we apply it to the space-time fractional Fokas equation, the space-time fractional ( 2 + 1 ) -dimensional dispersive long wave equations and the space-time fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.

MSC:35Q51, 35Q53.

Keywords

  • fractional sub-equation method
  • fractional partial differential equations
  • exact solutions
  • fractional complex transformation

1 Introduction

Fractional differential equations are generalizations of classical differential equations of integer order. In recent decades, fractional differential equations have been the focus of many studies due to their frequent appearance in various applications in physics, biology, engineering, signal processing, systems identification, control theory, finance and fractional dynamics. Many articles have investigated some aspects of fractional differential equations such as the existence and uniqueness of solutions to Cauchy-type problems, the methods for explicit and numerical solutions, and the stability of solutions [18]. In [9], Jafari et al. applied the fractional sub-equation method to construct exact solutions of the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation. In [10], Baleanu et al. studied the existence and uniqueness of the solution for a nonlinear fractional differential equation boundary-value problem by using fixed-point methods. In [11], Nyamoradi et al. investigated the existence of solutions for the multipoint boundary value problem of a fractional order differential inclusion.

Among the investigations for fractional differential equations, research into seeking exact solutions and numerical solutions of fractional differential equations is an important topic. Many powerful and efficient methods have been proposed to obtain numerical solutions and exact solutions of fractional differential equations so far. For example, these methods include the Adomian decomposition method [1214], the variational iterative method [1522], the homotopy perturbation method [2326], the differential transformation method [27], the finite difference method [28], the finite element method [29], the fractional Riccati sub-equation method [3032] and so on. In these investigations, we note that many authors have sought exact and numerical solutions for fractional partial differential equations (FPDEs) in the sense of modified Riemann-Liouville derivative (for example, see [16, 17, 3034]). Based on these methods, a variety of fractional differential equations have been investigated.

In this paper, we propose a new fractional sub-equation method to establish exact solutions for fractional partial differential equations (FPDEs) in the sense of modified Riemann-Liouville derivative defined by Jumarie [35], which is a fractional version of the known (G′/G) method [3639]. This method is based on the following fractional ODE:
D ξ 2 α G ( ξ ) + λ D ξ α G ( ξ ) + μ G ( ξ ) = 0 ,
(1)

where D ξ α G ( ξ ) denotes the modified Riemann-Liouville derivative of order α for G ( ξ ) with respect to ξ.

The rest of this paper is organized as follows. In Section 2, we present some definitions and properties of Jumarie’s modified Riemann-Liouville derivative and the expression for D ξ α G ( ξ ) G ( ξ ) related to Eq. (1). In Section 3, we give the description of the fractional sub-equation method for solving FPDEs. Then in Section 4 we apply this method to establish exact solutions for the space-time fractional Fokas equation, the space-time fractional ( 2 + 1 ) -dimensional dispersive long wave equations and the space-time fractional fifth-order Sawada-Kotera equation. Some conclusions are presented at the end of the paper.

2 Jumarie’s modified Riemann-Liouville derivative and general expression for D ξ α G ( ξ ) G ( ξ )

Jumarie’s modified Riemann-Liouville derivative of order α is defined by the following expression [35]:
D t α f ( t ) = { 1 Γ ( 1 α ) d d t 0 t ( t ξ ) α ( f ( ξ ) f ( 0 ) ) d ξ , 0 < α < 1 , ( f ( n ) ( t ) ) ( α n ) , n α < n + 1 , n 1 .
We list some important properties for the modified Riemann-Liouville derivative as follows (see [[35], Eqs. (3.10)-(3.13)]):
D t α t r = Γ ( 1 + r ) Γ ( 1 + r α ) t r α ,
(2)
D t α ( f ( t ) g ( t ) ) = g ( t ) D t α f ( t ) + f ( t ) D t α g ( t ) ,
(3)
D t α f [ g ( t ) ] = f g [ g ( t ) ] D t α g ( t ) = D g α f [ g ( t ) ] ( g ( t ) ) α .
(4)
In order to obtain the general solutions for Eq. (1), we suppose G ( ξ ) = H ( η ) and a nonlinear fractional complex transformation η = ξ α Γ ( 1 + α ) . Then by Eq. (2) and the first equality in Eq. (4), Eq. (1) can be turned into the following second ordinary differential equation
H ( η ) + λ H ( η ) + μ H ( η ) = 0 .
(5)
By the general solutions of Eq. (5), we have
H ( η ) H ( η ) = { λ 2 + λ 2 4 μ 2 ( C 1 sinh λ 2 4 μ 2 η + C 2 cosh λ 2 4 μ 2 η C 1 cosh λ 2 4 μ 2 η + C 2 sinh λ 2 4 μ 2 η ) , λ 2 4 μ > 0 , λ 2 + 4 μ λ 2 2 ( C 1 sin 4 μ λ 2 2 η + C 2 cos 4 μ λ 2 2 η C 1 cos 4 μ λ 2 2 η + C 2 sin 4 μ λ 2 2 η ) , λ 2 4 μ < 0 , λ 2 + C 2 C 1 + C 2 η , λ 2 4 μ = 0 ,
(6)

where C 1 , C 2 are arbitrary constants.

