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The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics
Journal of the Association of Arab Universities for Basic and Applied Sciences (2014) xxx, xxx–xxx
University of Bahrain
Journal of the Association of Arab Universities for Basic and Applied Sciences www.elsevier.com/locate/jaaubas www.sciencedirect.com
ORIGINAL ARTICLE
The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics Elsayed M.E. Zayed *, Yasser A. Amer, Reham M.A. Shohib Mathematics Department, Faculty of Sciences, Zagazig University, Zagazig, Egypt Received 24 November 2013; revised 19 June 2014; accepted 23 June 2014
KEYWORDS Nonlinear fractional partial differential equations; Modified extended tanh-function method; Nonlinear fractional complex transformation; Exact solutions
Abstract In this article, the modified extended tanh-function method is employed to solve fractional partial differential equations in the sense of the modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into nonlinear ordinary differential equations of integer orders. For illustrating the validity of this method, we apply it to four nonlinear equations namely, the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space– time fractional nonlinear Whitham–Broer–Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations. ª 2014 Production and hosting by Elsevier B.V. on behalf of University of Bahrain.
1. Introduction Fractional differential equations are the generalizations of classical differential equations with integer orders. In recent years, nonlinear fractional differential equations in mathematical physics are playing a major role in various fields, such as physics, biology, engineering, signal processing, and control theory, finance and fractal dynamics (Miller and Ross, 1993; Kilbas et al., 2006; Podlubny, 1999). Finding approximate and exact solutions to the fractional differential equations is an important task. A large amount of literatures were developed concerning the solutions of the fractional differential equations in nonlinear dynamics (El-sayed et al., 2009). Many * Corresponding author. E-mail addresses: [email protected] (E.M.E. [email protected] (Y.A. Amer). Peer review under responsibility of University of Bahrain.
Zayed),
powerful and efficient methods have been proposed to obtain the numerical and exact solutions of fractional differential equations. For example, these methods include the variational iteration method (Safari et al., 2009); Wu and Lee, 2010; Yang and Baleanu, 2012; Guo and Mei, 2011), the Lagrange characteristic method (Jumarie, 2006a), the homotopy analysis method (Song and Zhang, 2009), the Adomian decomposition method (El-Sayed and Gaber, 2006; El-sayed et al., 2010), the homotopy perturbation method (He, 1999; He, 2000; Yildirim and Gulkanat, 2010), the differential transformation method (Odibat and Momani, 2008), the finite difference method (Cui, 2009), the finite element method (Huang et al., 2009), the fractional sub-equation method (Zhang and Zhang, 2011; Guo et al., 2012; Lu, 2012), the (G0 /G)-expansion method (Zheng, 2012; Gepreel and Omran, 2011; Younis and Zafar, 2013), the modified extended tanh-function method (El-Wakil et al., 2005; El-Wakil et al., 2002; Soliman, 2006; Dai and Wang, 2014), the fractional complex transformation
http://dx.doi.org/10.1016/j.jaubas.2014.06.008 1815-3852 ª 2014 Production and hosting by Elsevier B.V. on behalf of University of Bahrain. Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
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E.M.E. Zayed et al.
method (Li and He, 2010; Li, 2010; He and Li, Li et al., 2012; Hristov, 2010), the exp-function method (He, 2013), the similarity transformation method (Dai et al., 2013; Zhu, 2013), the Hirota method (Liu et al., 2013) and so on. The objective of this paper is to apply the modified extended tanh-function method for solving fractional partial differential equations in the sense of the modified Riemann–Liouville derivative which has been derived by (Jumarie, 2006b). These equations can be reduced into nonlinear ordinary differential equations (ODE) with integer orders using some fractional complex transformations. Jumarie’s modified Riemann–Liouville derivative of order a is defined by the following expression: ( 1 d Rt ðt gÞa ½fðgÞ fð0Þdg; 0 < a 6 1; Cð1aÞ dt 0 a Dt fðtÞ ¼ ðanÞ fðnÞ ðtÞ ; n 6 a < n þ 1; n P 1 We list some important properties for the modified Riemann–Liouville derivative as follows: Cð1 þ rÞ ra t ;r > 0 Cð1 þ r aÞ
ð1Þ
Dat ½fðtÞgðtÞ ¼ fðtÞDat gðtÞ þ gðtÞDat fðtÞ
ð2Þ
Dat ½fðgðtÞÞ
ð3Þ
Dat tr ¼
¼
f0g ðgðtÞÞDat gðtÞ
Dat ½fðgðtÞÞ ¼ Dag fðgðtÞÞ½g0ðtÞa
Pðu; u0 ; u00 ; :::Þ ¼ 0;
ð7Þ
where P is a polynomial in u(n) and its total derivatives d2 u u0 ,u00 ,u000 ,. . . such that u0 ¼ du ; u00 ¼ dn 2 ; . . ... dn Step 2: We suppose that the formal solution of the ODE (7) can be expressed as follows: uðnÞ ¼ a0 þ
N X ½ai /i ðnÞ þ bi /i ðnÞ;
ð8Þ
i¼1
where /(n) satisfies the Riccati equation /0 ¼ b þ /2 ;
ð9Þ
where b is a constant. Fortunately, Eq. (9) admits several types of the following solutions: (i) If b < 0, we have the hyperbolic solutions; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi /ðnÞ ¼ b tanhð bnÞ; /ðnÞ ¼ b cothð bnÞ:
ð10Þ
(ii) If b > 0, we have the trigonometric solutions; pffiffiffi pffiffiffi pffiffiffi pffiffiffi /ðnÞ ¼ b tanð bnÞ; /ðnÞ ¼ b cotð bnÞ:
ð11Þ
(iii) If b=0, we have the rational solutions;
ð4Þ
where C denotes the Gamma function. The rest of this paper is organized as follows: In Section 2, the description of the modified extended tanh-function method for solving nonlinear fractional partial differential equations is given. In Section 3, we apply this method to establish the exact solutions for the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space–time fractional nonlinear Whitham–Broer– Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations. In Section 4 physical explanations of some obtained solutions are given. In Section 5, some conclusions are obtained.
/ðnÞ ¼
1 ; nþd
ð12Þ
where d is a constant. Step 3: We determine the positive integer N in (8) by balancing the highest nonlinear terms and the highest order derivatives of u(n) in Eq. (7). Step 4: We substitute (8) along with Eq. (9) into Eq. (7) and equate all the coefficients of /i(i = 0, ± 1, ± 2,...) to zero to yield a system of algebraic equations for ai,bi,c,k, b. Step 5: We solve the algebraic equations obtained in Step 4 using Mathematica or Maple, and use the well- known solutions (10)–(12) of Eq. (9) to obtain the exact solutions of Eq. (5). 3. Applications
2. Description the modified extended tanh-function method for solving nonlinear fractional partial differential equations
In this section, we construct the exact solutions of the following four nonlinear fractional partial differential equations using the proposed method of Section 2:
Suppose we have the following nonlinear fractional partial differential equation:
Example 1. The Space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations.
