- Email: [email protected]
The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics
Journal of the Association of Arab Universities for Basic and Applied Sciences (2017) 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
The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics Ozkan Guner a,*, Ahmet Bekir b a Cankiri Karatekin University, Faculty of Economics and Administrative Sciences, Department of International Trade, Cankiri, Turkey b Eskisehir Osmangazi University, Art-Science Faculty, Department of Mathematics-Computer, Eskisehir, Turkey
Received 3 October 2016; revised 21 November 2016; accepted 9 December 2016
KEYWORDS Exact solution; Space–time fractional Telegraph equation; Space–time fractional KPP equation; Exp-function method
Abstract Using the Exp-function method, we derive exact solutions of the nonlinear space–time fractional Telegraph equation and space–time fractional KPP equation. As a result, we obtain many exact analytical solutions including hyperbolic function. The fractional derivative is described in Jumarie’s modified Riemann–Liouville sense. This method is very effective and convenient for solving nonlinear fractional differential equations. Ó 2016 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders. Fractional differential equations (FDEs) are used in various fields of physics, mathematics and engineering. So, they have gained much attention many mathematicians. Applications of FDEs for instance; signal processing, control theory, viscoelastic materials, systems identification, biomedical sciences, biology, finance, fractional dynamics as well as other disciplines (Miller and Ross, 1993;Podlubny, 1999; Kilbas et al., 2006). In the last two decades, a lot of attention is paid to finding appropriate solutions of FDEs. There are many effective methods to obtain numerical and analytical solutions of these kinds * Corresponding author. E-mail addresses: [email protected] (O. [email protected] (A. Bekir). Peer review under responsibility of University of Bahrain.
Guner),
of FDEs, such as homotopy analysis method, adomian decomposition method, variational iteration method, homotopy perturbation method (Daftardar-Gejji and Bhalekar, 2008; Erturk et al., 2008; Gepreel and Mohamed, 2013; Sweilam et al., 2007; Gepreel, 2011; Mohamed et al., 2012) and the tanh method, the Exp-function method, the first integral method, the functional variable method, the ðG0=GÞ-expansion method, the sub-equation method, the modified trial equation method, the simplest equation method, the generalized Kudryashov method, the ansatz method and so on (Naher et al., 2013; Zheng, 2012, 2013; Bekir and Guner, 2013, 2014; Bekir et al., 2015a,b,c; Bekir et al., 2016; Bulut et al., 2013; Yan and Xu, 2015; Zhang et al., 2010; Guner, 2015; Guner and Bekir, 2016; Guner et al., 2015a,b; Taghizadeh et al., 2013; Khan and Akbar, 2013, 2014; Lu, 2012; Eslami et al., 2014; Zhang and Zhang, 2011; Liu and Chen, 2013). The Exp-function method was first proposed by He and Wu in 2006 (He, 2006) and systematically studied in He and Abdou (2007), Noor et al. (2008), Ebaid (2012) and