Since D ξ α G ( ξ ) = D ξ α H ( η ) = H ( η ) D ξ α η = H ( η ) , we obtain
D ξ α G ( ξ ) G ( ξ ) = { λ 2 + λ 2 4 μ 2 ( C 1 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α C 1 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α ) , λ 2 4 μ > 0 , λ 2 + 4 μ λ 2 2 ( C 1 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α C 1 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α ) , λ 2 4 μ < 0 , λ 2 + C 2 Γ ( 1 + α ) C 1 Γ ( 1 + α ) + C 2 ξ α , λ 2 4 μ = 0 .
(7)

3 Description of the fractional sub-equation method

In this section we describe the main steps of the fractional sub-equation method for finding exact solutions of FPDEs.

Suppose that a fractional partial differential equation, say in the independent variables t , x 1 , x 2 , , x n , is given by
P ( u 1 , u k , D t α u 1 , , D t α u k , D x 1 α u 1 , , D x 1 α u k , , D x n α u 1 , , D x n α u k , D t 2 α u 1 , , D t 2 α u k , D x 1 2 α u 1 , ) = 0 ,
(8)

where u i = u i ( t , x 1 , x 2 , , x n ) , i = 1 , , k , are unknown functions, P is a polynomial in u i and their various partial derivatives include fractional derivatives.

Step 1. Suppose that
u i ( t , x 1 , x 2 , , x n ) = U i ( ξ ) , ξ = c t + k 1 x 1 + k 2 x 2 + + k n x n + ξ 0 .
(9)
Then by the second equality in Eq. (4), Eq. (8) can be turned into the following fractional ordinary differential equation with respect to the variable ξ:
P ˜ ( U 1 , , U k , c α D ξ α U 1 , , c α D ξ α U k , k 1 α D ξ α U 1 , , k 1 α D ξ α U k , , k n α D ξ α U 1 , , k n α D ξ α U k , c 2 α D ξ 2 α U 1 , , c 2 α D ξ 2 α U k , k 1 2 α D ξ 2 α U 1 , ) = 0 .
(10)
Step 2. Suppose that the solution of (10) can be expressed by a polynomial in ( D ξ α G G ) as follows:
U j ( ξ ) = i = 0 m j a j , i ( D ξ α G G ) i , j = 1 , 2 , , k ,
(11)

where G = G ( ξ ) satisfies Eq. (1), and a j , i , i = 0 , 1 , , m , j = 1 , 2 , , k , are constants to be determined later with a j , m 0 . The positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (10).

Step 3. Substituting (11) into (10) and using (1), collecting all terms with the same order of ( D ξ α G G ) together, the left-hand side of (10) is converted into another polynomial in ( D ξ α G G ) . Equating each coefficient of this polynomial to zero yields a set of algebraic equations for a j , i , i = 0 , 1 , , m , j = 1 , 2 , , k .

Step 4. Solving the equation system in Step 3 and using (7), we can construct a variety of exact solutions for Eq. (8).

Remark 1 If we set α = 1 in Eq. (1), then it becomes G ( ξ ) + λ G ( ξ ) + μ G ( ξ ) = 0 , which is the foundation of the known (G′/G) method for solving partial differential equations (PDEs). So, in this way, the described fractional sub-equation method above is the extension of the (G′/G) method to fractional case.

Remark 2 The idea of the transformation from n independent variables to one independent variable denoted in Eq. (9) is similar to that in [[30], Eq. (12)], [[31], Eq. (8)], and [[33], Eq. (6)]. After applying this transformation to Eq. (8), by use of the second equality of Eq. (4), the original fractional partial differential equation can be transformed into another fractional ordinary differential equation in one independent variable.

4 Applications of the method

4.1 Space-time fractional Fokas equation

We consider the space-time fractional Fokas equation
4 2 α q t α x 1 α 4 α q x 1 3 α x 2 α + 4 α q x 2 3 α x 1 α + 12 α q x 1 α α q x 2 α + 12 q 2 α q x 1 α x 2 α 6 2 α q y 1 α y 2 α = 0 , 0 < α 1 .
(12)

In [30], the authors solved Eq. (12) by a fractional Riccati sub-equation method and obtained some exact solutions for it. Now we will apply the method described in Section 3 to Eq. (12).

