Indian J Phys DOI 10.1007/s12648-013-0407-0

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Exact solutions of nonlinear partial fractional differential equations using fractional sub-equation method K A Gepreel1,2* and A A Al-Thobaiti2 1

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

2

Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia Received: 30 July 2013 / Accepted: 01 October 2013

Abstract: In this article, we use fractional sub-equation method to find the exact solutions for some nonlinear partial fractional differential equations, namely space–time fractional coupled Sakharov–Kuznetsov (Z–K) equations, space–time fractional nonlinear coupled Korteweg de Vries (KdV) equations and space–time fractional Hirota–Satsuma KdV system. As a result, three families of exact analytical solutions are obtained. Proposed method is more effective and powerful to construct the exact solutions for nonlinear partial fractional differential equations. Also, we use fractional complex transform and exp function method to find some of the exact solutions for nonlinear partial fractional differential equations. Keywords:

Fractional sub-equation method; Exact solutions; Fractional derivatives

PACS No.:

02.30.Jr

1. Introduction Fractional differential equations (FDEs) have been at focus of many studies due to their frequent appearance in various applications. Several fields of application to fractional differentiation and integration are already well established while 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]. Numerical and analytical approximate methods have included Adomian decomposition method (ADM) [12, 13], variational iteration method (VIM) [14], homotopy perturbation method [15–17] and homotopy analysis method [18]. There are many methods for finding the exact solutions for nonlinear differential equations [19, 20]. Fractional differential equations have caught much attention recently due to the exact description of nonlinear phenomena. No analytical method has been available before 1998 for such equations even for linear fractional differential equations. Fractional complex transform and the chain rule for fractional calculus

are elucidated geometrically [21]. He et al. [21] have shown that FDEs can best describe discontinuous media and the fractional order is equivalent to its fractional dimensions. The fractional complex transform [22–24] is the simplest approach; it is to convert the FDEs into ordinary differential equations and we can use the direct method such as expfunction method and sub-equation method to solve the obtained ordinary differential equations directly. More recently, Zhang et al. [25] have introduced a new method called fractional sub-equation method to look for the traveling wave solutions of the nonlinear partial fractional differential equations. In this article, we have used the fractional subequation method [25, 26] to obtain the exact solutions of some nonlinear fractional PDEs, namely the space–time fractional coupled Sakharov–Kuznetsov (Z–K) equations, the space– time fractional nonlinear coupled KdV equations and the space–time fractional Hirota–Satsuma KdV system. Also, we have used the fractional complex transform and the exp function method to find some exact solutions for nonlinear partial fractional differential equations.

2. Preliminaries and notation

*Corresponding author, E-mail: [email protected]

In this section, we give some basic definitions of fractional calculus theory which are to be used in this work. Ó 2013 IACS

K A Gepreel and A A Al-Thobaiti

Local fractional derivative of f ðxÞ order a in interval [a,b] is defined by [27, 28]  d a f  Da ðf ðxÞ  f ðx0 Þ a D f ðx0 Þ ¼ a  ¼ lim ; ð1Þ x!x0 dx ðx  x Þa 0

x¼x0

where Da ðf ðxÞ  f ðx0 ÞÞ ¼ Cða þ 1ÞDðf ðxÞ  f ðx0 ÞÞ Also the inverse of local fractional derivative to f ðxÞ of order a in interval [a,b] is defined by [28, 29] a a Ib f ðxÞ

1 ¼ Cð1 þ aÞ

Zb

f ðtÞðdtÞa

a

ð2Þ

N1 X 1 lim ¼ f ðtj ÞðDtj Þa ; Cð1 þ aÞ Dt!0 j¼0

where Dtj ¼ tjþ1  tj ; Dt ¼ maxfDt1 ; Dt2 ; . . .g; j ¼ 0; 1; ; . . .N  1; t0 ¼ a; tN ¼ b are the partition of the interval [a,b].

We outline the main steps of this method as follows [24, 26]: We consider the following nonlinear partial fractional differential equations

Step 1

ð3Þ

We use the travelling wave transformation

uðx; tÞ ¼ uðnÞ;

n ¼ x þ ct;