http://dx.doi.org/10.1016/j.jaubas.2016.12.002 1815-3852 Ó 2016 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002
2
O. Guner, A. Bekir
Navickas et al. (2010). The Exp-function method was originally proposed to solve PDEs. However, its application to fractional calculus is rare and primary. In 2013, this method was successfully extended to fractional calculus, and it becomes an effective tool for fractional differential equations (see He, 2013; Guner and Bekir, 2017). There are different kinds of fractional derivative operators. The most famous one is the Caputo definition that the function should be differentiable (Caputo, 1967). Recently, Jumarie derived definitions for the fractional derivative called modified Riemann–Liouville, which are suitable for continuous and non-differentiable functions. This paper adopts the Jumarieı´ s derivative, which has been disadvantages, and has already been updated by some new ones, for example AtanganaBaleanu derivative and He derivative (Atangana and Koca, 2016;Atangana and Baleanu, 2016; Sayevand and Pichaghchi, 2016; He, 2011). The order a of Jumarie’s derivative is defined by Jumarie (2006, 2009): ( Daw fðwÞ ¼
1 d Cð1aÞ dw
Rw 0
;
h 6 a < h þ 1; h P 1: ð1:1Þ
where f : R ! R; w ! fðwÞ denotes a continuous (but not necessarily differentiable) function and satisfying above properties Daw wc ¼
Cð1 þ cÞ wca Cð1 þ c aÞ
c > 0;
ð1:2Þ
Daw ðCfðwÞÞ ¼ CDaw fðwÞ;
ð1:3Þ
Daw fafðwÞ þ bgðwÞg ¼ aDaw fðwÞ þ bDaw gðwÞ;
ð1:4Þ
Daw C ¼ 0;
ð1:5Þ
where a; b and C are constant. The outline of the present paper is as follows. In Section 2, we have a brief description of the Exp-function method for solving fractional partial differential equations. In Section 3, we apply the Exp-function method on the nonlinear space–time fractional Telegraph equation and the nonlinear fractional Kolmogorov Petrovskii-Piskunov (KPP) equation respectively. Finally, some conclusions are given in Section 4. 2. Description of the Exp-function method for solving fractional partial differential equations We take into consideration the following general nonlinear FDE of the type a b 2b Fðu; Dat u; Dbx u; D2a t u; Dt Dx u; Dx u; . . .Þ ¼ 0;
uðx; yÞ ¼ UðnÞ; n¼
kxb cta ; Cð1 þ bÞ Cð1 þ aÞ
0 < a; b 6 1 ð2:1Þ
where F is a polynomial of u; u is an unknown function and Da partial fractional derivative of u, in which the highest order derivatives and the nonlinear terms are involved. The fractional complex transform was proposed by He and Li to convert a fractal space (time) to its continuous partner, the physical basis was illustrated by He in his recent review articles (Li and He, 2010;He et al., 2012; Hu and He, 2016). Wu and Liang gave a very clear mathematic analysis of relationship between fractal dimensions and fractional calculus in Wu and Liang (2017).
ð2:2Þ ð2:3Þ
where k – 0 and c – 0 are constants. The fractional chain rule is defined as dU a Dn dn t dU a D n Dax u ¼ rx dn x Dat u ¼ rt
ð2:4Þ
where rt and rx are named as sigma indexes (Aksoy et al., 2016). We can choose rt ¼ rx ¼ L, where L is a constant. When we substitute, (2.2) with (1.2) and (2.4) into (2.1), we can rewrite Eq. (2.1) in the following NODE; QðU; U0 ; U00 ; U000 ; . . .Þ ¼ 0:
ðw nÞa ðfðnÞ fð0ÞÞdn; 0 < a < 1
ðahÞ
ðfðhÞ ðwÞÞ
Using the fractional complex transform
ð2:5Þ
where UðnÞ is the nth derivative of U with respect to n. Suppose that the solution of NODE (2.5) can be expressed in the following form Pd an exp ½nn UðnÞ ¼ Pqn¼c ð2:6Þ b m¼p m exp ½mn where p; q; c and d are positive integers which are known to be further determined, an and bm are unknown constants. To determine the value of c and p, we balance the linear term of the lowest order of equation Eq. (2.5) with the lowest order nonlinear term. Similarly, the value of d and q can be determined. 3. Applications of the proposed method Example 1: We consider the nonlinear space–time fractional Telegraph equation (Jafari et al., 2014) 2a a 3 D2a tt u Dxx u þ Dt u þ cu þ bu ¼ 0;
ð3:1Þ
a is a parameter describing the order of the fractional space and time derivative. When a ¼ 1, Eq. (3.1) is called the nonlinear Telegraph equation. Wang and Li, presented to construct the traveling wave solutions of this equation (Wang and Li, 2008). Also, Jafari et al. applied the sub equation method Eq. (3.1) and obtained the many exact solutions. Zayed and Al-Nowehy solved the equation by the generalized Kudryashov method and obtained the exact solutions of nonlinear PDEs in mathematical physics (Zayed and Al-Nowehy, 2016). Using the fractional complex transform, uðx; tÞ ¼ UðnÞ; n¼
kxa cta ; Cð1 þ aÞ Cð1 þ aÞ
ð3:2Þ ð3:3Þ
where k and c are constants. When we substitute (3.3) with (1.2) and (2.4) into (3.1), this Eq. (3.1) can reduced to an ODE ðc2 k2 ÞU00 cU0 þ cU þ bU3 ¼ 0; 0
where U ¼
ð3:4Þ
dU . dn
Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002
The Exp-function method for in mathematical physics
3
Balancing the order of U00 and U3 in Eq. (3.4), we obtain c1 exp½ðc þ 3pÞn þ ; U ¼ c2 exp½4pn þ 00
ð3:5Þ
If we take b1 ¼ 1 and b0 ¼ 1 Eq. (3.16) becomes u3;4 ðx; tÞ ¼
pffiffiffi pffiffiffi b þ b cosh
pffiffiffiffiffiffiffiffiffiffi
pffiffiffi i c
9c2 2cxa 3cta 2Cð1þaÞ
pffiffiffi b sinh
pffiffiffiffiffiffiffiffiffiffi
9c2 2cxa 3cta 2Cð1þaÞ
ð3:17Þ
and U3 ¼
:
c3 exp½3cn þ ; c4 exp½3pn þ
ð3:6Þ
where ci is determined coefficient only for simplicity. Balancing the lowest order of Exp-function in Eqs. (3.5) and (3.6) we obtain ðc þ 3pÞ ¼ ð3c þ pÞ;
ð3:7Þ
which leads to the result p ¼ c:
ð3:8Þ
In the same way also we have 3q þ d ¼ 4d;
ð3:9Þ
and this gives q ¼ d:
ð3:10Þ
If we set p ¼ c ¼ 1 and q ¼ d ¼ 1, Eq. (2.5) reduces to 0 þa1 expðnÞ UðnÞ ¼ ab11 expðnÞþa : expðnÞþb0 þb1 expðnÞ
ð3:11Þ
When we substitute Eq. (3.11) into Eq. (3.1), by the help of Maple, we get the following results, Case 1: qffiffiffiffiffiffiffi a1 ¼ 0; a1 ¼ b1 bc ; a0 ¼ 0; b1 ¼ b1 ;
b0 ¼ 0; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 4 :
c ¼ 3c4 ;
ð3:12Þ
b1 ¼ b1 ;
where b1 is free parameter. When substitute these results into (3.11), we get the following exact solution pffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffi 9c2 2cxa 3cta bc exp 4Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi : 9c2 2cxa 3cta 9c2 2cxa 3cta þ b1 exp 4Cð1þaÞ 4Cð1þaÞ