Fðu; Dat u; Dax u; :::Þ ¼ 0; 0 < a 6 1;
These equations are well-known (Guo et al., 2012; Zheng, 2012) and have the forms:
ð5Þ
where Dat u and Dax u are the modified Riemann–Liouville derivatives and F is a polynomial in u = u(x,t) and its fractional derivatives. In the following, we give the main steps of this method: Step 1: Using the nonlinear fractional complex transformation (Li and He, 2010; Li, 2010; He and Li, 2012; Li et al., 2012; Hristov, 2010). uðx; tÞ ¼ uðnÞ; n ¼
kxa cta þ þ n0 ; Cð1 þ aÞ Cð1 þ aÞ
ð6Þ
where k,c,n0 are constants with k,c „ 0, to reduce Eq. (5) to the following ODE of integer order with respect to the variable n :
1 Dat u D3a u þ 3uDax u 3Dax ðvwÞ ¼ 0; 2 x a Dat v þ D3a x v 3uDx v ¼ 0; a Dat w þ D3a x w 3uDx w ¼ 0;
ð13Þ ð14Þ ð15Þ
where 0 < a 6 1. Eqs. (13)–(15) can be used to describe the iteration of two long waves with different dispersion relations (Abazari and Abazari, 2012). When a = 1, Eqs. (13)–(15) were first proposed in (Wu and Geng, 1999). When 0 < a 6 1, Eqs. (13)–(15) have been discussed in (Zheng, 2012) using the (G0 /G)expansion method and in ( Guo et al, 2012) using the fractional sub-equation method. Let us now solve Eqs. (13)–(15) using the
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
The fractional complex transformation for nonlinear fractional partial differential equations /3 :
proposed method of Section 2. To this end, we suppose that u(x,t) = U(n),v(x,t) = V(n),w(x,t) = W(n) where n is given by the fractional complex transformation (6). Then by use of Eqs. (1) and (3), the system (13)–(15) can be turned into the following system of ODEs with integer orders: 1 3 cU k3 U00 þ kU2 3kVW ¼ 0; 2 2 cV0 þ k3 V000 3kUV0 ¼ 0;
1
WðnÞ ¼ c0 þ c1 / þ c2 /
/4 :
ð19Þ
2
2
ð20Þ
2
2
ð21Þ
þ b3 / þ b4 / ; þ c3 / þ c4 / ;
6b2 b3 k3 3kð2a2 b4 b þ a4 b2 bÞ ¼ 0 6c2 b3 k3 3kð2a2 c4 b þ a4 c2 bÞ ¼ 0 /5 :
24c3 k3 6kc3 a3 ¼ 0 24b3 k3 6kb3 a3 ¼ 0
/4 :
3a3 k3 þ 32 ka23 3kb3 c3 ¼ 0
24b4 b3 k3 6kb4 a4 ¼ 0 24c4 b3 k3 6kc4 ba4 ¼ 0
Solving the above set of algebraic equations by using Maple or Mathematica , we get the following results: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 8 b ¼ b;c ¼ 4ck 3 64 bc þ 32c þ 34b2 c23 120bc0 c3 þ 90c20 ; 3 0 3 3
where ai,bi,ci (i = 0,1,2,3,4) are constants to be determined later. We substitute (19)–(21) along with Eq. (9) into Eqs. (16)–(18) and collect all the terms with the same power of /j,(j = 0, ± 1, ± 2, ± 3,...). Equating each coefficient to zero yields a set of the following algebraic equations:
/5 :
3a4 b2 k3 32 ka24 þ 3kb4 c4 ¼ 0
ð17Þ
UðnÞ ¼ a0 þ a1 / þ a2 /1 þ a3 /2 þ a4 /2 ; VðnÞ ¼ b0 þ b1 / þ b2 /
2cbc4 þ 40c4 b2 k3 3kða2 c2 b þ 2a4 c4 þ 2a0 c4 bÞ ¼ 0
ð18Þ
Balancing the order of U00 with U2, V000 with UV0 and W000 with UW0 in Eqs. (16)–(18), we deduce that the formal solutions of Eqs. (16)–(18) have the forms: 1
a2 b2 k3 3a2 a4 k þ 3kðb2 c4 þ c2 b4 Þ ¼ 0 2cbb4 þ 40b4 b2 k3 3kða2 b2 b þ 2a4 b4 þ 2a0 b4 bÞ ¼ 0
ð16Þ
cW0 þ k3 W000 3kUW0 ¼ 0:
3
k ¼ k;a3 ¼ 4k2 ; 2
a0 ¼ 8ck 3
64 bc3 þ 643 c0 þ 169 9
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 34b2 c23 120bc0 c3 þ 90c20 ;
4
b3 ¼ 4kc3 ;c0 ¼ c0 ; a1 ¼ a2¼ a4 ¼ b1 ¼ b2 ¼ b4 ¼ c1 ¼ c2 ¼ c4 ¼ 0;c3 ¼ c3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 80 52 16 34b2 c23 120bc0 c3 þ 90c20 b0 ¼ k4 bc þ c þ 3 0 2 9 3 9 c 3
6b1 k3 3kð2a1 b3 þ b1 a3 Þ ¼ 0 6c1 k3 3kð2a1 c3 þ b1 c3 Þ ¼ 0 /3 :
a1 k3 þ 3a1 a3 k 3kðb1 c3 þ c1 b3 Þ ¼ 0 2cb3 þ 40b3 bk3 3kð2a0 b3 þ a1 b1 þ 2a3 b3 bÞ ¼ 0 2cc3 þ 40c3 bk3 3kð2a0 c3 þ a1 c1 þ 2a3 c3 bÞ ¼ 0
/2 :
ca3 4a3 bk3 þ 32 kða21 þ 2a0 a3 Þ 3kðb0 c3 þ b1 c1 þ b3 c0 Þ ¼ 0 b1 c þ 8b1 bk3 3kða0 b1 þ 2a1 b3 b þ 2a2 b3 þ a3 b1 b a3 b2 Þ ¼ 0 c1 c þ 8c1 bk3 3kða0 c1 þ 2a1 c3 b þ 2a2 c3 þ a3 c1 b a3 c2 Þ ¼ 0
/:
ca1 a1 bk3 þ 3kða0 a1 þ a2 a3 Þ 3kðb0 c1 þ b1 c0 þ b2 c3 þ b3 c2 Þ ¼ 0 2bcb3 þ 16b3 b2 k3 3kð2a0 b3 b þ a1 b1 b b2 a1 þ a2 b1 2a3 b4 þ 2a4 b3 Þ ¼ 0 2bcc3 þ 16c3 b2 k3 3kð2a0 c3 b þ a1 c1 b c2 a1 þ a2 c1 2a3 c4 þ 2a4 c3 Þ ¼ 0
/0 :
ca0 k3 ða3 b2 þ a4 Þ þ 32 kða20 þ 2a1 a2 þ a2 a4 Þ 3kðb0 c0 þ b1 c2 þ b2 c1 þ b3 c4 þ b4 c3 Þ ¼ 0 cðb1 b b2 Þ þ k3 ð2b1 b2 2b2 bÞ 3kða0 b1 b a0 b2 2a1 b4 þ 2a2 b3 b a3 b2 b þ a4 b1 Þ ¼ 0 cðc1 b c2 Þ þ k3 ð2c1 b2 2c2 bÞ 3kða0 c1 b a0 c2 2a1 c4 þ 2a2 c3 b a3 c2 b þ a4 c1 Þ ¼ 0
/1 :
ca2 a2 bk3 þ 3kða0 a2 þ a1 a4 Þ 3kðb0 c2 þ b1 c4 þ b2 c0 þ b4 c2 Þ ¼ 0 2cb4 þ 16b4 bk3 þ 3kð2a0 b4 a1 b2 b þ a2 b1 b b2 a2 2a3 b4 b þ 2a4 b3 bÞ ¼ 0 2cc4 þ 16c4 bk3 þ 3kð2a0 c4 a1 c2 b þ a2 c1 b c2 a2 2a3 c4 b þ 2a4 c3 bÞ ¼ 0
/2 :
ca4 4a4 bk3 þ 32 kða22 þ 2a0 a4 Þ 3kðb0 c4 þ b2 c2 þ b4 c0 Þ ¼ 0 b2 bc 8b2 b2 k3 þ 3kða0 b2 b þ 2a1 b4 b þ 2a2 b4 a4 b1 b þ a4 b2 Þ ¼ 0 c2 bc 8c2 b2 k3 þ 3kða0 c2 b þ 2a1 c4 b þ 2a2 c4 a4 c1 b þ a4 c2 Þ ¼ 0
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
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E.M.E. Zayed et al.
Now, we have the following exact solutions:
(3), the system (27), (28) after integrating once can be turned into the following system of ODEs with integer orders:
(i) If b < 0, we have the hyperbolic solutions 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 > > uðx; tÞ ¼ 8ck 3 ½649 bc3 þ 643 c0 þ 169 34b2 c23 120bc0 c3 þ 90c20 4k2 btanh2 ð bnÞ > > > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 4 4 bc3 þ 523 c0 þ 169 34b2 c23 120bc0 c3 þ 90c20 4bk tanh2 ð bnÞ vðx; tÞ ¼ kc2 ½80 > 9 c3 > 3 > > > pffiffiffiffiffiffiffi : wðx; tÞ ¼ c0 c3 btanh2 ð bnÞ
ð22Þ 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi > 2 2 2 2 k2 64 64 16 2 > > > uðx; tÞ ¼ 8c3 ½ 9 bc3 þ 3 c0 þ 9 34b c3 120bc0 c3 þ 90c0 4k bcoth ð bnÞ > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 4 4 bc3 þ 523 c0 þ 169 34b2 c23 120bc0 c3 þ 90c20 4bk coth2 ð bnÞ vðx; tÞ ¼ kc2 ½80 > 9 c3 > 3 > > > pffiffiffiffiffiffiffi : wðx; tÞ ¼ c0 c3 bcoth2 ð bnÞ
ð23Þ
(ii) If b > 0, we have the trigonometric solutions 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi > 2 2 2 k2 64 64 16 2 2 > > > uðx; tÞ ¼ 8c3 ½ 9 bc3 þ 3 c0 þ 9 34b c3 120bc0 c3 þ 90c0 þ 4k b tan ð bnÞ > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 4 bc3 þ 523 c0 þ 169 34b2 c23 120bc0 c3 þ 90c20 þ 4bk tan2 ð bnÞ vðx; tÞ ¼ kc2 ½80 > 9 c3 > 3 > > > pffiffiffi : wðx; tÞ ¼ c0 þ c3 b tan2 ð bnÞ
1 cU þ kU2 þ kV þ bk2 U0 ¼ 0; 2 cV þ kUV bk2 V0 þ ck3 U00 ¼ 0;
UðnÞ ¼ a0 þ a1 / þ a2 /1 ; VðnÞ ¼ b0 þ b1 / þ b2 /
/2 :
1 2 ak 2 1
þ kb3 þ a1 bk2 ¼ 0
/:
a1 c þ a0 a1 k þ kb1 ¼ 0 b1 c þ kða0 b1 þ a1 b0 þ a2 b3 Þ 2bk2 bb3 þ 2a1 ck3 b ¼ 0
/0 :
a0 c þ 12 kða20 þ 2a1 a2 Þ þ kb0 þ bk2 ða1 b a2 Þ ¼ 0 b0 c þ kða0 b0 þ a1 b2 þ a2 b1 Þ bk2 ðbb1 b3 Þ ¼ 0 a2 c þ a0 a2 k þ kb2 ¼ 0
ð26Þ
/2 :
b4 c þ kða0 b4 þ a2 b2 Þ þ b2 bk2 ¼ 0 1 2 a k þ kb4 a2 bbk2 ¼ 0 2 2
/3 : a2 b4 k þ 2bb4 bk2 þ 2a2 b2 ck3 ¼ 0 On solving the above algebraic equations by using Maple or Mathematica, we have the following results: Case 1 2
b ¼ b; c ¼ 4k These equations are well-known (Xu et al., 2007; Xu et al., 2007; Ping, 2010) and have the forms:
þ
þ
cD3a x u
¼ 0;
ð27Þ ð28Þ
where 0 < a 6 1, and c,b are constants. In these equations u(x,t) is the field of horizontal velocity , v(x,t) is the height deviating from the equilibrium position of liquid, while c and b represent different diffusion powers. When a = 1, Eqs. (27) and (28) are the generalization of nonlinear Whitham–Broer–Kaup equations, which can be used to describe the dispersive long wave in shallow water ( Xu et al, 2007; Ping, 2010). Eqs. (27) and (28) have been discussed in Guo and Mei (2012) using the improved fractional sub-equation method. Let us now solve Eqs. (27), (28) using the proposed method of Section 2. To this end, we suppose that u(x,t) = U(n),v(x,t) = V(n). Then by the use of Eqs. (1) and
pffiffiffiffi bðb1Þ b ; a0 3b
pffiffiffiffi b
¼ 4kbðb1Þ 3b
a2 ¼ 2kbðb1Þ ; b1 ¼ b2 ¼ 0; b0 ¼ 4k 3 2 2
b4 ¼ 2k bD2a x v
ð32Þ
b2 c þ kða0 b2 þ a1 b4 þ a2 b0 Þ þ 2bk2 b4 þ 2ca2 bk3 ¼ 0
Example 2. The Space–time fractional nonlinear Whitham– Broer–Kaup equations
Dax ðuvÞ
þ b3 / þ b4 / ;
b3 c þ kða0 b3 þ a1 b1 Þ b1 bk2 ¼ 0
/1 :
On comparing our results (22)–(26) with the results in (Younis and Zafar, 2013) and (El-Wakil et al., 2005) we conclude that our results are new.