ð4Þ

where c is an arbitrary constant to be determined later. The transformation given by Eq. (4) permits us to convert the partial fractional differential Eq. (3) to the fractional ODE in the following form Pðu; Dan u; D2a n u; . . .Þ ¼ 0;

uðnÞ ¼

Step 6 Zhang et al. [29] have derived some exact solutions of fractional Riccati equation given by Eq. (7) in the following form pffiffiffiffiffiffiffi 8 pffiffiffiffiffiffiffi pr tanh ½pr > ffiffiffiffiffiffi ffi ffiffiffiffiffiffiffi n; a; r\0; > > >  r coth ½ < pffiffiffi pffiffiffi r n; a; r\0; r tan ½ r n; r [ 0; /ðnÞ ¼ pffiffiffi p ffiffiffi a ; > >  r cot ½ r n; a  ; r [ 0; > > : Cð1þaÞ  na þx ; x is a const:; r ¼ 0; ð8Þ where the generalized hyperbolic and trigonometric functions are defined as Ea ðina Þ  Ea ðina Þ ; 2i a a Ea ðin Þ þ Ea ðin Þ cosðn; aÞ ¼ ; 2i sinðn; aÞ cosðn; aÞ tanðn; aÞ ¼ ; cotðn; aÞ ¼ ; cosðn; aÞ sinðn; aÞ ð9Þ Ea ðna Þ  Ea ðna Þ sinhðn; aÞ ¼ ; 2 Ea ðna Þ þ Ea ðna Þ coshðn; aÞ ¼ ; 2 sinhðn; aÞ coshðn; aÞ tanhðn; aÞ ¼ ; cothðn; aÞ ¼ ; coshðn; aÞ sinhðn; aÞ P nk where Ea ðnÞ ¼ 1 is Mittag–Leffler k¼0 Cð1þkaÞ; a [ 0 function.

ð5Þ

where P is a polynomial in u and its total derivatives with respect to n Step 2

Step 5 We solve the over-determined system of algebraic equations to determine ai ði ¼ 0; 1; 2; . . .Þ and c.

sinðn; aÞ ¼

3. Summary of fractional sub-equation method

Uðu; ux ; ut ; Dat u; Dax u; . . .Þ ¼ 0:

coefficients of ½/ðnÞi ði ¼ 0; 1; 2; . . .Þ to be zero, we get an over-determined system of algebraic equations with respect to ai ði ¼ 0; 1; 2; . . .Þ and c.

4. Fractional sub-equation method for nonlinear fractional PDEs

We suppose that Eq. (5) has the following solution N X

ai / i ;

ð6Þ

i¼N

where ai are arbitrary constants to be determined later, while /ðnÞ satisfies the following fractional Riccati equation: Dan /ðnÞ ¼ r þ /ðnÞ2 ;

ð7Þ

where r is an arbitrary constant. Step 3 We determine positive integer N of formal polynomial solution given in Eq. (6) by balancing nonlinear terms and highest-order fractional derivative in Eq. (5). Step 4 Substituting Eqs. (6) and (7) into Eq. (5) and collecting the coefficients of /ðnÞ and setting the

We use fractional sub-equation method to construct the exact solutions for some nonlinear partial fractional differential equations, namely space–time fractional coupled Sakharov–Kuznetsov (Z–K) equations, space–time fractional nonlinear coupled Korteweg de Vries (KdV) equations and space–time fractional Hirota–Satsuma KdV system, which have played an important role on the mathematical physics. 4.1. Example I We construct exact solutions for space– time fractional coupled Sakharov–Kuznetsov (S–K) equations in the following form

Fractional sub-equation method a 2a Dat u þ D3a x u þ Dx Dy u

 6u Dax u  Dax v ¼ 0; a 2a a Dat v þ k1 D3a x v þ k2 Dx Dy v þ k3 Dx v



6k4 v Dax v



k5 Dax u

k2 þ 2k4 ca þ 32k4 r  k1 ; 12 k4 2 a2 ¼ 4; a4 ¼ 4r ; a0 ¼

ð10Þ

k3 ðk2 þ k1 Þ þ 16k2 rk1  2k5 k4 þ ca ðk2 þ k1 Þ þ 8rðk22 þ k12 Þ ; 6 k4 ðk2 þ k1 Þ 2 ðk2 þ k1 Þ 2 r2 ðk2 þ k1 Þ b2 ¼ ; b4 ¼ ; k4 k4 a1 ¼ a3 ¼ b1 ¼ b3 ¼ 0:

b0 ¼

¼ 0;

which is a transformed generalization of coupled Sakharov– Kuznetsov equations [30]. Coupled Sakharov–Kuznetsov equation is the model describing two weakly interacting nonlinear waves in anisotropic background stratified fluid flows [30]. We use the travelling wave transformation, u ¼ uðnÞ; v ¼ vðnÞ; n ¼ x þ y þ c t: ð11Þ

Eqs. (13) and (15) and general solutions of Eq. (7) lead to the following exact solutions of space–time fractional (S–K) Eq. (10) given by:

The travelling wave transformations (11) leads to write Eq. (10) into the following form

Solitary wave solutions [31] for space–time fractional (S–K) equations in the following form

ð15Þ

a a ca Dan u þ 2 D3a n u  6 u Dn u  Dn v ¼ 0; a a a ca Dan v þ ð k1 þ k2 Þ D3a n v þ k3 Dn v  6 k4 v Dn v  k5 Dn u ¼ 0;

ð12Þ where c is an arbitrary constant. Balancing the highest order fractional derivatives and nonlinear terms in Eq. (12), we get the solutions of Eq. (12) have the following form uðnÞ ¼ a0 þ a1 /ðnÞ þ a2 /2 ðnÞ þ a3 /1 ðnÞ þ a4 /2 ðnÞ; vðnÞ ¼ b0 þ b1 /ðnÞ þ b2 /2 ðnÞ þ b3 /1 ðnÞ þ b4 /2 ðnÞ; ð13Þ where ai ; bi ði ¼ 0; 1; 2; 3; 4Þ are arbitrary constants and /ðnÞ satisfies the fractional Riccati Eq. (7). We substitute Eqs. (13) and (7) into Eq. (12) and set the coefficients of /n ðnÞ n ¼ 0; 1; 2; 3; . . . to be zero. We yield a set of over-determined system of algebraic equations with respect to ai ; bi ði ¼ 0; 1; 2; 3; 4Þ and c. On using Maple software package, we solve over-determined algebraic equations to get the following cases : Case (i) k2 þ2k4 ca þ32k4 rk1 ; 12k4 2r2 ðk2 þk1 Þ a4 ¼4r2 ; b4 ¼ ; k4 k3 ðk2 þk1 Þþ16k2 rk1 2k5 k4 þca ðk2 þk1 Þþ8rðk22 þk12 Þ b0 ¼ ; 6k4 ðk2 þk1 Þ a1 ¼a2 ¼a3 ¼b1 ¼b2 ¼b3 ¼0: a0 ¼

ð14Þ Case (ii)

u1 ðx; y; tÞ ¼

pffiffiffiffiffiffiffi  k2 þ 2k4 ca þ 32k4 r  k1  4r coth2 r n; a ; 12 k4

v1 ðx; y; tÞ k3 ðk2 þ k1 Þ þ 16k2 rk1  2k5 k4 þ ca ðk2 þ k1 Þ þ 8rðk22 þ k12 Þ 6 k4 ðk2 þ k1 Þ   p ffiffiffiffiffiffi ffi 2 ðk2 þ k1 Þ r  coth2 r n; a ; r\0: k4 ¼

ð16Þ a

k2 þ 2k4 c þ 32k4 r  k1 12 k4 pffiffiffiffiffiffiffi  2  4r tanh r n; a ; v2 ðx; y; tÞ u2 ðx; y; tÞ ¼

k3 ðk2 þ k1 Þ þ 16k2 rk1  2k5 k4 þ ca ðk2 þ k1 Þ þ 8rðk22 þ k12 Þ 6 k4 ðk2 þ k1 Þ   p ffiffiffiffiffiffi ffi 2 ðk2 þ k1 Þ r  tanh2 r n; a ; r\0: k4 ¼

ð17Þ Periodic trigonometric wave solutions [31] for space– time fractional (S–K) equations take the following form k2 þ 2k4 ca þ 32k4 r  k1 u3 ðx; y; tÞ ¼ 12 k4 pffiffiffi  þ 4r cot2 r n; a ; v3 ðx; y; tÞ k3 ðk2 þ k1 Þ þ 16k2 rk1  2k5 k4 þ ca ðk2 þ k1 Þ þ 8rðk22 þ k12 Þ 6 k4 ðk2 þ k1 Þ   p ffiffiffi 2 ðk2 þ k1 Þ r þ cot2 r n; a ; r [ 0: k4 ¼

ð18Þ

K A Gepreel and A A Al-Thobaiti

u4 ðx; y; tÞ ¼

pffiffiffi  k2 þ 2k4 ca þ 32k4 r  k1 þ 4r tan2 r n; a ; 12 k4

v4 ðx; y; tÞ ¼

k3 ðk2 þ k1 Þ þ 16k2 rk1  2k5 k4 þ ca ðk2 þ k1 Þ þ 8rðk22 þ k12 Þ 6 k4 ðk2 þ k1 Þ   p ffiffiffi 2 ðk2 þ k1 Þ r þ tan2 r n; a ; r [ 0: k4