b1 pffiffiffiffiffiffiffiffiffiffi
u1;2 ðx;tÞ ¼ b1 exp
ð3:13Þ
If we take b1 ¼ 1 and b1 ¼ 1 Eq. (3.13) becomes pffiffiffi i c cosh u1;2 ðx;tÞ ¼
pffiffiffiffiffiffiffiffiffiffi
4Cð1þaÞ
pffiffiffi 2 b cosh
pffiffiffiffiffiffiffiffiffiffi
pffiffiffi þ i c sinh pffiffiffiffiffiffiffiffiffiffi
9c2 2cxa 3cta
9c2 2cxa 3cta 4Cð1þaÞ
9c2 2cxa 3cta 4Cð1þaÞ
:
b1 ¼ 0; c ¼ 3c2 ;
b0 ¼ b0 ; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 2 :
a1 ¼ 0; b1 ¼ b1 ;
a0 ¼ 0;
a1 ¼ b1
b1 ¼ 0;
b0 ¼ b0 ; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 2 :
b1 ¼ b1 ;
c¼
3c2 ;
qffiffiffiffiffiffiffi bc ; ð3:18Þ
where b1 is free parameter. Similarly, we obtain pffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi 9c2 2cxa þ3cta b1 bc exp 2Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi : u5;6 ðx; tÞ ¼ 2 9c 2cxa þ3cta b0 þ b1 exp 2Cð1þaÞ
ð3:19Þ
Eq. (3.19) becomes, if we take b1 ¼ 1 and b0 ¼ 1 pffiffiffiffiffiffiffiffiffiffi pffiffiffi 9c2 2cxa þ3cta i c sinh 2Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi : u5;6 ðx;tÞ ¼ pffiffiffi pffiffiffi pffiffiffi 9c2 2cxa þ3cta 9c2 2cxa þ3cta b sinh b þ b cosh 2Cð1þaÞ 2Cð1þaÞ pffiffiffi i c cosh
pffiffiffiffiffiffiffiffiffiffi
9c2 2cxa þ3cta 2Cð1þaÞ
ð3:20Þ
where i ¼ 1. Case 4: qffiffiffiffiffiffiffi a1 ¼ b1 bc ; 2
b1 ¼ b1 ; c ¼ 3c2 ;
a0 ¼ 0;
a1 ¼ 0;
b0 ¼ b0 ; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 2 :
b1 ¼ 0;
where b1 is free parameter. In the same way we get pffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi 9c2 2cxa 3cta b1 bc exp 2Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi : u7;8 ðx; tÞ ¼ 9c2 2cxa 3cta b0 þ b1 exp 2Cð1þaÞ
ð3:21Þ
ð3:22Þ
Eq. (3.22) becomes, when we take b1 ¼ 1 and b0 ¼ 1 pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi pffiffiffi 9c2 2cxa 3cta þ i c sinh 2Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi : u7;8 ðx;tÞ ¼ pffiffiffi pffiffiffi pffiffiffi 9c2 2cxa 3cta 9c2 2cxa 3cta þ b sinh b þ b cosh 2Cð1þaÞ 2Cð1þaÞ
9c2 2cxa 3cta 2Cð1þaÞ
ð3:23Þ
where i2 ¼ 1. Case 2: a1 ¼ 0;
a1 ¼ 0;
pffiffiffi i c cosh
ð3:14Þ
qffiffiffiffiffiffiffi a0 ¼ b0 bc ;
where i2 ¼ 1. Case 3:
ð3:15Þ
where b0 is free parameter. Substituting these results into (3.11), we get qffiffiffiffiffiffiffi b0 bc pffiffiffiffiffiffiffiffiffiffi : u3;4 ðx; tÞ ¼ ð3:16Þ 9c2 2cxa 3cta b0 þ b1 exp 2Cð1þaÞ
where i ¼ 1. Case 5: 2
a1 ¼ 0;
a0 ¼ 0;
a1 ¼ b1
b1 ¼ b1 ;
b0 ¼ 0; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 4 :
b1 ¼ b1 ;
c¼
3c4 ;
qffiffiffiffiffiffiffi bc ; ð3:24Þ
where b1 is free parameter. These results give pffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffi 9c2 2cxa þ3cta b1 bc exp 4Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi : u9;10 ðx;tÞ ¼ 9c2 2cxa þ3cta 9c2 2cxa þ3cta b1 exp þ b1 exp 4Cð1þaÞ 4Cð1þaÞ ð3:25Þ
Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002
4
O. Guner, A. Bekir If we take b1 ¼ 1 and b1 ¼ 1 Eq. (3.25) becomes pffiffiffi i c cosh
pffiffiffiffiffiffiffiffiffiffi
u9;10 ðx;tÞ ¼
pffiffiffi 2 b cosh
pffiffiffiffiffiffiffiffiffiffi
pffiffiffi i c sinh pffiffiffiffiffiffiffiffiffiffi