Suppose that q ( x , y , t ) = U ( ξ ) , where ξ = c t + k 1 x 1 + k 2 x 2 + l 1 y 1 + l 2 y 2 + ξ 0 , k 1 , k 2 , l 1 , l 2 , c, ξ 0 are all constants with k 1 , k 2 , l 1 , l 2 , c 0 . Then by use of the second equality in Eq. (4), Eq. (12) can be turned into
4 c α k 1 α D ξ 2 α U k 1 3 α k 2 α D ξ 4 α U + k 2 3 α k 1 α D ξ 4 α U + 12 k 1 α k 2 α ( D ξ α U ) 2 + 12 k 1 α k 2 α U D ξ 2 α U 6 l 1 α l 2 α D ξ 2 α U = 0 .
(13)
Suppose that the solution of Eq. (13) can be expressed by
U ( ξ ) = i = 0 m a i ( D ξ α G G ) i ,
(14)
where G = G ( ξ ) satisfies Eq. (1). By balancing the order between the highest order derivative term and nonlinear term in Eq. (13), we can obtain m = 2 . So, we have
U ( ξ ) = a 0 + a 1 ( D ξ α G G ) + a 2 ( D ξ α G G ) 2 .
(15)
Substituting (15) into (13) and collecting all the terms with the same power of ( D ξ α G G ) together, equating each coefficient to zero, yield a set of algebraic equations. Solving these equations yields
a 0 = k 1 3 α k 2 α λ 2 k 1 α k 2 3 α λ 2 + 8 k 1 3 α k 2 α μ 8 k 1 α k 2 3 α μ 4 c α k 1 α + 6 l 1 α l 2 α 12 k 1 α k 2 α , a 1 = λ ( k 1 2 α k 2 2 α ) , a 2 = k 1 2 α k 2 2 α .

Substituting the result above into Eq. (15) and combining with (7), we can obtain the following exact solutions to Eq. (12).

When λ 2 4 μ > 0 ,
q 1 ( t , x 1 , x 2 , y 1 , y 2 ) = k 1 3 α k 2 α λ 2 k 1 α k 2 3 α λ 2 + 8 k 1 3 α k 2 α μ 8 k 1 α k 2 3 α μ 4 c α k 1 α + 6 l 1 α l 2 α 12 k 1 α k 2 α + λ ( k 1 2 α k 2 2 α ) [ λ 2 + λ 2 4 μ 2 ( C 1 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α C 1 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α ) ] + ( k 1 2 α k 2 2 α ) [ λ 2 + λ 2 4 μ 2 ( C 1 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α C 1 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α ) ] 2 ,
(16)

where ξ = c t + k 1 x 1 + k 2 x 2 + l 1 y 1 + l 2 y 2 + ξ 0 .

When λ 2 4 μ < 0 ,
q 2 ( t , x 1 , x 2 , y 1 , y 2 ) = k 1 3 α k 2 α λ 2 k 1 α k 2 3 α λ 2 + 8 k 1 3 α k 2 α μ 8 k 1 α k 2 3 α μ 4 c α k 1 α + 6 l 1 α l 2 α 12 k 1 α k 2 α + λ ( k 1 2 α k 2 2 α ) [ λ 2 + 4 μ λ 2 2 ( C 1 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α C 1 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α ) ] + ( k 1 2 α k 2 2 α ) [ λ 2 + 4 μ λ 2 2 ( C 1 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α C 1 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α ) ] 2 ,
(17)

where ξ = c t + k 1 x 1 + k 2 x 2 + l 1 y 1 + l 2 y 2 + ξ 0 .

When λ 2 4 μ = 0 ,
q 3 ( t , x 1 , x 2 , y 1 , y 2 ) = k 1 3 α k 2 α λ 2 k 1 α k 2 3 α λ 2 + 8 k 1 3 α k 2 α μ 8 k 1 α k 2 3 α μ 4 c α k 1 α + 6 l 1 α l 2 α 12 k 1 α k 2 α + λ ( k 1 2 α k 2 2 α ) [ λ 2 + C 2 Γ ( 1 + α ) C 1 Γ ( 1 + α ) + C 2 ξ α ] + ( k 1 2 α k 2 2 α ) [ λ 2 + C 2 Γ ( 1 + α ) C 1 Γ ( 1 + α ) + C 2 ξ α ] 2 ,
(18)

where ξ = c t + k 1 x 1 + k 2 x 2 + l 1 y 1 + l 2 y 2 + ξ 0 .

Remark 3 As one can see, the established solutions for the space-time fractional Fokas equation above are different from the results in [30] and are new exact solutions so far to our best knowledge.

Remark 4 The method used above in solving Eq. (12) can also be used to obtain solutions to initial or boundary value problems. For example, if we add the initial value condition
q ( 0 , x 1 , x 2 , y 1 , y 2 ) = tan 2 ( 2 x 1 + x 2 + y 1 + y 2 Γ ( 1.5 ) ) ,
then, after substituting this initial value condition to the trigonometric solutions obtained in Eq. (17) and fulfilling some basic comparison and computation, one can see that α = 0.5 , k 1 = 2 , k 2 = 1 , l 1 = l 2 = 1 , ξ 0 = 0 , C 2 = 0 , λ = 0 , μ = 1 , c = 41 8 + 3 2 . So, we obtain the solution to the initial value problem as
q ( t , x 1 , x 2 , y 1 , y 2 ) = tan 2 ( 2 x 1 + x 2 + y 1 + y 2 + ( 41 8 + 3 2 ) t Γ ( 1.5 ) ) .