ð19Þ where n ¼ x þ y þ ct. Also Eqs. (13) and (15) and the general solutions of fractional Riccati equation given by Eq. (7) lead to find the following exact solutions of Eq. (10): k2 þ 2k4 ca þ 32k4 r  k1 u5 ðx; y; tÞ ¼ 12 k4  pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi  2  4r tanh r n; a þ coth2 r n; a ; k3 ðk2 þ k1 Þ þ 16k2 rk1  2k5 k4 þ ca ðk2 þ k1 Þ þ 8rðk22 þ k12 Þ 6 k4 ðk2 þ k1 Þ    pffiffiffiffiffiffiffi  p ffiffiffiffiffiffiffi 2 ðk2 þ k1 Þ r tanh2 r n; a þ coth2 r n; a ; r\0:  k4 ¼

ð20Þ a

k2 þ 2k4 c þ 32k4 r  k1 u6 ðx; y; tÞ ¼ 12 k4  pffiffiffi  pffiffiffi  2 þ 4r tan r n; a þ cot2 r n; a ; v6 ðx; y; tÞ k3 ðk2 þ k1 Þ þ 16k2 rk1  2k5 k4 þ ca ðk2 þ k1 Þ þ 8rðk22 þ k12 Þ 6 k4 ðk2 þ k1 Þ   pffiffiffi  p ffiffiffi 2 ðk2 þ k1 Þ r  þ tan2 r n; a þ cot2 r n; a ; r [ 0: k4 ¼

ð21Þ u7 ðx; y; tÞ ¼

ð22Þ

n ¼ x þ c t:

ð24Þ

Travelling wave transformations given by Eq. (24) lead to write Eq. (23) into the following form a ca Dan u þ L1 Dan u þ L2 u Dan u þ L3 D3a n u þ L4 Dn v ¼ 0; a ca Dan v þ L5 Dan v þ L6 v Dan v þ L7 D3a n v þ L8 Dn u ¼ 0;

ð25Þ

where c is an arbitrary constant. Balancing highest order fractional derivatives and nonlinear terms in Eq. (25), we get the solutions of Eq. (25) which have the following form vðnÞ ¼ b0 þ b1 /ðnÞ þ b2 /2 ðnÞ þ b3 /1 ðnÞ þ b4 /2 ðnÞ;

ð26Þ

where ai ; bi ði ¼ 0; 1; 2; 3; 4Þ are arbitrary constants and /ðnÞ satisfies fractional Riccati equation given by Eq. (7). We substitute Eqs. (7) and (26) into Eq. (25) and set the coefficients of /n ðnÞn ¼ 0; 1; 2; 3; . . . to be zero. Then we yield a set of over-determined system of algebraic equations with respect to ai ; bi ði ¼ 0; 1; 2; 3; 4Þ and c. On using Maple software package, we solve over-determined algebraic equations to get the following cases : Case (i) 8 L23 L6 r þ L1 L3 L6 þ ca L3 L6 þ L4 L7 L2 ; L3 L6 L2 12L3 r2 a4 ¼  ; L2 a0 ¼ 

8 L27 L2 r þ L8 L3 L6 þ ca L7 L2 þ L5 L7 L2 ; L7 L6 L2 12L7 r2 b4 ¼  ; L6 a1 ¼ a2 ¼ a3 ¼ b1 ¼ b2 ¼ b3 ¼ 0:

ð27Þ

Case (ii) 8 L23 L6 r þ L1 L3 L6 þ ca L3 L6 þ L4 L7 L2 ; L3 L6 L2 12L3 12L3 r2 ¼ ; a4 ¼  ; L2 L2 8 L2 L2 r þ L8 L3 L6 þ ca L7 L2 þ L5 L7 L2 ¼ 7 ; L7 L6 L2 12L7 12L7 r2 ¼ ; b4 ¼  ; L6 L6 ¼ a3 ¼ b1 ¼ b3 ¼ 0:

a0 ¼ 

4.2. Example II

a2

We construct the exact solutions for space–time fractional nonlinear coupled KdV equations in the following form a Dat v þ L5 Dax v þ L6 v Dax v þ L7 D3a x v þ L8 Dx u ¼ 0;

v ¼ vðnÞ;

b0 ¼ 

where n ¼ x þ y þ ct

a Dat u þ L1 Dax u þ L2 u Dax u þ L3 D3a x u þ L4 Dx v ¼ 0;

u ¼ uðnÞ;

uðnÞ ¼ a0 þ a1 /ðnÞ þ a2 /2 ðnÞ þ a3 /1 ðnÞ þ a4 /2 ðnÞ;

v5 ðx; y; tÞ

k2 þ 2k4 ca  k1 C2 ð1 þ aÞ þ4 a ; 12 k4 ðn þ xÞ2 k3 ðk2 þ k1 Þ  2k5 k4 þ ca ðk2 þ k1 Þ v7 ðx; y; tÞ ¼ 6 k4 ðk2 þ k1 Þ " # 2 2 ðk2 þ k1 Þ C ð1 þ aÞ þ ; r ¼ 0; k4 ðna þ xÞ2

difference in linear long wave speed of uncoupled system, L4 and L8 are coupling parameters, while L2 ; L6 and L3 ; L7 are nonlinear and linear dispersive coefficients, respectively [32]. We use travelling wave transformation,