Dat v
ð31Þ 2
2
/3 : a1 b3 k 2b3 bk2 þ 2ca1 k3 ¼ 0
ðnþdÞ
Dat u þ uDax u þ Dax v þ bD2a x u ¼ 0;
1
where ai(i = 0,1,2),bi(i = 0,1,2,3,4) are constants to be determined later. We substitute (31) and (32) along with Eq. (9) into Eqs. (29) and (30) and collect all the terms with the same power of /j,(j = 0, ± 1, ± 2, ± 3). Equating each coefficient to zero yields a set of the following algebraic equations:
ð25Þ
(iii) If b=0, we have the rational solutions 8 pffiffiffiffiffi 2 4k2 > uðx; tÞ ¼ 3ck 3 ð8 þ 2 10Þc0 þ ðnþdÞ 2 > > < p ffiffiffiffiffi 4 4bk4 vðx; tÞ ¼ 3ck 2 ð52 þ 16 10Þc0 þ c ðnþdÞ 2 3 > 3 > > : wðx; tÞ ¼ c0 þ c3 2
ð30Þ
with zero constants of integration. Balancing the highest order derivatives and highest nonlinear terms in Eqs. (29) and (30), we have the following formal solutions:
ð24Þ 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi > 2 2 2 k2 64 64 16 2 2 > > > uðx; tÞ ¼ 8c3 ½ 9 bc3 þ 3 c0 þ 9 34b c3 120bc0 c3 þ 90c0 þ 4k bcot ð bnÞ > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 4 bc3 þ 523 c0 þ 169 34b2 c23 120bc0 c3 þ 90c20 þ 4bk cot2 ð bnÞ vðx; tÞ ¼ kc2 ½80 > 9 c3 > 3 > > > pffiffiffi : wðx; tÞ ¼ c0 þ c3 bcot2 ð bnÞ
ð29Þ
b ð2b2 b1Þ ;c 9
¼ b
2
2 2
; b < 0; a1 ¼ 2kbðb1Þ ; 3b
2
b ð2b b1Þ ; b3 9b
2 2
¼ 2k
b ð2b2 b1Þ ; 9b2
ð8b2 þ2b1Þ 9b2
In this case we have the hyperbolic solutions 8 pffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi < uðx; tÞ ¼ 2kbðb1Þ b 1 ½tanhð bnÞ þ cothð bnÞ 3b ffi pffiffiffiffiffiffiffi : vðx; tÞ ¼ 2k2 b2 ð2b2 b1Þ 2 ½tanh2 ðpffiffiffiffiffiffi bnÞ þ coth2 ð bnÞ 3b
ð33Þ Case 2 pffiffi pffiffi 2 2k bðb1Þ b ; a0 3b 2kbðb1Þ 1Þ; a2 ¼ 3 ;
b ¼ b; b ¼ b; c ¼ 2 a1 ¼
2 kbðb 3b
b0 ¼ b1 ¼ b2 ¼ 0; b3 ¼ 2k
2 2
pffiffi pffiffi 2kbðb1Þ b ;b 3b
¼ 2
b ð4b2 5b1Þ ;c 9b2
2
¼ b
> 0;
ð8b2 þ2b1Þ ; 9b2
2 2
b4 ¼ 2k
b ð2b2 b1Þ 9
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
The fractional complex transformation for nonlinear fractional partial differential equations In this case we have the trigonometric solutions 8 pffiffiffi pffiffiffi pffiffiffi < uðx; tÞ ¼ 2kbðb1Þ pffiffi 2 þ ½tanð bnÞ þ cotð bnÞ 3 b : vðx;tÞ ¼ 2k2 b2 ðb1Þ ð4b 1Þ tan2 ðpffiffibffinÞ ð2b þ 1Þcot2 ðpffiffibffinÞ 9b ð34Þ and 8 pffiffiffi pffiffiffi pffiffiffi < uðx; tÞ ¼ 2kbðb1Þ pffiffi 2 ½tanð bnÞ þ cotð bnÞ 3 b : vðx;tÞ ¼ 2k2 b2 ðb1Þ ð4b 1Þcot2 ðpffiffibffinÞ ð2b þ 1Þtan2 ðpffiffibffinÞ 9b ð35Þ On comparing our results (33)–(35) with the results obtained in (Younis and Zafar, 2013; Dai et al., 2013; Zhu, 2013) we conclude that our results are new.
where 0 < a 6 1 and p, q are constants. Eqs. (36) and (37) have been discussed in (Zhao et al, 2012) using the extended fractional sub-equation method. Let us now solve Eqs. (36) and (37) using the proposed method of Section 2. To this end we suppose that u(x,t) = U(n), v(x,t) = V(n) . Then by the use of Eqs. (1) and (3), the system (36) and (37) can be turned into the following system of ODEs with integer orders: cU0 k2 U00 þ 2kUU0 þ pkðUVÞ0 ¼ 0;
ð38Þ
cV0 k2 V00 þ 2kVV0 þ qkðUVÞ0 ¼ 0;
ð39Þ
Integrating Eqs. (38) and (39) with vanishing the constants of integration, we get cU k2 U0 þ kU2 þ pkUV ¼ 0; 2
0
2
cV k V þ kV þ qkUV ¼ 0; Example 3. The space–time fractional nonlinear coupled Burgers equations These equations are well-known (Zhao et al, 2012) and have the forms: ð36Þ
a a Dat v D2a x v þ 2vDx v þ qDx ðuvÞ ¼ 0;
ð37Þ
Fig. 1
ð40Þ ð41Þ
Balancing the highest order of derivatives and highest nonlinear terms in Eqs. (40) and (41), we have the following formal solutions: UðnÞ ¼ a0 þ a1 / þ a2 /1 ; 1
a a Dat u D2a x u þ 2uDx u þ pDx ðuvÞ ¼ 0;
5
VðnÞ ¼ b0 þ b1 / þ b2 / ;
ð42Þ ð43Þ
where ai,bi(i = 0,1,2) are constants to be determined later. We substitute (42) and (43) along with Eq. (9) into Eqs. (40) and
The plots of solutions (22) of the space time fractional generalized nonlinear Hirota–Statsuma coupled KdV equations.
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
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Fig. 2
The plots of solutions (24) of the space time fractional generalized nonlinear Hirota–Statsuma coupled KdV equations.
Fig. 3
The plots of solutions (33) of the space time fractional nonlinear Whitham–Broer–Kaup equations.
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
The fractional complex transformation for nonlinear fractional partial differential equations (41) and collect all the terms with the same power of /j,(j = 0, ± 1, ± 2). Equating each coefficient to zero yields a set of the following algebraic equations: /2 :
k2 a1 þ ka21 þ a1 b1 pk ¼ 0 k2 b1 þ kb21 þ a1 b1 qk ¼ 0
/: b1 cþ2b0 b1 kþqkða0 b1 þa1 b0 Þ¼ 0 a1 cþ2a0 a1 kþpkða0 b1 þa1 b0 Þ¼ 0 /0 : a0 ck2 ða1 ba2 Þþkða20 þ2a1 a2 Þþpkða0 b0 þa1 b2 þa2 b1 Þ ¼0 b0 ck2 ðb1 bb2 Þþkðb20 þ2b1 b2 Þþqkða0 b0 þa1 b2 þa2 b1 Þ ¼0 /1 :
2
/
:
a2 c þ 2a0 a2 k þ pkða0 b2 þ a2 b0 Þ ¼ 0 b2 c þ 2b0 b2 k þ qkða0 b2 þ a2 b0 Þ ¼ 0 2
k b2 b þ k2 a2 b þ
kb22 ka22
Fig. 4
þ a2 b2 qk ¼ 0 þ a2 b2 pk ¼ 0
7
By solving the above set of algebraic equations by using Maple or Mathematica, we get the results Case 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi b < 0; c ¼ 2k2 b; a0 ¼ k b; p ¼ q ¼ a1 ¼ b2 ¼ 0; pffiffiffiffiffiffiffi b0 ¼ k b; a2 ¼ bk; b1 ¼ k In this case, we have the hyperbolic solutions: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ( uðx; tÞ ¼ k b þ pbkffiffiffiffi cothð bnÞ b pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi vðx; tÞ ¼ k b 1 tanhð bnÞ and pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ( uðx; tÞ ¼ k b þ pbkffiffiffiffi tanhð bnÞ b pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi vðx; tÞ ¼ k b 1 cothð bnÞ
ð44Þ
ð45Þ
Case 2
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi b < 0; p ¼ 0; q ¼ 2; c ¼ 2k2 b; a0 ¼ k b; p ¼ a1 ¼ b0 ¼ 0; a2 ¼ kb; b2 ¼ bk; b1 ¼ k
The plots of solutions (34) of the space time fractional nonlinear Whitham–Broer–Kaup equations.