4.2 Space-time fractional ( 2 + 1 ) -dimensional dispersive long wave equations

We consider the following space-time fractional ( 2 + 1 ) -dimensional dispersive long wave equations
{ 2 α u y α t α + 2 α v x 2 α + α ( u α u x α ) y α = 0 , α v t α + α u x α + α ( u v ) x α + 3 α u x 2 α y α = 0 , 0 < α 1 ,
(19)
which are the known ( 2 + 1 ) -dimensional dispersive long wave equations [4053]:
{ u y t + v x x + ( u u x ) y = 0 , v t + u x + ( u v ) x + u x x y = 0 .
(20)

Some types of exact solutions for Eqs. (20) have been obtained in [4053] by use of various methods including the Riccati sub-equation method [40, 41, 46], the nonlinear transformation method [42], the Jacobi function method [44, 45, 53], the (G′/G)-expansion method [43], the modified CK’s direct method [47], the EXP-function method [48], the Hopf-Cole transformation method [49], the modified extended Fan’s sub-equation method [50, 51], the generalized algebraic method [52]. But we notice that so far no research has been pursued for Eqs. (19). In the following, we will apply the proposed fractional sub-equation method to Eqs. (19).

To begin with, we suppose u ( x , y , t ) = U ( ξ ) , v ( x , y , t ) = V ( ξ ) , where ξ = k 1 x + k 2 y + c t + ξ 0 , k 1 , k 2 , c, ξ 0 are all constants with k 1 , k 2 , c 0 . Then by use of the second equality in Eq. (4), Eqs. (19) can be turned into
{ c α k 2 α D ξ 2 α U + k 1 2 α D ξ 2 α V + k 1 α k 2 α ( U D ξ 2 α U + ( D ξ α U ) 2 ) = 0 , c α D ξ α V + k 1 α D ξ α U + k 1 α D ξ α ( U V ) + k 1 2 α k 2 α D ξ 3 α U = 0 .
(21)
Suppose that the solution of Eqs. (21) can be expressed by
{ U ( ξ ) = i = 0 m 1 a i ( D ξ α G G ) i , V ( ξ ) = i = 0 m 2 b i ( D ξ α G G ) i .
(22)
Balancing the order of D ξ 2 α V and U D ξ 2 α U , D ξ α ( U V ) and D ξ 3 α U in (21), we can obtain m 1 = 1 , m 2 = 2 . So, we have
{ U ( ξ ) = a 0 + a 1 ( D ξ α G G ) , V ( ξ ) = b 0 + b 1 ( D ξ α G G ) + b 2 ( D ξ α G G ) 2 .
(23)
Substituting (23) into (21), using Eq. (1) and collecting all the terms with the same power of ( D ξ α G G ) together, equating each coefficient to zero, yield a set of algebraic equations. Solving these equations yields
a 0 = ± k 1 2 α λ c α k 1 α , a 1 = ± 2 k 1 α , b 0 = 2 k 1 α k 2 α μ 1 , b 1 = 2 k 1 α k 2 α λ , b 2 = 2 k 1 α k 2 α .

Substituting the result above into Eq. (23) and combining with (7), we can obtain the following exact solutions to Eqs. (19).

When λ 2 4 μ > 0 ,
{ u 1 ( x , y , t ) = ± k 1 2 α λ c α k 1 α ± 2 k 1 α [ λ 2 + λ 2 4 μ 2 ( C 1 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α C 1 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α ) ] u 1 ( x , y , t ) = 3 2 k 1 2 α [ λ 2 + λ 2 4 μ 2 ( C 1 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α C 1 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α ) ] 2 , v 1 ( x , y , t ) = 2 k 1 α k 2 α μ 1 v 1 ( x , y , t ) = 2 k 1 α k 2 α λ [ λ 2 + λ 2 4 μ 2 ( C 1 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α C 1 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α ) ] v 1 ( x , y , t ) = 2 k 1 α k 2 α [ λ 2 + λ 2 4 μ 2 ( C 1 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α C 1 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α ) ] 2 ,
(24)

where ξ = k 1 x + k 2 y + c t + ξ 0 .

When λ 2 4 μ < 0 ,
{ u 2 ( x , y , t ) = ± k 1 2 α λ c α k 1 α ± 2 k 1 α [ λ 2 + 4 μ λ 2 2 ( C 1 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α C 1 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α ) ] u 2 ( x , y , t ) = 3 2 k 1 2 α [ λ 2 + 4 μ λ 2 2 ( C 1 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α C 1 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α ) ] 2 , v 2 ( x , y , t ) = 2 k 1 α k 2 α μ 1 v 2 ( x , y , t ) = 2 k 1 α k 2 α λ [ λ 2 + 4 μ λ 2 2 ( C 1 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α C 1 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α ) ] v 2 ( x , y , t ) = 2 k 1 α k 2 α [ λ 2 + 4 μ λ 2 2 ( C 1 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α C 1 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α ) ] 2 ,
(25)

where ξ = k 1 x + k 2 y + c t + ξ 0 .