ð23Þ

which is a transformed generalization of nonlinear coupled KdV equations [32]. For case corresponding to a ¼ 1; L1 ; L5 are detaining parameters which measure the

b0 b2 a1

ð28Þ

From Eqs. (26), (27) and general solutions of fractional Riccati equation given by Eq. (7), we get exact solutions of

Fractional sub-equation method

space–time fractional nonlinear coupled KdV equations given by Eq. (23) to take the following forms: 8 L23 L6 r þ L1 L3 L6 þ ca L3 L6 þ L4 L7 L2 L3 L6 L2   p ffiffiffiffiffiffiffi 12L3 r þ coth2 r n; a ; L2 8 L2 L2 r þ L8 L3 L6 þ ca L7 L2 þ L5 L7 L2 v1 ðx; tÞ ¼  7 L7 L6 L2   p ffiffiffiffiffiffiffi 12L7 r þ coth2 r n; a ; r\0: L6

ð29Þ

where r [ 0;

a

þ L1 L3 L6 þ c L3 L6 þ L4 L7 L2 L3 L6 L2 pffiffiffiffiffiffiffi  12L3 r 2 r n; a ; þ tanh L2 8 L27 L2 r þ L8 L3 L6 þ ca L7 L2 þ L5 L7 L2 v2 ðx; tÞ ¼  L7 L6 L2   p ffiffiffiffiffiffiffi 12L7 r þ tanh2 r n; a ; r\0: L6 u2 ðx; tÞ ¼ 

8 L23 L6 r þ L1 L3 L6 þ ca L3 L6 þ L4 L7 L2 L3 L6 L2   pffiffiffi  p ffiffiffi 12L3 r   tan2 r n; a þ cot2 r n; a ; L2 8 L2 L2 r þ L8 L3 L6 þ ca L7 L2 þ L5 L7 L2 v6 ðx; tÞ ¼  7 L7 L6 L2   pffiffiffi  p ffiffiffi 12L7 r   tan2 r n; a þ cot2 r n; a ; ð34Þ L6

u6 ðx; tÞ ¼ 

u1 ðx; tÞ ¼ 

8 L23 L6 r

n ¼ x þ ct:

where r\0;

u7 ðx; tÞ ¼ 

n ¼ x þ ct:

L1 L3 L6 þ ca L3 L6 þ L4 L7 L2 12L3 C2 ð1 þ aÞ  ; L3 L6 L2 L2 ðna þ xÞ2

v7 ðx; tÞ ¼ 

L8 L3 L6 þ ca L7 L2 þ L5 L7 L2 12L7 C2 ð1 þ aÞ  ; L7 L6 L2 L6 ðna þ xÞ2

ð35Þ ð30Þ

8 L23 L6 r þ L1 L3 L6 þ ca L3 L6 þ L4 L7 L2  L3 L6 L2 pffiffiffi  12L3 r cot2 r n; a ; L2 8 L2 L2 r þ L8 L3 L6 þ ca L7 L2 þ L5 L7 L2 v3 ðx; tÞ ¼  7  L7 L6 L2 pffiffiffi  12L7 r ð31Þ cot2 r n; a ; r [ 0: L6

where r ¼ 0;

n ¼ x þ ct:

4.3. Example III

u3 ðx; tÞ ¼ 

8 L2 L6 r þ L1 L3 L6 þ ca L3 L6 þ L4 L7 L2 u4 ðx; tÞ ¼  3 L3 L6 L2 pffiffiffi  12L3 r 2 r n; a ;  tan L2 8 L27 L2 r þ L8 L3 L6 þ ca L7 L2 þ L5 L7 L2 v4 ðx; tÞ ¼  L7 L6 L2   p ffiffiffi 12L7 r  tan2 r n; a ; r [ 0: L6