Fig. 5
The plots of solutions (44) of the space time fractional nonlinear coupled Burgers equations.
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In this case the following solutions 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi < uðx; tÞ ¼ k b pbkffiffiffiffi cothð bnÞ b ffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi : vðx; tÞ ¼ kpffiffiffiffiffiffi tanhð bnÞ b cothð bnÞ pbkffiffiffiffi b and 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi < uðx; tÞ ¼ k b pbkffiffiffiffi tanhð bnÞ b ffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi : vðx; tÞ ¼ kpffiffiffiffiffiffi cothð bnÞ b tanhð bnÞ pbkffiffiffiffi b
ð46Þ
ð47Þ
On comparing our results (44)–(47) with the results obtained in (Liu et al., 2013) we conclude that our results are new. Example 4. The space–time fractional nonlinear coupled mKdV equations These equations are well-known (Zhao et al., 2012) and have the forms:
1 3 2a 2 a a a Dat u ¼ D3a x u 3u Dx u þ Dx u þ 3Dx ðuvÞ 3kDx u; 2 2
ð48Þ
a a a a 2 a Dat v ¼ D3a x v 3vDx v 3Dx uDx v þ 3u Dx v þ 3kDx v;
ð49Þ
where 0 < a 6 1 and k is constant. Eqs. (48) and (49) have been discussed in (Zhao et al, 2012) using the extended fractional sub-equation method. Let us now solve Eqs. (48) and (49) using the proposed method of Section 2. To this end we suppose that u(x,t) = U(n),v(x,t) = V(n). Then by the use of Eqs. (1) and (3), the system (48) and (49) can be turned into the following system of ODEs with integer orders: 1 3 ðc þ 3kkÞU k3 U00 þ kU3 k2 U0 3kUV ¼ 0; 2 2 cV0 ¼ k3 V000 3kVV0 3k2 U0 V0 þ 3kU2 V0 þ 3kkV0 ;
ð50Þ ð51Þ
Balancing the highest order of derivatives and highest nonlinear terms in Eqs. (50) and (51), we have the following formal solutions:
Fig. 6
The plots of solutions (46) of the space time fractional nonlinear coupled Burgers equations.
Fig. 7
The plots of solutions (54) of the space time fractional nonlinear coupled mKdV equations.
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
The fractional complex transformation for nonlinear fractional partial differential equations UðnÞ ¼ a0 þ a1 / þ a2 /1 ; 1
VðnÞ ¼ b0 þ b1 / þ b2 / ;
ð52Þ
Case 2.
ð53Þ
1 1 1 b ¼ ; c ¼ ; k ¼ 1; k ¼ k; a0 ¼ ; 4 2 2 1 a1 ¼ b1 ¼ 0; b0 ¼ k; b2 ¼ : 4
where ai,bi(i = 0,1,2) are constants to be determined later. We substitute (52) and (53) along with Eq. (9) into Eqs. (50) and (51) and collect all the terms with the same power of /j,(j = 0, ± 1, ± 2, ± 3). Equating each coefficient to zero yields the following set of algebraic equations: /4 : 6a1 k3 þ 3a1 b1 k3 3a21 b1 k ¼ 0 3
3
/ :
k a1 þ ¼0 3b21 k 6a0 a1 b1 k ¼ 0
/2 :
ðc 3kkÞb1 þ 8a1 bk3 þ 3b0 b1 k
and 8 pffiffiffiffiffiffiffi < uðx; tÞ ¼ 14 ½2 þ p1ffiffiffiffi tanhð bnÞ; b ffi : vðx; tÞ ¼ k ½4 þ p1ffiffiffiffi tanhðpffiffiffiffiffiffi bnÞ: 4 b
2
þ3k ð2a1 b1 b a1 b2 a2 b1 Þ ¼ 0 3a0 a21 k 32 k2 a1 3a1 b1 k ¼ 0 / : 3kb21 b 3kð2a0 a1 b1 b 2a0 a1 b2 þ 2a0 a2 b1 Þ ¼ 0 ðc þ 3kkÞa1 a1 bk3 þ kð3a20 a1 þ 3a2 a21 Þ 3kða0 b1 þ a1 b0 Þ ¼ 0
/0 :
Hence the solutions of the space–time fractional coupled mKdV Eqs. (48), (49) are given by: 8 pffiffiffiffiffiffiffi < uðx; tÞ ¼ 14 ½2 þ p1ffiffiffiffi cothð bnÞ; b ffi : vðx; tÞ ¼ k ½4 þ p1ffiffiffiffi cothðpffiffiffiffiffiffi bnÞ 4 b
ka31
9
ð56Þ
ð57Þ
On comparing our results (54)–(57) with the results obtained in (Liu et al., 2013) we conclude that our results are new.
ðc 3kkÞðb1 b b2 Þ þ k3 ð2a1 b2 2a2 bÞ þ 3k2 ð2a1 b2 b þ a1 b1 b2 2a2 b1 b þ a2 b2 Þ þ3kðb0 b1 b b0 b2 Þ 3kða21 b2 b þ a20 b1 b a20 b2 þ 2a1 a2 b1 b 2a1 a2 b2 þ a22 b1 Þ ¼ 0 ðc þ 3kkÞa0 þ kð3a30 þ 6a0 a2 a1 Þ 32 k2 ða1 b a2 Þ 3kða0 b0 þ a1 b2 þ a2 b1 Þ ¼ 0
/1 : 3b22 kþ3kð2a0 a1 bb2 2a0 a2 bb1 þ2a0 a2 b2 Þ¼ 0 ðcþ3kkÞa2 a2 bk3 þkð3a20 a2 þ3a1 a22 Þ3kða0 b2 þa2 b0 Þ¼0 /2 :
ðc 3kkÞb2 b þ 8a2 b2 k3 þ 3b0 b2 bk þ 3kða1 b2 b2 2a2 b2 b þ a2 b1 b2 Þ þ3kða20 b2 b 2a1 a2 b2 b þ a22 b1 b a22 b2 Þ ¼ 0 3a0 a22 k þ 32 k2 a2 b 3a2 b2 k ¼ 0
/3 :
3kbb22 þ 6a0 a2 bb2 k ¼ 0
4. Physical explanations of some obtained solutions
k3 a2 b2 þ ka32 ¼ 0 /4 : 6a2 b3 k3 þ 3a2 b2 b2 k2 þ 3a22 b2 bk ¼ 0 By solving the above set of algebraic equations by using Maple or Mathematica, we get the results Case 1
4.1. The Space–time fractional generalized nonlinear Hirota– Satsuma coupled KdV equations
1 k 1 ; c ¼ ; k ¼ k; k ¼ k; a0 ¼ ; 2 2 2 4k a1 ¼ k; a2 ¼ b2 ¼ 0; b0 ¼ k; b1 ¼ k: b¼
Hence the solutions of the space–time fractional coupled (mKdV) Eqs. (48), (49) are given by: 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi < uðx; tÞ ¼ 12 þ k b tanhð bnÞ; ffi pffiffiffiffiffiffiffi : vðx; tÞ ¼ k þ kpffiffiffiffiffiffi b tanhð bnÞ 2 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi < uðx; tÞ ¼ 12 þ k b cothð bnÞ; ffi pffiffiffiffiffiffiffi : vðx; tÞ ¼ k þ kpffiffiffiffiffiffi b cothð bnÞ 2
In this paper, we have obtained three types of solutions namely, hyperbolic, trigonometric and rational function solutions. In this section, we have presented some graphs of these types of solutions to visualize the underlying mechanism of the original equations. Using the mathematical software Maple 15, we will give some plots for these solutions.
ð54Þ
The obtained solutions of these equations incorporate three types of explicit solutions namely hyperbolic, trigonometric and rational function solutions (22)–(26) respectively. From these explicit results, the solutions (22) are kink solutions, (23) are singular kink solutions, while (24) and (25) are periodic solutions and (26) are rational solutions. For more convenience the graphical representations of (22) and (24) are shown in Figs. 1 and 2 respectively. 4.2. The Space–time fractional nonlinear Whitham–Broer–Kaup equations
ð55Þ
The obtained solutions of these equations incorporate two types of explicit solutions namely hyperbolic and trigonometric
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
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Fig. 8
The plots of solutions (57) of the space time fractional nonlinear coupled mKdV equations.
function solutions (33)–(35) respectively. From these explicit results, the solutions (33) represent kink solutions while (34) and (35) are periodic solutions. For more convenience the graphical representations of (33) and (34) are shown in Figs. 3 and 4 respectively.
Eq. (9). Besides, as this method is based on the homogeneous balancing principle, it can also be applied to other nonlinear fractional partial differential equations, where the homogeneous balancing principle is satisfied. Acknowledgments
4.3. The space–time fractional nonlinear coupled Burgers equations The obtained solutions of these equations incorporate hyperbolic function solutions (44)–(46) which represent the kink and singular kink solutions. For more convenience the graphical representations of (44) and (46) are shown in Figs. 5 and 6 respectively. 4.4. The space–time fractional nonlinear coupled mKdV equations The obtained solutions of these equations incorporate hyperbolic function solutions (54)–(57) which represent the kink and singular kink solutions. For more convenience the graphical representations of (54) and (57) are shown in Figs. 7 and 8 respectively. 5. Some conclusions In this paper, we have extended successfully the modified extended tanh- function method to solve four nonlinear fractional partial differential equations. As applications, abundant new exact solutions for the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space–time fractional nonlinear Whitham–Broer–Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations have been successfully found. As one can see, the nonlinear fractional complex transformation (6) for n is very important, which ensures that a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order, whose solutions can be expressed in the form (8) where /(n) satisfies the Riccati
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ORIGINAL ARTICLE
The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics Elsayed M.E. Zayed *, Yasser A. Amer, Reham M.A. Shohib Mathematics Department, Faculty of Sciences, Zagazig University, Zagazig, Egypt Received 24 November 2013; revised 19 June 2014; accepted 23 June 2014
KEYWORDS Nonlinear fractional partial differential equations; Modified extended tanh-function method; Nonlinear fractional complex transformation; Exact solutions
Abstract In this article, the modified extended tanh-function method is employed to solve fractional partial differential equations in the sense of the modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into nonlinear ordinary differential equations of integer orders. For illustrating the validity of this method, we apply it to four nonlinear equations namely, the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space– time fractional nonlinear Whitham–Broer–Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations. ª 2014 Production and hosting by Elsevier B.V. on behalf of University of Bahrain.