When λ 2 4 μ = 0 ,
{ u 3 ( x , y , t ) = ± k 1 2 α λ c α k 1 α ± 2 k 1 α [ λ 2 + C 2 Γ ( 1 + α ) C 1 Γ ( 1 + α ) + C 2 ξ α ] 3 2 k 1 2 α [ λ 2 + C 2 Γ ( 1 + α ) C 1 Γ ( 1 + α ) + C 2 ξ α ] 2 , v 3 ( x , y , t ) = 2 k 1 α k 2 α μ 1 2 k 1 α k 2 α λ [ λ 2 + C 2 Γ ( 1 + α ) C 1 Γ ( 1 + α ) + C 2 ξ α ] v 3 ( x , y , t ) = 2 k 1 α k 2 α [ λ 2 + C 2 Γ ( 1 + α ) C 1 Γ ( 1 + α ) + C 2 ξ α ] 2 ,
(26)

where ξ = k 1 x + k 2 y + c t + ξ 0 .

Remark 5 The established solutions in Eqs. (24)-(26) are new exact solutions for the space-time fractional ( 2 + 1 ) -dimensional dispersive long wave equations.

4.3 Space-time fractional fifth-order Sawada-Kotera equation

We consider the space-time fractional fifth-order Sawada-Kotera equation
D t α u + D x 5 α u + 45 u 2 D x α u + 15 ( D x α u D x 2 α u + u D x 3 α u ) = 0 , 0 < α 1 ,
(27)
which is the variation of the fifth-order Sawada-Kotera equation [54]
u t + u x x x x x + 45 u x u 2 + 15 ( u x u x x + u u x x x ) = 0 .
(28)
Now we apply the proposed method to Eq. (27). To begin with, we suppose u ( x , t ) = U ( ξ ) , where ξ = k x + c t + ξ 0 , k, c, ξ 0 are all constants with k , c 0 . Then, by use of the second equality in Eq. (4), Eq. (27) can be turned into
c α D ξ α U + k 5 α D ξ 5 α U + 45 k α U D ξ α U + 15 k 3 α ( D ξ α U D ξ 2 α U + U D ξ 3 α U ) = 0 .
(29)
Suppose that the solution of Eq. (29) can be expressed by
U ( ξ ) = i = 0 m a i ( D ξ α G G ) i .
(30)
Balancing the order of U ( 5 ) and U D ξ 3 α U in Eq. (30), we have m = 2 . So,
U ( ξ ) = a 0 + a 1 ( D ξ α G G ) + a 2 ( D ξ α G G ) 2 .
(31)
Substituting (31) into (29), using Eq. (1) and collecting all the terms with the same power of ( D ξ α G G ) together, equating each coefficient to zero, yield a set of algebraic equations. Solving these equations yields
a 0 = 40 k 3 α μ + 5 k 3 α λ 2 ± 80 k 6 α μ 2 40 k 6 α μ λ 2 + 5 k 6 α λ 4 20 k α c α 30 k α , a 1 = 2 k 2 α λ , a 2 = 2 k 2 α .

Substituting the result above into Eq. (31) and combining with (7), we can obtain corresponding exact solutions to Eq. (27).

When λ 2 4 μ > 0 ,
u 1 ( x , t ) = 40 k 3 α μ + 5 k 3 α λ 2 ± 80 k 6 α μ 2 40 k 6 α μ λ 2 + 5 k 6 α λ 4 20 k α c α 30 k α 2 k 2 α λ [ λ 2 + λ 2 4 μ 2 ( C 1 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α C 1 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α ) ] 2 k 2 α [ λ 2 + λ 2 4 μ 2 ( C 1 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α C 1 cosh λ 2 4 μ 2 Γ ( 1 + α ) ξ α + C 2 sinh λ 2 4 μ 2 Γ ( 1 + α ) ξ α ) ] 2 ,
(32)

where ξ = k x + c t + ξ 0 .

When λ 2 4 μ < 0 ,
u 2 ( x , t ) = 40 k 3 α μ + 5 k 3 α λ 2 ± 80 k 6 α μ 2 40 k 6 α μ λ 2 + 5 k 6 α λ 4 20 k α c α 30 k α 2 k 2 α λ [ λ 2 + 4 μ λ 2 2 ( C 1 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α C 1 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α ) ] 2 k 2 α [ λ 2 + 4 μ λ 2 2 ( C 1 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α C 1 cos 4 μ λ 2 2 Γ ( 1 + α ) ξ α + C 2 sin 4 μ λ 2 2 Γ ( 1 + α ) ξ α ) ] 2 ,
(33)

where ξ = k x + c t + ξ 0 .