1 u þ 3 u Dax u  6 vDax v; Dat u ¼ D3a 4 x 1 v  3 u Dax v; Dat v ¼  D3a 2 x

ð36Þ

which is a transformed generalization of Hirota–Satsuma KdV system [33]. For the case corresponding to a ¼ 1; this system describes interactions of two long waves with different dispersion relations [33]. We use the travelling wave transformation, u ¼ uðnÞ; ð32Þ

where n ¼ x þ ct. Eqs. (9), (26) and (28) lead to get the exact solutions of space–time fractional nonlinear coupled KdV Eq. (23) in the following form: 8 L23 L6 r þ L1 L3 L6 þ ca L3 L6 þ L4 L7 L2 L3 L6 L2 pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi  12L3 r  2 r n; a þ coth2 r n; a ; þ tanh L2 8 L2 L2 r þ L8 L3 L6 þ ca L7 L2 þ L5 L7 L2 v5 ðx; tÞ ¼  7 L7 L6 L2   pffiffiffiffiffiffiffi  p ffiffiffiffiffiffiffi 12L7 r  þ tanh2 r n; a þ coth2 r n; a ; L6 ð33Þ

u5 ðx; tÞ ¼ 

We construct the exact solutions for space–time fractional Hirota–Satsuma KdV system in the following form

v ¼ vðnÞ;

n ¼ x þ c t:

ð37Þ

The travelling wave transformations given by Eq. (37) lead to write Eq. (36) into the following form 1 ca Dan u ¼ D3a u þ 3 u Dan u  6 vDan v; 4 n 1 v  3 u Dan v; ca Dan v ¼  D3a 2 n

ð38Þ

where c is an arbitrary constant. By balancing highest order fractional derivatives and nonlinear terms in Eq. (38), we get solutions of Eq. (38) take the following form uðnÞ ¼ a0 þ a1 /ðnÞ þ a2 /2 ðnÞ þ a3 /1 ðnÞ þ a4 /2 ðnÞ; vðnÞ ¼ b0 þ b1 /ðnÞ þ b2 /2 ðnÞ þ b3 /1 ðnÞ þ b4 /2 ðnÞ; ð39Þ

K A Gepreel and A A Al-Thobaiti

where ai ; bi ði ¼ 0; 1; 2; 3; 4Þ are arbitrary constants and /ðnÞ satisfies fractional Riccati Eq. (7). We substitute Eqs. (7) and (39) into Eq. (38) and set the coefficients of /n ðnÞn ¼ 0; 1; 2; 3; . . . to be zero. Then we yield a set of over-determined system of algebraic equations with respect to ai ; bi ði ¼ 0; 1; 2; 3; 4Þ and c. On using Maple software package, we solve over-determined algebraic equations to get the following cases: Case (i) ca 4 r ; a2 ¼ 2; a4 ¼ 2 r2 ;  3 3  a  2c 2r þ b0 ¼  ; b2 ¼ 1; b4 ¼ r2 3 3 a0 ¼ 

ð40Þ

ð45Þ

 pffiffiffi  c 4r  r tan2 r n; a  3 3  pffiffiffi þ cot2 r n; a ; r [ 0: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ca 2 r pffiffiffi pffiffiffi þ v5 ðx; tÞ ¼  r tan r n; a 3 3 pffiffiffi  þ cot r n; a ;

ð46Þ

a

u5 ðx; tÞ ¼ 

ca C2 ð1 þ aÞ  a ; r ¼ 0; 3 ðn þ xÞ2 rffiffiffiffiffiffiffiffi   2 ca Cð1 þ aÞ v6 ðx; tÞ ¼  ; 3 ðna þ xÞ

a1 ¼ a3 ¼ b1 ¼ b3 ¼ 0:

u6 ðx; tÞ ¼ 

Case (ii) a

c 4r ; a2 ¼ 1; a4 ¼  r2 ;  3 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ca 2 r 2 ca 2 r b1 ¼  þ ; b3 ¼  þ r; 3 3 3 3 a1 ¼ a3 ¼ b0 ¼ b2 ¼ b4 ¼ 0:

 pffiffiffiffiffiffiffi  ca 4 r þ r tanh2 r n; a  3 3  pffiffiffiffiffiffiffi þ coth2 r n; a ; r\0: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi  2 ca 2 r pffiffiffiffiffiffiffi þ v4 ðx; tÞ ¼  r tanh r n; a 3 3 pffiffiffiffiffiffiffi  þ coth r n; a ;

u4 ðx; tÞ ¼ 

a0 ¼ 

ð41Þ

ð47Þ

where r ¼ 0; n ¼ x þ ct:

5. Exp function method for nonlinear fractional PDEs From case (i), Eq. (39) and the general solutions of fractional Riccati Eq. (7), we get exact solutions of space– time fractional Hirota–Satsuma KdV system to take the following form  pffiffiffiffiffiffiffi  ca 4 r þ 2r tanh2 r n; a  3 3  pffiffiffiffiffiffiffi þ coth2 r n; a ; r\0:  a  2c 2r þ v1 ðx; tÞ ¼   3 3    pffiffiffiffiffiffiffi  p ffiffiffiffiffiffiffi  r tanh2 r n; a þ coth2 r n; a ; u1 ðx; tÞ ¼ 

ð42Þ  pffiffiffi  ca 4 r  2r tan2 r n; a u2 ðx; tÞ ¼   3  3  pffiffiffi þ cot2 r n; a ; r [ 0:  a   pffiffiffi  2c 2r þ v2 ðx; tÞ ¼   r tan2 r n; a 3 3 pffiffiffi  2 r n; a ; þ cot ca C2 ð1 þ aÞ 2 a ; r ¼ 0; 3 ðn þ xÞ2  a 2c C2 ð1 þ aÞ ; v3 ðx; tÞ ¼   a 3 ðn þ xÞ2

Consider the following nonlinear partial fractional differential equations U ðu; ut ; ux ; uy ; uz ; Dat u; Dbx u; Dcy u; Ddz u; D2a t u; . . .Þ ¼ 0; 0\a; b; c; d  1;

ð48Þ

where u is an unknown function and U is a polynomial of u and its partial fractional derivatives, in which highest order derivatives and nonlinear terms are involved. In the following, we give main steps of the exp-function method. Step (i) Li and He [34] have suggested a fractional complex transform to convert FDE into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily applied to fractional calculus. Traveling wave variable uðx; y; z; tÞ ¼ uðnÞ;

ð43Þ

u3 ðx; tÞ ¼ 

ð44Þ

where n ¼ x þ ct: From Eqs. (9), (39) and (41), exact solutions of the space–time fractional Hirota–Satsuma KdV system given by Eq. (36) take the following form:

n¼

M xb N yc L ta þ þ ; Cðb þ 1Þ Cðc þ 1Þ Cða þ 1Þ

ð49Þ

where M; N; K and L are non zero arbitrary constants, which permit us to reduce Eq. (48) to an ODE of uðnÞ in the form PðU; U 0 ; U 00 ; U 000 ; . . .Þ ¼ 0:

ð50Þ

Step (ii) According to exp-function method [31, 35], we assume that wave solution can be expressed in the following form

Fractional sub-equation method

Pd an expðn nÞ P ; UðnÞ ¼ qn¼c b m¼p m expðm nÞ

ð51Þ

where p; q; c and d are positive integers which are to be determined later and an ; bn are unknown constants. Eq. (51) can be written in the following form ac expðc nÞ þ    þ ad expðd nÞ UðnÞ ¼ : ð52Þ ap expðp nÞ þ    þ aq expðq nÞ Step (iii) To determine values of c and p, we balance the linear term of highest order of Eq. (50) with highest order nonlinear term. Similarly, to determine value of d and q, we balance linear term of lowest order of Eq. (50) with lowest order nonlinear term. Step (iv) We substitute Eq. (52) into Eq. (50) and collect all terms with the same order of enn together; left hand side of Eq. (50) is converted into a polynomial in enn . Then we set each coefficient of this polynomial to be zero to derive a set of algebraic equations for an ; bn . Step (v) We solve the algebraic system obtained in Step (iv). Then we can obtain the exact solutions of Eq. (48).

ð57Þ

which leads to get p ¼ c:

ð58Þ

Similarly, to determine values of d and q, we balance linear term of the lowest order in Eq. (55), so we get    þ d1 exp½ ð3q þ dÞ n ;    þ d2 exp½ 4q n    þ d3 exp½2d n    þ d3 exp½ð2d þ 2qÞ n U 2 ðnÞ ¼ ¼ ;    þ d4 exp½2q n    þ d4 exp½4q n U 00 ðnÞ ¼

ð59Þ where di are determined coefficients only for simplicity. Balancing highest order of exp function in Eq. (59), we obtain 3q þ d ¼ 2q þ 2d;

ð60Þ

q ¼ d:

We use the exp function method to find the exact solutions for space–time fractional coupled Sakharov-Kuznetsov equations given by Eq. (10). Firstly, we consider the following transformations ð53Þ