1. Introduction Fractional differential equations are the generalizations of classical differential equations with integer orders. In recent years, nonlinear fractional differential equations in mathematical physics are playing a major role in various fields, such as physics, biology, engineering, signal processing, and control theory, finance and fractal dynamics (Miller and Ross, 1993; Kilbas et al., 2006; Podlubny, 1999). Finding approximate and exact solutions to the fractional differential equations is an important task. A large amount of literatures were developed concerning the solutions of the fractional differential equations in nonlinear dynamics (El-sayed et al., 2009). Many * Corresponding author. E-mail addresses: [email protected] (E.M.E. [email protected] (Y.A. Amer). Peer review under responsibility of University of Bahrain.
Zayed),
powerful and efficient methods have been proposed to obtain the numerical and exact solutions of fractional differential equations. For example, these methods include the variational iteration method (Safari et al., 2009); Wu and Lee, 2010; Yang and Baleanu, 2012; Guo and Mei, 2011), the Lagrange characteristic method (Jumarie, 2006a), the homotopy analysis method (Song and Zhang, 2009), the Adomian decomposition method (El-Sayed and Gaber, 2006; El-sayed et al., 2010), the homotopy perturbation method (He, 1999; He, 2000; Yildirim and Gulkanat, 2010), the differential transformation method (Odibat and Momani, 2008), the finite difference method (Cui, 2009), the finite element method (Huang et al., 2009), the fractional sub-equation method (Zhang and Zhang, 2011; Guo et al., 2012; Lu, 2012), the (G0 /G)-expansion method (Zheng, 2012; Gepreel and Omran, 2011; Younis and Zafar, 2013), the modified extended tanh-function method (El-Wakil et al., 2005; El-Wakil et al., 2002; Soliman, 2006; Dai and Wang, 2014), the fractional complex transformation
http://dx.doi.org/10.1016/j.jaubas.2014.06.008 1815-3852 ª 2014 Production and hosting by Elsevier B.V. on behalf of University of Bahrain. Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
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E.M.E. Zayed et al.
method (Li and He, 2010; Li, 2010; He and Li, Li et al., 2012; Hristov, 2010), the exp-function method (He, 2013), the similarity transformation method (Dai et al., 2013; Zhu, 2013), the Hirota method (Liu et al., 2013) and so on. The objective of this paper is to apply the modified extended tanh-function method for solving fractional partial differential equations in the sense of the modified Riemann–Liouville derivative which has been derived by (Jumarie, 2006b). These equations can be reduced into nonlinear ordinary differential equations (ODE) with integer orders using some fractional complex transformations. Jumarie’s modified Riemann–Liouville derivative of order a is defined by the following expression: ( 1 d Rt ðt gÞa ½fðgÞ fð0Þdg; 0 < a 6 1; Cð1aÞ dt 0 a Dt fðtÞ ¼ ðanÞ fðnÞ ðtÞ ; n 6 a < n þ 1; n P 1 We list some important properties for the modified Riemann–Liouville derivative as follows: Cð1 þ rÞ ra t ;r > 0 Cð1 þ r aÞ
ð1Þ
Dat ½fðtÞgðtÞ ¼ fðtÞDat gðtÞ þ gðtÞDat fðtÞ
ð2Þ
Dat ½fðgðtÞÞ
ð3Þ
Dat tr ¼
¼
f0g ðgðtÞÞDat gðtÞ
Dat ½fðgðtÞÞ ¼ Dag fðgðtÞÞ½g0ðtÞa
Pðu; u0 ; u00 ; :::Þ ¼ 0;
ð7Þ
where P is a polynomial in u(n) and its total derivatives d2 u u0 ,u00 ,u000 ,. . . such that u0 ¼ du ; u00 ¼ dn 2 ; . . ... dn Step 2: We suppose that the formal solution of the ODE (7) can be expressed as follows: uðnÞ ¼ a0 þ
N X ½ai /i ðnÞ þ bi /i ðnÞ;
ð8Þ
i¼1
where /(n) satisfies the Riccati equation /0 ¼ b þ /2 ;
ð9Þ
where b is a constant. Fortunately, Eq. (9) admits several types of the following solutions: (i) If b < 0, we have the hyperbolic solutions; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi /ðnÞ ¼ b tanhð bnÞ; /ðnÞ ¼ b cothð bnÞ:
ð10Þ
(ii) If b > 0, we have the trigonometric solutions; pffiffiffi pffiffiffi pffiffiffi pffiffiffi /ðnÞ ¼ b tanð bnÞ; /ðnÞ ¼ b cotð bnÞ:
ð11Þ
(iii) If b=0, we have the rational solutions;
ð4Þ
where C denotes the Gamma function. The rest of this paper is organized as follows: In Section 2, the description of the modified extended tanh-function method for solving nonlinear fractional partial differential equations is given. In Section 3, we apply this method to establish the exact solutions for the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space–time fractional nonlinear Whitham–Broer– Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations. In Section 4 physical explanations of some obtained solutions are given. In Section 5, some conclusions are obtained.
/ðnÞ ¼
1 ; nþd
ð12Þ
where d is a constant. Step 3: We determine the positive integer N in (8) by balancing the highest nonlinear terms and the highest order derivatives of u(n) in Eq. (7). Step 4: We substitute (8) along with Eq. (9) into Eq. (7) and equate all the coefficients of /i(i = 0, ± 1, ± 2,...) to zero to yield a system of algebraic equations for ai,bi,c,k, b. Step 5: We solve the algebraic equations obtained in Step 4 using Mathematica or Maple, and use the well- known solutions (10)–(12) of Eq. (9) to obtain the exact solutions of Eq. (5). 3. Applications
2. Description the modified extended tanh-function method for solving nonlinear fractional partial differential equations
In this section, we construct the exact solutions of the following four nonlinear fractional partial differential equations using the proposed method of Section 2:
Suppose we have the following nonlinear fractional partial differential equation:
Example 1. The Space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations.
Fðu; Dat u; Dax u; :::Þ ¼ 0; 0 < a 6 1;
These equations are well-known (Guo et al., 2012; Zheng, 2012) and have the forms:
ð5Þ
where Dat u and Dax u are the modified Riemann–Liouville derivatives and F is a polynomial in u = u(x,t) and its fractional derivatives. In the following, we give the main steps of this method: Step 1: Using the nonlinear fractional complex transformation (Li and He, 2010; Li, 2010; He and Li, 2012; Li et al., 2012; Hristov, 2010). uðx; tÞ ¼ uðnÞ; n ¼
kxa cta þ þ n0 ; Cð1 þ aÞ Cð1 þ aÞ
ð6Þ
where k,c,n0 are constants with k,c „ 0, to reduce Eq. (5) to the following ODE of integer order with respect to the variable n :
1 Dat u D3a u þ 3uDax u 3Dax ðvwÞ ¼ 0; 2 x a Dat v þ D3a x v 3uDx v ¼ 0; a Dat w þ D3a x w 3uDx w ¼ 0;
ð13Þ ð14Þ ð15Þ
where 0 < a 6 1. Eqs. (13)–(15) can be used to describe the iteration of two long waves with different dispersion relations (Abazari and Abazari, 2012). When a = 1, Eqs. (13)–(15) were first proposed in (Wu and Geng, 1999). When 0 < a 6 1, Eqs. (13)–(15) have been discussed in (Zheng, 2012) using the (G0 /G)expansion method and in ( Guo et al, 2012) using the fractional sub-equation method. Let us now solve Eqs. (13)–(15) using the