When λ 2 4 μ = 0 ,
u 3 ( x , t ) = 40 k 3 α μ + 5 k 3 α λ 2 ± 80 k 6 α μ 2 40 k 6 α μ λ 2 + 5 k 6 α λ 4 20 k α c α 30 k α 2 k 2 α λ [ λ 2 + C 2 Γ ( 1 + α ) C 1 Γ ( 1 + α ) + C 2 ξ α ] 2 k 2 α [ λ 2 + C 2 Γ ( 1 + α ) C 1 Γ ( 1 + α ) + C 2 ξ α ] 2 ,
(34)

where ξ = k x + c t + ξ 0 .

Remark 6 The established solutions in Eqs. (32)-(34) are new exact solutions for the space-time fractional fifth-order Sawada-Kotera equation.

5 Conclusions

We have proposed a new fractional sub-equation method for solving FPDEs successfully, which is the fractional version of the known (G′/G) method. As one can see, the two nonlinear fractional complex transformations for ξ and η used here are very important. The first transformation ensures that a certain fractional partial differential equation can be turned into another fractional ordinary differential equation, the solutions of which can be expressed by a polynomial in ( D ξ α G G ) , where G satisfies the fractional ODE D ξ 2 α G ( ξ ) + λ D ξ α G ( ξ ) + μ G ( ξ ) = 0 . The general expression for ( D ξ α G G ) related to this fractional ODE can be obtained due to the second fractional complex transformations for η. Finally, we note that with this kind of nonlinear fractional complex transformations, it is worth to investigate the applications of other algebraic methods to fractional partial differential equations such as the Exp-function method, F-expansion method, Jacobi elliptic function method and so on.

Declarations

Acknowledgements

The authors would like to thank the reviewers very much for their valuable suggestions on the paper.

Authors’ Affiliations

(1)
School of Science, Shandong University of Technology, Zibo, Shandong, 255049, China