9c2 2cxa þ3cta 4Cð1þaÞ
9c2 2cxa þ3cta 4Cð1þaÞ
9c2 2cxa þ3cta 4Cð1þaÞ
2 3 Dat u D2a x u þ lu þ cu þ du ¼ 0;
: ð3:26Þ
where i ¼ 1. Case 6: 2
qffiffiffiffiffiffiffi bc ;
a1 ¼ 0;
a0 ¼ b0
b1 ¼ b1 ;
b0 ¼ b0 ; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 2 :
c ¼ 3c2 ;
a1 ¼ 0; ð3:27Þ
b1 ¼ 0;
where b0 is free parameter. Substituting these results into (3.11), we have qffiffiffiffiffiffiffi b0 bc pffiffiffiffiffiffiffiffiffiffi : ð3:28Þ u11;12 ðx; tÞ ¼ 9c2 2cxa þ3cta b0 þ b1 exp 2Cð1þaÞ If we take b1 ¼ 1 and b0 ¼ 1 Eq. (3.28) becomes u3;4 ðx; tÞ ¼
pffiffiffi pffiffiffi b þ b cosh
pffiffiffiffiffiffiffiffiffiffi
pffiffiffi i c
9c2 2cxa þ3cta 2Cð1þaÞ
pffiffiffi þ b sinh
pffiffiffiffiffiffiffiffiffiffi
9c2 2cxa þ3cta 2Cð1þaÞ
:
ð3:29Þ
where i2 ¼ 1. Case 7: a1 ¼ a1 ; b1 ¼ ia1
qffiffi
b ; c
c ¼ c;
a0 ¼ a0 ; b0 ¼ ia0
qffiffi
b ; c
a1 ¼ b1
qffiffiffiffiffiffiffi bc ; ð3:30Þ
b1 ¼ b1 ;
k ¼ k:
where a1 ; a0 and b1 are free parameters. Substituting these results into (3.11) we get a a qffiffi cta cta þ a0 ib1 bc exp kx a1 exp kx Cð1þaÞ Cð1þaÞ qffiffi qffiffi a a : u13;14 ðx; tÞ ¼ cta cta ia0 bc þ b1 exp kx ia1 bc exp kx Cð1þaÞ Cð1þaÞ ð3:31Þ where i2 ¼ 1 and from ((3.14)), ((3.17)), ((3.24)), ((3.23)), ((3.25)), ((3.31)) it is possible to see that the solution will exist provided c – 0 and c – 29. Example 2: Nonlinear fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation has the form (Zayed and Amer, 2015):
u1;2 ðx; tÞ ¼
pffiffiffiffiffiffiffiffiffiffiffi
c2 4dlc b1 2d
ð3:32Þ
where 0 < a 6 1 and l; c; d are non zero constants. Nonlinear fractional KPP equation is important in the physical fields, and it includes the fractional Klein–Gordon equation, fractional Fitzhugh–Nagumo equation, fractional Newell–Whitehead equation, fractional Burgers equation, fractional Fisher equation, fractional Cahn–Allen equation, fractional Chaffee–Infanfe equation and fractional Huxlay equation. Zayed and Amer applied the improved Riccati equation method and found more general exact solutions which include the solitary wave solutions, the periodic solutions and the rational function solutions. Zayed et al. applied the fractional ðDan G=GÞ-expansion method Eq. (3.32) and obtained three types of solutions via the solitary, trigonometric and rational solutions (Zayed et al., 2015). Song and Wang have implemented the differential transform method for the approximate solution of this equation (Song and Wang, 2012). When a ¼ 1 Eq. (3.32) is called the nonlinear KPP equation and it has been discussed in (Feng et al., 2011) using the ðG0=GÞ-expansion method. In Zayed and Ibrahim (2014), Zayed and Ibrahim have obtained the solitary wave solutions via the modified simple equation method. Hariharan has obtained the analytical/numerical solutions of this equation with the Homotopy analysis method (HAM) for linear and nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) and fractional KPP equations (Hariharan, 2013). Now, we solve Eq. (3.32) using the proposed method of Section 2. To reduce this equation to the following ODE with integer order, we use the expression (2.3): cLU0 k2 L2 U00 þ lu þ cu2 þ du3 ¼ 0;