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
The fractional complex transformation for nonlinear fractional partial differential equations /3 :
proposed method of Section 2. To this end, we suppose that u(x,t) = U(n),v(x,t) = V(n),w(x,t) = W(n) where n is given by the fractional complex transformation (6). Then by use of Eqs. (1) and (3), the system (13)–(15) can be turned into the following system of ODEs with integer orders: 1 3 cU k3 U00 þ kU2 3kVW ¼ 0; 2 2 cV0 þ k3 V000 3kUV0 ¼ 0;
1
WðnÞ ¼ c0 þ c1 / þ c2 /
/4 :
ð19Þ
2
2
ð20Þ
2
2
ð21Þ
þ b3 / þ b4 / ; þ c3 / þ c4 / ;
6b2 b3 k3 3kð2a2 b4 b þ a4 b2 bÞ ¼ 0 6c2 b3 k3 3kð2a2 c4 b þ a4 c2 bÞ ¼ 0 /5 :
24c3 k3 6kc3 a3 ¼ 0 24b3 k3 6kb3 a3 ¼ 0
/4 :
3a3 k3 þ 32 ka23 3kb3 c3 ¼ 0
24b4 b3 k3 6kb4 a4 ¼ 0 24c4 b3 k3 6kc4 ba4 ¼ 0
Solving the above set of algebraic equations by using Maple or Mathematica , we get the following results: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 8 b ¼ b;c ¼ 4ck 3 64 bc þ 32c þ 34b2 c23 120bc0 c3 þ 90c20 ; 3 0 3 3
where ai,bi,ci (i = 0,1,2,3,4) are constants to be determined later. We substitute (19)–(21) along with Eq. (9) into Eqs. (16)–(18) and collect all the terms with the same power of /j,(j = 0, ± 1, ± 2, ± 3,...). Equating each coefficient to zero yields a set of the following algebraic equations:
/5 :
3a4 b2 k3 32 ka24 þ 3kb4 c4 ¼ 0
ð17Þ
UðnÞ ¼ a0 þ a1 / þ a2 /1 þ a3 /2 þ a4 /2 ; VðnÞ ¼ b0 þ b1 / þ b2 /
2cbc4 þ 40c4 b2 k3 3kða2 c2 b þ 2a4 c4 þ 2a0 c4 bÞ ¼ 0
ð18Þ
Balancing the order of U00 with U2, V000 with UV0 and W000 with UW0 in Eqs. (16)–(18), we deduce that the formal solutions of Eqs. (16)–(18) have the forms: 1
a2 b2 k3 3a2 a4 k þ 3kðb2 c4 þ c2 b4 Þ ¼ 0 2cbb4 þ 40b4 b2 k3 3kða2 b2 b þ 2a4 b4 þ 2a0 b4 bÞ ¼ 0
ð16Þ
cW0 þ k3 W000 3kUW0 ¼ 0:
3
k ¼ k;a3 ¼ 4k2 ; 2
a0 ¼ 8ck 3
64 bc3 þ 643 c0 þ 169 9
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 34b2 c23 120bc0 c3 þ 90c20 ;
4
b3 ¼ 4kc3 ;c0 ¼ c0 ; a1 ¼ a2¼ a4 ¼ b1 ¼ b2 ¼ b4 ¼ c1 ¼ c2 ¼ c4 ¼ 0;c3 ¼ c3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 80 52 16 34b2 c23 120bc0 c3 þ 90c20 b0 ¼ k4 bc þ c þ 3 0 2 9 3 9 c 3
6b1 k3 3kð2a1 b3 þ b1 a3 Þ ¼ 0 6c1 k3 3kð2a1 c3 þ b1 c3 Þ ¼ 0 /3 :
a1 k3 þ 3a1 a3 k 3kðb1 c3 þ c1 b3 Þ ¼ 0 2cb3 þ 40b3 bk3 3kð2a0 b3 þ a1 b1 þ 2a3 b3 bÞ ¼ 0 2cc3 þ 40c3 bk3 3kð2a0 c3 þ a1 c1 þ 2a3 c3 bÞ ¼ 0
/2 :
ca3 4a3 bk3 þ 32 kða21 þ 2a0 a3 Þ 3kðb0 c3 þ b1 c1 þ b3 c0 Þ ¼ 0 b1 c þ 8b1 bk3 3kða0 b1 þ 2a1 b3 b þ 2a2 b3 þ a3 b1 b a3 b2 Þ ¼ 0 c1 c þ 8c1 bk3 3kða0 c1 þ 2a1 c3 b þ 2a2 c3 þ a3 c1 b a3 c2 Þ ¼ 0
/:
ca1 a1 bk3 þ 3kða0 a1 þ a2 a3 Þ 3kðb0 c1 þ b1 c0 þ b2 c3 þ b3 c2 Þ ¼ 0 2bcb3 þ 16b3 b2 k3 3kð2a0 b3 b þ a1 b1 b b2 a1 þ a2 b1 2a3 b4 þ 2a4 b3 Þ ¼ 0 2bcc3 þ 16c3 b2 k3 3kð2a0 c3 b þ a1 c1 b c2 a1 þ a2 c1 2a3 c4 þ 2a4 c3 Þ ¼ 0
/0 :
ca0 k3 ða3 b2 þ a4 Þ þ 32 kða20 þ 2a1 a2 þ a2 a4 Þ 3kðb0 c0 þ b1 c2 þ b2 c1 þ b3 c4 þ b4 c3 Þ ¼ 0 cðb1 b b2 Þ þ k3 ð2b1 b2 2b2 bÞ 3kða0 b1 b a0 b2 2a1 b4 þ 2a2 b3 b a3 b2 b þ a4 b1 Þ ¼ 0 cðc1 b c2 Þ þ k3 ð2c1 b2 2c2 bÞ 3kða0 c1 b a0 c2 2a1 c4 þ 2a2 c3 b a3 c2 b þ a4 c1 Þ ¼ 0
/1 :
ca2 a2 bk3 þ 3kða0 a2 þ a1 a4 Þ 3kðb0 c2 þ b1 c4 þ b2 c0 þ b4 c2 Þ ¼ 0 2cb4 þ 16b4 bk3 þ 3kð2a0 b4 a1 b2 b þ a2 b1 b b2 a2 2a3 b4 b þ 2a4 b3 bÞ ¼ 0 2cc4 þ 16c4 bk3 þ 3kð2a0 c4 a1 c2 b þ a2 c1 b c2 a2 2a3 c4 b þ 2a4 c3 bÞ ¼ 0
/2 :
ca4 4a4 bk3 þ 32 kða22 þ 2a0 a4 Þ 3kðb0 c4 þ b2 c2 þ b4 c0 Þ ¼ 0 b2 bc 8b2 b2 k3 þ 3kða0 b2 b þ 2a1 b4 b þ 2a2 b4 a4 b1 b þ a4 b2 Þ ¼ 0 c2 bc 8c2 b2 k3 þ 3kða0 c2 b þ 2a1 c4 b þ 2a2 c4 a4 c1 b þ a4 c2 Þ ¼ 0
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
4
E.M.E. Zayed et al.
Now, we have the following exact solutions:
(3), the system (27), (28) after integrating once can be turned into the following system of ODEs with integer orders:
(i) If b < 0, we have the hyperbolic solutions 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 > > uðx; tÞ ¼ 8ck 3 ½649 bc3 þ 643 c0 þ 169 34b2 c23 120bc0 c3 þ 90c20 4k2 btanh2 ð bnÞ > > > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 4 4 bc3 þ 523 c0 þ 169 34b2 c23 120bc0 c3 þ 90c20 4bk tanh2 ð bnÞ vðx; tÞ ¼ kc2 ½80 > 9 c3 > 3 > > > pffiffiffiffiffiffiffi : wðx; tÞ ¼ c0 c3 btanh2 ð bnÞ
ð22Þ 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi > 2 2 2 2 k2 64 64 16 2 > > > uðx; tÞ ¼ 8c3 ½ 9 bc3 þ 3 c0 þ 9 34b c3 120bc0 c3 þ 90c0 4k bcoth ð bnÞ > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 4 4 bc3 þ 523 c0 þ 169 34b2 c23 120bc0 c3 þ 90c20 4bk coth2 ð bnÞ vðx; tÞ ¼ kc2 ½80 > 9 c3 > 3 > > > pffiffiffiffiffiffiffi : wðx; tÞ ¼ c0 c3 bcoth2 ð bnÞ
ð23Þ
(ii) If b > 0, we have the trigonometric solutions 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi > 2 2 2 k2 64 64 16 2 2 > > > uðx; tÞ ¼ 8c3 ½ 9 bc3 þ 3 c0 þ 9 34b c3 120bc0 c3 þ 90c0 þ 4k b tan ð bnÞ > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 4 bc3 þ 523 c0 þ 169 34b2 c23 120bc0 c3 þ 90c20 þ 4bk tan2 ð bnÞ vðx; tÞ ¼ kc2 ½80 > 9 c3 > 3 > > > pffiffiffi : wðx; tÞ ¼ c0 þ c3 b tan2 ð bnÞ
1 cU þ kU2 þ kV þ bk2 U0 ¼ 0; 2 cV þ kUV bk2 V0 þ ck3 U00 ¼ 0;
UðnÞ ¼ a0 þ a1 / þ a2 /1 ; VðnÞ ¼ b0 þ b1 / þ b2 /
/2 :
1 2 ak 2 1
þ kb3 þ a1 bk2 ¼ 0
/:
a1 c þ a0 a1 k þ kb1 ¼ 0 b1 c þ kða0 b1 þ a1 b0 þ a2 b3 Þ 2bk2 bb3 þ 2a1 ck3 b ¼ 0
/0 :
a0 c þ 12 kða20 þ 2a1 a2 Þ þ kb0 þ bk2 ða1 b a2 Þ ¼ 0 b0 c þ kða0 b0 þ a1 b2 þ a2 b1 Þ bk2 ðbb1 b3 Þ ¼ 0 a2 c þ a0 a2 k þ kb2 ¼ 0
ð26Þ
/2 :
b4 c þ kða0 b4 þ a2 b2 Þ þ b2 bk2 ¼ 0 1 2 a k þ kb4 a2 bbk2 ¼ 0 2 2
/3 : a2 b4 k þ 2bb4 bk2 þ 2a2 b2 ck3 ¼ 0 On solving the above algebraic equations by using Maple or Mathematica, we have the following results: Case 1 2
b ¼ b; c ¼ 4k These equations are well-known (Xu et al., 2007; Xu et al., 2007; Ping, 2010) and have the forms:
þ
þ
cD3a x u
¼ 0;
ð27Þ ð28Þ
where 0 < a 6 1, and c,b are constants. In these equations u(x,t) is the field of horizontal velocity , v(x,t) is the height deviating from the equilibrium position of liquid, while c and b represent different diffusion powers. When a = 1, Eqs. (27) and (28) are the generalization of nonlinear Whitham–Broer–Kaup equations, which can be used to describe the dispersive long wave in shallow water ( Xu et al, 2007; Ping, 2010). Eqs. (27) and (28) have been discussed in Guo and Mei (2012) using the improved fractional sub-equation method. Let us now solve Eqs. (27), (28) using the proposed method of Section 2. To this end, we suppose that u(x,t) = U(n),v(x,t) = V(n). Then by the use of Eqs. (1) and
pffiffiffiffi bðb1Þ b ; a0 3b
pffiffiffiffi b
¼ 4kbðb1Þ 3b
a2 ¼ 2kbðb1Þ ; b1 ¼ b2 ¼ 0; b0 ¼ 4k 3 2 2
b4 ¼ 2k bD2a x v
ð32Þ
b2 c þ kða0 b2 þ a1 b4 þ a2 b0 Þ þ 2bk2 b4 þ 2ca2 bk3 ¼ 0
Example 2. The Space–time fractional nonlinear Whitham– Broer–Kaup equations
Dax ðuvÞ
þ b3 / þ b4 / ;
b3 c þ kða0 b3 þ a1 b1 Þ b1 bk2 ¼ 0
/1 :
On comparing our results (22)–(26) with the results in (Younis and Zafar, 2013) and (El-Wakil et al., 2005) we conclude that our results are new.