References

  1. Saadatmandi A, Dehghan AM: A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 2010, 59: 1326-1336. 10.1016/j.camwa.2009.07.006MathSciNetView ArticleGoogle Scholar
  2. Zhou Y, Jiao F, Li J: Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Anal. 2009, 71: 2724-2733. 10.1016/j.na.2009.01.105MathSciNetView ArticleGoogle Scholar
  3. Galeone L, Garrappa R: Explicit methods for fractional differential equations and their stability properties. J. Comput. Appl. Math. 2009, 228: 548-560. 10.1016/j.cam.2008.03.025MathSciNetView ArticleGoogle Scholar
  4. Trigeassou JC, Maamri N, Sabatier J, Oustaloup A: A Lyapunov approach to the stability of fractional differential equations. Signal Process. 2011, 91: 437-445. 10.1016/j.sigpro.2010.04.024View ArticleGoogle Scholar
  5. Deng W: Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal. 2010, 72: 1768-1777. 10.1016/j.na.2009.09.018MathSciNetView ArticleGoogle Scholar
  6. Ghoreishi F, Yazdani S: An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis. Comput. Math. Appl. 2011, 61: 30-43. 10.1016/j.camwa.2010.10.027MathSciNetView ArticleGoogle Scholar
  7. Edwards JT, Ford NJ, Simpson AC: The numerical solution of linear multi-term fractional differential equations: systems of equations. J. Comput. Appl. Math. 2002, 148: 401-418. 10.1016/S0377-0427(02)00558-7MathSciNetView ArticleGoogle Scholar
  8. Muslim M: Existence and approximation of solutions to fractional differential equations. Math. Comput. Model. 2009, 49: 1164-1172. 10.1016/j.mcm.2008.07.013MathSciNetView ArticleGoogle Scholar
  9. Jafari H, Tajadodi H, Baleanu D, Al-Zahrani AA, Alhamed YA, Zahid AH: Fractional sub-equation method for the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation. Cent. Eur. J. Phys. 2013. 10.2478/s11534-013-0203-7Google Scholar
  10. Baleanu D, Rezapour S, Mohammadi H: Some existence results on nonlinear fractional differential equations. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 2013., 371(1990): Article ID 20120144Google Scholar
  11. Nyamoradi N, Baleanu D, Agarwal RP: On a multipoint boundary value problem for a fractional order differential inclusion on an infinite interval. Adv. Math. Phys. 2013., 2013: Article ID 823961Google Scholar
  12. El-Sayed AMA, Gaber M: The Adomian decomposition method for solving partial differential equations of fractal order in finite domains. Phys. Lett. A 2006, 359: 175-182. 10.1016/j.physleta.2006.06.024MathSciNetView ArticleGoogle Scholar
  13. El-Sayed AMA, Behiry SH, Raslan WE: Adomian’s decomposition method for solving an intermediate fractional advection-dispersion equation. Comput. Math. Appl. 2010, 59: 1759-1765. 10.1016/j.camwa.2009.08.065MathSciNetView ArticleGoogle Scholar
  14. Daftardar-Gejji V, Jafari H: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 2005, 301(2):508-518. 10.1016/j.jmaa.2004.07.039MathSciNetView ArticleGoogle Scholar
  15. He JH: A new approach to nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 1997, 2: 230-235. 10.1016/S1007-5704(97)90007-1View ArticleGoogle Scholar
  16. Wu G, Lee EWM: Fractional variational iteration method and its application. Phys. Lett. A 2010, 374: 2506-2509. 10.1016/j.physleta.2010.04.034MathSciNetView ArticleGoogle Scholar
  17. Guo S, Mei L: The fractional variational iteration method using He’s polynomials. Phys. Lett. A 2011, 375: 309-313. 10.1016/j.physleta.2010.11.047MathSciNetView ArticleGoogle Scholar
  18. Wu GC, Baleanu D: Variational iteration method for the Burgers’ flow with fractional derivatives - new Lagrange multipliers. Appl. Math. Model. 2013, 37: 6183-6190. 10.1016/j.apm.2012.12.018MathSciNetView ArticleGoogle Scholar
  19. Jafari H, Tajadodi H, Baleanu D: A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials. Fract. Calc. Appl. Anal. 2013, 16(1):109-122.MathSciNetView ArticleGoogle Scholar
  20. Jafari H, Kadem A, Baleanu D, Yilmaz T: Solutions of the fractional Davey-Stewartson equations with variational iteration method. Rom. Rep. Phys. 2012, 64(2):337-346.Google Scholar
  21. Jafari H, Khalique CM: Analytical solutions of nonlinear fractional differential equations using variational iteration method. J. Nonlinear Syst. Appl. 2011, 2(3-4):148-151.Google Scholar
  22. Jafari H, Tajadodi H: He’s variational iteration method for solving fractional Riccati differential equation. Int. J. Differ. Equ. 2010., 2010: Article ID 764738Google Scholar
  23. He JH: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 1999, 178: 257-262. 10.1016/S0045-7825(99)00018-3View ArticleGoogle Scholar
  24. He JH: A coupling method of homotopy technique and a perturbation technique for non-linear problems. Int. J. Non-Linear Mech. 2000, 35: 37-43. 10.1016/S0020-7462(98)00085-7View ArticleGoogle Scholar
  25. Ganji ZZ, Ganji DD, Jafari H, Rostamian M: Application of the homotopy perturbation method for coupled system of partial differential equations with time fractional derivatives. Topol. Methods Nonlinear Anal. 2008, 31(2):341-348.MathSciNetGoogle Scholar
  26. Jafari H, Momani Sh: Solving fractional diffusion and wave equations by modified homotopy perturbation method. Phys. Lett. A 2007, 370: 388-396. 10.1016/j.physleta.2007.05.118MathSciNetView ArticleGoogle Scholar
  27. Odibat Z, Momani S: Fractional Green function for linear time-fractional equations of fractional order. Appl. Math. Lett. 2008, 21: 194-199. 10.1016/j.aml.2007.02.022MathSciNetView ArticleGoogle Scholar
  28. Cui M: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 2009, 228: 7792-7804. 10.1016/j.jcp.2009.07.021MathSciNetView ArticleGoogle Scholar
  29. Huang Q, Huang G, Zhan H: A finite element solution for the fractional advection-dispersion equation. Adv. Water Resour. 2008, 31: 1578-1589. 10.1016/j.advwatres.2008.07.002View ArticleGoogle Scholar