 C U 0 þ 2U 000  6 UU 0  V 0 ¼ 0;  C V 0 þ ðk1 þ k2 Þ V 000 þ k3 V 0  6 k4 VV 0  k5 U 0 ¼ 0; ð54Þ 0 dV where U 0 ¼ dU dn and V ¼ dn . Integrating Eq. (54) with respect to n and setting the integrating constant to be zero, we have  C U þ 2U 00  3 U 2  V ¼ 0;  C V þ ðk1 þ k2 Þ V 00 þ k3 V  3 k4 V 2  k5 U ¼ 0: ð55Þ Balancing highest order derive U 00 and the nonlinear term U 2 in Eq. (55), we get c1 exp½ ð3p þ cÞn þ  ; c2 exp½4p n þ   c3 exp½2c n þ  c3 exp½ð2c þ 2pÞn þ  ¼ ; ð56Þ U 2 ðnÞ ¼ c4 exp½2pn þ   c4 exp½4p n þ  

ð61Þ

For simplicity, we set p ¼ c ¼ 1 and q ¼ d ¼ 1; which reduces to UðnÞ ¼

a1 expðnÞ þ a0 þ a1 expðnÞ : b1 expðnÞ þ b0 þ b1 expðnÞ

ð62Þ

In the same way, we can get VðnÞ ¼

where C is a nonzero constant. Substituting Eq. (53) into Eq. (15), we get:

U 00 ðnÞ ¼

3p þ c ¼ 2p þ 2c;

which leads to the result that

5.1. Exp function method for space–time fractional coupled Sakharov–Kuznetsov equations

uðx; y; tÞ ¼ UðnÞ; vðx; y; tÞ ¼ VðnÞ xa ya C ta n¼ þ  ; Cða þ 1Þ Cða þ 1Þ Cða þ 1Þ

where ci are determined coefficients only for simplicity. Balancing highest order of exp function in Eq. (56), we obtain

c1 expðnÞ þ c0 þ c1 expðnÞ : d1 expðnÞ þ d0 þ d1 expðnÞ

ð63Þ

Substituting Eqs. (62) and (63) into Eq. (55) and collecting the coefficient of enn : Setting each coefficient of enn to be zero, we get a system of algebraic equations. Solving this system of algebraic equations, we get the following results a0 ¼ 

2 b1 d0 ; d1

c0 ¼ 2 d0 ðC  2Þ;

b0 ¼

b1 d0 ; d1

d1 ¼

b1 ¼

b1 d02 ; 4 d12

d02 ; 4 d1

k1 ¼ 2 C k4  k2 þ 4k4 ; k5 ¼ ðC  2ÞðC  k3 þ 2 C k4  4k4 Þ; a1 ¼ a1 ¼ c1 ¼ c1 ¼ 0; ð64Þ where b1 ; d0 ; d1 ; k2 ; k3 ; k4 are arbitrary constants. Eqs. (62)–(64) lead to obtain the following exact solutions

K A Gepreel and A A Al-Thobaiti

2 b1 d0

UðnÞ ¼

b1 d1 expðnÞ þ b1 d0 þ

b1 d02 4 d1

expðnÞ

2 d0 ðC  2Þ

VðnÞ ¼

References

; ð65Þ ;

d2

d1 expðnÞ þ d0 þ 4 d0 expðnÞ 1

where n¼

xa ya C ta þ  : Cða þ 1Þ Cða þ 1Þ Cða þ 1Þ

ð66Þ

In the special case, when we set b1 ¼ d0 ¼ 1; and d1 ¼ 12 in Eq. (65), we can obtain the following soliton solutions UðnÞ ¼

2 ; coshðnÞ þ 1

VðnÞ ¼

2ðC  2Þ ; coshðnÞ þ 1

ð67Þ

where n¼

xa ya C ta þ  : Cða þ 1Þ Cða þ 1Þ Cða þ 1Þ

ð68Þ

Exp-function method is more powerful than fractional subequation method. But exp-function method is complicated for solving nonlinear partial fractional differential equations. Fractional sub-equation method is simple method for solving complicated nonlinear partial fractional differential equations. Also exp function method can not possibly solve all the nonlinear partial fractional differential Eqs. (23) and (36). We must use fractional complex transformation, when we use exp-function method, because fractional derivative to the rational function not defined.

6. Conclusions In this paper, we have constructed exact solutions for nonlinear fractional differential equation by using fractional sub- equation method as suggested by Guo et al. [20]. This method allows us to solve more complicated nonlinear fractional differential equations in mathematical physics via space–time fractional coupled Sakharov- Kuznetsov (Z–K) equations, space–time fractional nonlinear coupled KdV equations and space–time fractional Hirota–Satsuma KdV system. As a result, many new types of exact solutions are obtained. These solutions include generalized hyperbolic function solutions, generalized trigonometric function solutions, and rational function solutions.

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