Dat v
ð31Þ 2
2
/3 : a1 b3 k 2b3 bk2 þ 2ca1 k3 ¼ 0
ðnþdÞ
Dat u þ uDax u þ Dax v þ bD2a x u ¼ 0;
1
where ai(i = 0,1,2),bi(i = 0,1,2,3,4) are constants to be determined later. We substitute (31) and (32) along with Eq. (9) into Eqs. (29) and (30) and collect all the terms with the same power of /j,(j = 0, ± 1, ± 2, ± 3). Equating each coefficient to zero yields a set of the following algebraic equations:
ð25Þ
(iii) If b=0, we have the rational solutions 8 pffiffiffiffiffi 2 4k2 > uðx; tÞ ¼ 3ck 3 ð8 þ 2 10Þc0 þ ðnþdÞ 2 > > < p ffiffiffiffiffi 4 4bk4 vðx; tÞ ¼ 3ck 2 ð52 þ 16 10Þc0 þ c ðnþdÞ 2 3 > 3 > > : wðx; tÞ ¼ c0 þ c3 2
ð30Þ
with zero constants of integration. Balancing the highest order derivatives and highest nonlinear terms in Eqs. (29) and (30), we have the following formal solutions:
ð24Þ 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi > 2 2 2 k2 64 64 16 2 2 > > > uðx; tÞ ¼ 8c3 ½ 9 bc3 þ 3 c0 þ 9 34b c3 120bc0 c3 þ 90c0 þ 4k bcot ð bnÞ > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 4 bc3 þ 523 c0 þ 169 34b2 c23 120bc0 c3 þ 90c20 þ 4bk cot2 ð bnÞ vðx; tÞ ¼ kc2 ½80 > 9 c3 > 3 > > > pffiffiffi : wðx; tÞ ¼ c0 þ c3 bcot2 ð bnÞ
ð29Þ
b ð2b2 b1Þ ;c 9
¼ b
2
2 2
; b < 0; a1 ¼ 2kbðb1Þ ; 3b
2
b ð2b b1Þ ; b3 9b
2 2
¼ 2k
b ð2b2 b1Þ ; 9b2
ð8b2 þ2b1Þ 9b2
In this case we have the hyperbolic solutions 8 pffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi < uðx; tÞ ¼ 2kbðb1Þ b 1 ½tanhð bnÞ þ cothð bnÞ 3b ffi pffiffiffiffiffiffiffi : vðx; tÞ ¼ 2k2 b2 ð2b2 b1Þ 2 ½tanh2 ðpffiffiffiffiffiffi bnÞ þ coth2 ð bnÞ 3b
ð33Þ Case 2 pffiffi pffiffi 2 2k bðb1Þ b ; a0 3b 2kbðb1Þ 1Þ; a2 ¼ 3 ;
b ¼ b; b ¼ b; c ¼ 2 a1 ¼
2 kbðb 3b
b0 ¼ b1 ¼ b2 ¼ 0; b3 ¼ 2k
2 2
pffiffi pffiffi 2kbðb1Þ b ;b 3b
¼ 2
b ð4b2 5b1Þ ;c 9b2
2
¼ b
> 0;
ð8b2 þ2b1Þ ; 9b2
2 2
b4 ¼ 2k
b ð2b2 b1Þ 9
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
The fractional complex transformation for nonlinear fractional partial differential equations In this case we have the trigonometric solutions 8 pffiffiffi pffiffiffi pffiffiffi < uðx; tÞ ¼ 2kbðb1Þ pffiffi 2 þ ½tanð bnÞ þ cotð bnÞ 3 b : vðx;tÞ ¼ 2k2 b2 ðb1Þ ð4b 1Þ tan2 ðpffiffibffinÞ ð2b þ 1Þcot2 ðpffiffibffinÞ 9b ð34Þ and 8 pffiffiffi pffiffiffi pffiffiffi < uðx; tÞ ¼ 2kbðb1Þ pffiffi 2 ½tanð bnÞ þ cotð bnÞ 3 b : vðx;tÞ ¼ 2k2 b2 ðb1Þ ð4b 1Þcot2 ðpffiffibffinÞ ð2b þ 1Þtan2 ðpffiffibffinÞ 9b ð35Þ On comparing our results (33)–(35) with the results obtained in (Younis and Zafar, 2013; Dai et al., 2013; Zhu, 2013) we conclude that our results are new.
where 0 < a 6 1 and p, q are constants. Eqs. (36) and (37) have been discussed in (Zhao et al, 2012) using the extended fractional sub-equation method. Let us now solve Eqs. (36) and (37) using the proposed method of Section 2. To this end we suppose that u(x,t) = U(n), v(x,t) = V(n) . Then by the use of Eqs. (1) and (3), the system (36) and (37) can be turned into the following system of ODEs with integer orders: cU0 k2 U00 þ 2kUU0 þ pkðUVÞ0 ¼ 0;
ð38Þ
cV0 k2 V00 þ 2kVV0 þ qkðUVÞ0 ¼ 0;
ð39Þ
Integrating Eqs. (38) and (39) with vanishing the constants of integration, we get cU k2 U0 þ kU2 þ pkUV ¼ 0; 2
0
2
cV k V þ kV þ qkUV ¼ 0; Example 3. The space–time fractional nonlinear coupled Burgers equations These equations are well-known (Zhao et al, 2012) and have the forms: ð36Þ
a a Dat v D2a x v þ 2vDx v þ qDx ðuvÞ ¼ 0;
ð37Þ
Fig. 1
ð40Þ ð41Þ
Balancing the highest order of derivatives and highest nonlinear terms in Eqs. (40) and (41), we have the following formal solutions: UðnÞ ¼ a0 þ a1 / þ a2 /1 ; 1
a a Dat u D2a x u þ 2uDx u þ pDx ðuvÞ ¼ 0;
5
VðnÞ ¼ b0 þ b1 / þ b2 / ;
ð42Þ ð43Þ
where ai,bi(i = 0,1,2) are constants to be determined later. We substitute (42) and (43) along with Eq. (9) into Eqs. (40) and
The plots of solutions (22) of the space time fractional generalized nonlinear Hirota–Statsuma coupled KdV equations.
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
6
E.M.E. Zayed et al.
Fig. 2
The plots of solutions (24) of the space time fractional generalized nonlinear Hirota–Statsuma coupled KdV equations.
Fig. 3
The plots of solutions (33) of the space time fractional nonlinear Whitham–Broer–Kaup equations.
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
The fractional complex transformation for nonlinear fractional partial differential equations (41) and collect all the terms with the same power of /j,(j = 0, ± 1, ± 2). Equating each coefficient to zero yields a set of the following algebraic equations: /2 :
k2 a1 þ ka21 þ a1 b1 pk ¼ 0 k2 b1 þ kb21 þ a1 b1 qk ¼ 0
/: b1 cþ2b0 b1 kþqkða0 b1 þa1 b0 Þ¼ 0 a1 cþ2a0 a1 kþpkða0 b1 þa1 b0 Þ¼ 0 /0 : a0 ck2 ða1 ba2 Þþkða20 þ2a1 a2 Þþpkða0 b0 þa1 b2 þa2 b1 Þ ¼0 b0 ck2 ðb1 bb2 Þþkðb20 þ2b1 b2 Þþqkða0 b0 þa1 b2 þa2 b1 Þ ¼0 /1 :
2
/
:
a2 c þ 2a0 a2 k þ pkða0 b2 þ a2 b0 Þ ¼ 0 b2 c þ 2b0 b2 k þ qkða0 b2 þ a2 b0 Þ ¼ 0 2
k b2 b þ k2 a2 b þ
kb22 ka22
Fig. 4
þ a2 b2 qk ¼ 0 þ a2 b2 pk ¼ 0
7
By solving the above set of algebraic equations by using Maple or Mathematica, we get the results Case 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi b < 0; c ¼ 2k2 b; a0 ¼ k b; p ¼ q ¼ a1 ¼ b2 ¼ 0; pffiffiffiffiffiffiffi b0 ¼ k b; a2 ¼ bk; b1 ¼ k In this case, we have the hyperbolic solutions: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ( uðx; tÞ ¼ k b þ pbkffiffiffiffi cothð bnÞ b pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi vðx; tÞ ¼ k b 1 tanhð bnÞ and pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ( uðx; tÞ ¼ k b þ pbkffiffiffiffi tanhð bnÞ b pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi vðx; tÞ ¼ k b 1 cothð bnÞ
ð44Þ
ð45Þ
Case 2
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi b < 0; p ¼ 0; q ¼ 2; c ¼ 2k2 b; a0 ¼ k b; p ¼ a1 ¼ b0 ¼ 0; a2 ¼ kb; b2 ¼ bk; b1 ¼ k
The plots of solutions (34) of the space time fractional nonlinear Whitham–Broer–Kaup equations.