  30. Zhang S, Zhang HQ: Fractional sub-equation method and its applications to nonlinear fractional PDEs. Phys. Lett. A 2011, 375: 1069-1073. 10.1016/j.physleta.2011.01.029MathSciNetView ArticleGoogle Scholar
  31. Guo SM, Mei LQ, Li Y, Sun YF: The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics. Phys. Lett. A 2012, 376(2012):407-411.MathSciNetGoogle Scholar
  32. Lu B: Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations. Phys. Lett. A 2012, 376: 2045-2048. 10.1016/j.physleta.2012.05.013MathSciNetView ArticleGoogle Scholar
  33. Lu B: The first integral method for some time fractional differential equations. J. Math. Anal. Appl. 2012, 395: 684-693. 10.1016/j.jmaa.2012.05.066MathSciNetView ArticleGoogle Scholar
  34. Zheng B: (G′/G)-Expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun. Theor. Phys. 2012, 58: 623-630. 10.1088/0253-6102/58/5/02View ArticleGoogle Scholar
  35. Jumarie G: Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 2006, 51: 1367-1376. 10.1016/j.camwa.2006.02.001MathSciNetView ArticleGoogle Scholar
  36. Wang ML, Li XZ, Zhang JL: The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 2008, 372: 417-423. 10.1016/j.physleta.2007.07.051MathSciNetView ArticleGoogle Scholar
  37. Wang ML, Zhang JL, Li XZ: Application of the (G′/G)-expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations. Appl. Math. Comput. 2008, 206: 321-326. 10.1016/j.amc.2008.08.045MathSciNetView ArticleGoogle Scholar
  38. Aslan I: Discrete exact solutions to some nonlinear differential-difference equations via the (G′/G)-expansion method. Appl. Math. Comput. 2009, 215: 3140-3147. 10.1016/j.amc.2009.09.056MathSciNetView ArticleGoogle Scholar
  39. Ayhan B, Bekir A: The (G′/G)-expansion method for the nonlinear lattice equations. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 3490-3498. 10.1016/j.cnsns.2012.01.009MathSciNetView ArticleGoogle Scholar
  40. Kong CC, Wang D, Song LN, Zhang HQ:New exact solutions to MKDV-Burgers equation and ( 2 + 1 ) -dimensional dispersive long wave equation via extended Riccati equation method. Chaos Solitons Fractals 2009, 39: 697-706. 10.1016/j.chaos.2007.01.083MathSciNetView ArticleGoogle Scholar
  41. Yan ZY:Generalized transformations and abundant new families of exact solutions for ( 2 + 1 ) -dimensional dispersive long wave equations. Comput. Math. Appl. 2003, 46: 1363-1372. 10.1016/S0898-1221(03)90225-2MathSciNetView ArticleGoogle Scholar
  42. Zhang JF, Han P:New multisoliton solutions of the ( 2 + 1 ) -dimensional dispersive long wave equations. Commun. Nonlinear Sci. Numer. Simul. 2001, 6: 178-182. 10.1016/S1007-5704(01)90002-4MathSciNetView ArticleGoogle Scholar
  43. Eslami M, Neyrame A, Ebrahimi M:Explicit solutions of nonlinear ( 2 + 1 ) -dimensional dispersive long wave equation. J. King Saud Univ, Comput. Inf. Sci. 2012, 24: 69-71. 10.1016/j.jksus.2010.08.003View ArticleGoogle Scholar
  44. Chen Y, Wang Q:A new general algebraic method with symbolic computation to construct new doubly-periodic solutions of the ( 2 + 1 ) -dimensional dispersive long wave equation. Appl. Math. Comput. 2005, 167: 919-929. 10.1016/j.amc.2004.06.119MathSciNetView ArticleGoogle Scholar
  45. Wang Q, Chen Y, Zhang HQ:An extended Jacobi elliptic function rational expansion method and its application to ( 2 + 1 ) -dimensional dispersive long wave equation. Phys. Lett. A 2005, 340: 411-426. 10.1016/j.physleta.2005.04.034MathSciNetView ArticleGoogle Scholar
  46. Chen Y, Li B:Symbolic computation and construction of soliton-like solutions to the ( 2 + 1 ) -dimensional dispersive long-wave equations. Int. J. Eng. Sci. 2004, 42: 715-724. 10.1016/j.ijengsci.2003.06.002View ArticleGoogle Scholar
  47. Liu N, Liu XQ, Lu HL:New exact solutions and conservation laws of the ( 2 + 1 ) -dimensional dispersive long wave equations. Phys. Lett. A 2009, 373: 214-220. 10.1016/j.physleta.2008.11.007View ArticleGoogle Scholar
  48. Zhang S, Tong JL, Wang W: Exp-function method for a nonlinear ordinary differential equation and new exact solutions of the dispersive long wave equations. Comput. Math. Appl. 2009, 58: 2294-2299. 10.1016/j.camwa.2009.03.020MathSciNetView ArticleGoogle Scholar
  49. Zhou YQ, Liu Q, Zhang J, Zhang WN:Exact solution for ( 2 + 1 ) -dimension nonlinear dispersive long wave equation. Appl. Math. Comput. 2006, 177: 495-499. 10.1016/j.amc.2005.11.026MathSciNetView ArticleGoogle Scholar
  50. Yomba E:The modified extended Fan’s sub-equation method and its application to ( 2 + 1 ) -dimensional dispersive long wave equation. Chaos Solitons Fractals 2005, 26: 785-794. 10.1016/j.chaos.2005.01.061MathSciNetView ArticleGoogle Scholar
  51. Chen Y, Wang Q:A series of new soliton-like solutions and double-like periodic solutions of a ( 2 + 1 ) -dimensional dispersive long wave equation. Chaos Solitons Fractals 2005, 23: 801-807. 10.1016/j.chaos.2004.05.024MathSciNetView ArticleGoogle Scholar
  52. Zeng X, Zeng J:Symbolic computation and new families of exact solutions to the ( 2 + 1 ) -dimensional dispersive long-wave equations. Chaos Solitons Fractals 2006, 29: 1115-1120. 10.1016/j.chaos.2005.08.069MathSciNetView ArticleGoogle Scholar
  53. Zhang S:The periodic wave solutions for the ( 2 + 1 ) -dimensional dispersive long water equations. Chaos Solitons Fractals 2007, 32: 847-854. 10.1016/j.chaos.2005.11.088MathSciNetView ArticleGoogle Scholar
  54. Liu CF, Dai ZD: Exact soliton solutions for the fifth-order Sawada-Kotera equation. Appl. Math. Comput. 2008, 206: 272-275. 10.1016/j.amc.2008.08.028MathSciNetView ArticleGoogle Scholar

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© Zheng and Wen; licensee Springer 2013

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