Fig. 5
The plots of solutions (44) of the space time fractional nonlinear coupled Burgers equations.
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
8
E.M.E. Zayed et al.
In this case the following solutions 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi < uðx; tÞ ¼ k b pbkffiffiffiffi cothð bnÞ b ffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi : vðx; tÞ ¼ kpffiffiffiffiffiffi tanhð bnÞ b cothð bnÞ pbkffiffiffiffi b and 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi < uðx; tÞ ¼ k b pbkffiffiffiffi tanhð bnÞ b ffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi : vðx; tÞ ¼ kpffiffiffiffiffiffi cothð bnÞ b tanhð bnÞ pbkffiffiffiffi b
ð46Þ
ð47Þ
On comparing our results (44)–(47) with the results obtained in (Liu et al., 2013) we conclude that our results are new. Example 4. The space–time fractional nonlinear coupled mKdV equations These equations are well-known (Zhao et al., 2012) and have the forms:
1 3 2a 2 a a a Dat u ¼ D3a x u 3u Dx u þ Dx u þ 3Dx ðuvÞ 3kDx u; 2 2
ð48Þ
a a a a 2 a Dat v ¼ D3a x v 3vDx v 3Dx uDx v þ 3u Dx v þ 3kDx v;
ð49Þ
where 0 < a 6 1 and k is constant. Eqs. (48) and (49) have been discussed in (Zhao et al, 2012) using the extended fractional sub-equation method. Let us now solve Eqs. (48) and (49) using the proposed method of Section 2. To this end we suppose that u(x,t) = U(n),v(x,t) = V(n). Then by the use of Eqs. (1) and (3), the system (48) and (49) can be turned into the following system of ODEs with integer orders: 1 3 ðc þ 3kkÞU k3 U00 þ kU3 k2 U0 3kUV ¼ 0; 2 2 cV0 ¼ k3 V000 3kVV0 3k2 U0 V0 þ 3kU2 V0 þ 3kkV0 ;
ð50Þ ð51Þ
Balancing the highest order of derivatives and highest nonlinear terms in Eqs. (50) and (51), we have the following formal solutions:
Fig. 6
The plots of solutions (46) of the space time fractional nonlinear coupled Burgers equations.
Fig. 7
The plots of solutions (54) of the space time fractional nonlinear coupled mKdV equations.
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
The fractional complex transformation for nonlinear fractional partial differential equations UðnÞ ¼ a0 þ a1 / þ a2 /1 ; 1
VðnÞ ¼ b0 þ b1 / þ b2 / ;
ð52Þ
Case 2.
ð53Þ
1 1 1 b ¼ ; c ¼ ; k ¼ 1; k ¼ k; a0 ¼ ; 4 2 2 1 a1 ¼ b1 ¼ 0; b0 ¼ k; b2 ¼ : 4
where ai,bi(i = 0,1,2) are constants to be determined later. We substitute (52) and (53) along with Eq. (9) into Eqs. (50) and (51) and collect all the terms with the same power of /j,(j = 0, ± 1, ± 2, ± 3). Equating each coefficient to zero yields the following set of algebraic equations: /4 : 6a1 k3 þ 3a1 b1 k3 3a21 b1 k ¼ 0 3
3
/ :
k a1 þ ¼0 3b21 k 6a0 a1 b1 k ¼ 0
/2 :
ðc 3kkÞb1 þ 8a1 bk3 þ 3b0 b1 k
and 8 pffiffiffiffiffiffiffi < uðx; tÞ ¼ 14 ½2 þ p1ffiffiffiffi tanhð bnÞ; b ffi : vðx; tÞ ¼ k ½4 þ p1ffiffiffiffi tanhðpffiffiffiffiffiffi bnÞ: 4 b
2
þ3k ð2a1 b1 b a1 b2 a2 b1 Þ ¼ 0 3a0 a21 k 32 k2 a1 3a1 b1 k ¼ 0 / : 3kb21 b 3kð2a0 a1 b1 b 2a0 a1 b2 þ 2a0 a2 b1 Þ ¼ 0 ðc þ 3kkÞa1 a1 bk3 þ kð3a20 a1 þ 3a2 a21 Þ 3kða0 b1 þ a1 b0 Þ ¼ 0
/0 :
Hence the solutions of the space–time fractional coupled mKdV Eqs. (48), (49) are given by: 8 pffiffiffiffiffiffiffi < uðx; tÞ ¼ 14 ½2 þ p1ffiffiffiffi cothð bnÞ; b ffi : vðx; tÞ ¼ k ½4 þ p1ffiffiffiffi cothðpffiffiffiffiffiffi bnÞ 4 b
ka31
9
ð56Þ
ð57Þ
On comparing our results (54)–(57) with the results obtained in (Liu et al., 2013) we conclude that our results are new.
ðc 3kkÞðb1 b b2 Þ þ k3 ð2a1 b2 2a2 bÞ þ 3k2 ð2a1 b2 b þ a1 b1 b2 2a2 b1 b þ a2 b2 Þ þ3kðb0 b1 b b0 b2 Þ 3kða21 b2 b þ a20 b1 b a20 b2 þ 2a1 a2 b1 b 2a1 a2 b2 þ a22 b1 Þ ¼ 0 ðc þ 3kkÞa0 þ kð3a30 þ 6a0 a2 a1 Þ 32 k2 ða1 b a2 Þ 3kða0 b0 þ a1 b2 þ a2 b1 Þ ¼ 0
/1 : 3b22 kþ3kð2a0 a1 bb2 2a0 a2 bb1 þ2a0 a2 b2 Þ¼ 0 ðcþ3kkÞa2 a2 bk3 þkð3a20 a2 þ3a1 a22 Þ3kða0 b2 þa2 b0 Þ¼0 /2 :
ðc 3kkÞb2 b þ 8a2 b2 k3 þ 3b0 b2 bk þ 3kða1 b2 b2 2a2 b2 b þ a2 b1 b2 Þ þ3kða20 b2 b 2a1 a2 b2 b þ a22 b1 b a22 b2 Þ ¼ 0 3a0 a22 k þ 32 k2 a2 b 3a2 b2 k ¼ 0
/3 :
3kbb22 þ 6a0 a2 bb2 k ¼ 0
4. Physical explanations of some obtained solutions
k3 a2 b2 þ ka32 ¼ 0 /4 : 6a2 b3 k3 þ 3a2 b2 b2 k2 þ 3a22 b2 bk ¼ 0 By solving the above set of algebraic equations by using Maple or Mathematica, we get the results Case 1
4.1. The Space–time fractional generalized nonlinear Hirota– Satsuma coupled KdV equations
1 k 1 ; c ¼ ; k ¼ k; k ¼ k; a0 ¼ ; 2 2 2 4k a1 ¼ k; a2 ¼ b2 ¼ 0; b0 ¼ k; b1 ¼ k: b¼
Hence the solutions of the space–time fractional coupled (mKdV) Eqs. (48), (49) are given by: 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi < uðx; tÞ ¼ 12 þ k b tanhð bnÞ; ffi pffiffiffiffiffiffiffi : vðx; tÞ ¼ k þ kpffiffiffiffiffiffi b tanhð bnÞ 2 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi < uðx; tÞ ¼ 12 þ k b cothð bnÞ; ffi pffiffiffiffiffiffiffi : vðx; tÞ ¼ k þ kpffiffiffiffiffiffi b cothð bnÞ 2
In this paper, we have obtained three types of solutions namely, hyperbolic, trigonometric and rational function solutions. In this section, we have presented some graphs of these types of solutions to visualize the underlying mechanism of the original equations. Using the mathematical software Maple 15, we will give some plots for these solutions.
ð54Þ
The obtained solutions of these equations incorporate three types of explicit solutions namely hyperbolic, trigonometric and rational function solutions (22)–(26) respectively. From these explicit results, the solutions (22) are kink solutions, (23) are singular kink solutions, while (24) and (25) are periodic solutions and (26) are rational solutions. For more convenience the graphical representations of (22) and (24) are shown in Figs. 1 and 2 respectively. 4.2. The Space–time fractional nonlinear Whitham–Broer–Kaup equations
ð55Þ
The obtained solutions of these equations incorporate two types of explicit solutions namely hyperbolic and trigonometric
Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008
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E.M.E. Zayed et al.
Fig. 8
The plots of solutions (57) of the space time fractional nonlinear coupled mKdV equations.
function solutions (33)–(35) respectively. From these explicit results, the solutions (33) represent kink solutions while (34) and (35) are periodic solutions. For more convenience the graphical representations of (33) and (34) are shown in Figs. 3 and 4 respectively.
Eq. (9). Besides, as this method is based on the homogeneous balancing principle, it can also be applied to other nonlinear fractional partial differential equations, where the homogeneous balancing principle is satisfied. Acknowledgments
4.3. The space–time fractional nonlinear coupled Burgers equations The obtained solutions of these equations incorporate hyperbolic function solutions (44)–(46) which represent the kink and singular kink solutions. For more convenience the graphical representations of (44) and (46) are shown in Figs. 5 and 6 respectively. 4.4. The space–time fractional nonlinear coupled mKdV equations The obtained solutions of these equations incorporate hyperbolic function solutions (54)–(57) which represent the kink and singular kink solutions. For more convenience the graphical representations of (54) and (57) are shown in Figs. 7 and 8 respectively. 5. Some conclusions In this paper, we have extended successfully the modified extended tanh- function method to solve four nonlinear fractional partial differential equations. As applications, abundant new exact solutions for the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space–time fractional nonlinear Whitham–Broer–Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations have been successfully found. As one can see, the nonlinear fractional complex transformation (6) for n is very important, which ensures that a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order, whose solutions can be expressed in the form (8) where /(n) satisfies the Riccati
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Please cite this article in press as: Zayed, E.M.E. et al., The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.06.008