ð3:33Þ
Same approach in the previous example, we have p ¼ c and q ¼ d. Substituting Eq. (3.11) into Eq. (3.33), and collecting all the terms with the same power of e j ; j ¼ 3; . . . ; 3 together and then equating each coefficient equal to zero, yields a set of algebraic equations. By solving this algebraic equations with help of symbolic computation, we get. Case 1: a1 ¼ 0;
a0 ¼ 0;
b1 ¼ b1 ; c¼
pffiffiffiffiffiffiffiffiffiffiffi
6ldþc2 c2 4dl ; 8Ld
a1 ¼
pffiffiffiffiffiffiffiffiffiffiffi
c2 4dlc b1 ; 2d
b0 ¼ 0; b1 ¼ b1 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi k¼
d c2 2dlþc 4Ld
c2 4dl
: ð3:34Þ
where a1 and b1 are free parameters. By substituting these results into (3.11), we get
0 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi d c2 2dlþc c2 4dl ta 6ldþc2 c2 4dl ta AA exp @@ 4LdCð1þaÞ
8LdCð1þaÞ
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi d c2 2dlþc c2 4dl ta 6ldþc2 c2 4dl ta A b1 exp @ 4LdCð1þaÞ 8LdCð1þaÞ
ð3:35Þ
ffi 0 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2dl þ c 2 4dl ta 2 a 2 c d c 6ld þ c c 4dl t CC B B B CC þ b1 exp B @@ AA 4LdCð1 þ aÞ 8LdCð1 þ aÞ
Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002
The Exp-function method for in mathematical physics
5
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If we take b1 ¼ 2; b1 ¼ 1 and c2 4dl ¼ 2d c Eq. (3.35) becomes 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d c2 2dl þ c 2d c ta 1 1 u1;2 ðx; tÞ ¼ tanh @ 2 2 4dLCð1 þ aÞ ! p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6ld þ c2 2d c ta : ð3:36Þ 8dLCð1 þ aÞ Case 2: a1 ¼ 0;
a0 ¼ a0 ;
a1 ¼
pffiffiffiffiffiffiffiffiffiffiffi
c2 4dlc b1 ; 2d
pffiffiffiffiffiffiffiffiffiffiffi db0 2da0 þcb0 þb0 c2 4dl pffiffiffiffiffiffiffiffiffiffiffi ; b0 ¼ b0 ; b1 ¼ b1 ¼ b1 ; b1 c2 2dlþc c2 4dl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffi d c2 2dlþc c2 4dl 6ldþc2 c2 4dl c¼ ; k¼ : 4Ld 2Ld
ð3:37Þ
where a0 ; b0 and b1 are free parameters. Substituting these results into (3.11), we get
a0 þ u3;4 ðx; tÞ ¼
pffiffiffiffiffiffiffiffiffiffiffi
c2 4dlc b1 2d
in (3.14), (3.17), (3.20), (3.23), (3.26), (3.31), (3.36) and (3.39) are new exact solutions to the these equations. References Aksoy, E., Kaplan, M., Bekir, A., 2016. Exponential rational functionmethod for space–time fractional differential equations. Waves Random Complex Media 26 (2), 142–151. Atangana, A., Koca, I., 2016. Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos Solitons Fractals 89, 447–454. Atangana, A., Baleanu, D., 2016. New fractional derivatives with nonlocal and non-singular kernel. Theory Appl. Heat Transfer Model Thermal Sci. 20 (2), 763–769. Bekir, A., Guner, O., 2013. Exact solutions of nonlinear fractional differential equations by ðG0 =GÞ-expansion method. Chin. Phys. B 22 (11), 110202. Bekir, A., Guner, O., 2014. Exact solutions of distinct physical structures to the fractional potential Kadomtsev-Petviashvili equation. Computat. Methods Differ. Eq. 2 (1), 26–36.
0 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi d c2 2dlþc c2 4dl ta 6ldþc2 c2 4dl ta AA exp @@ 2dLCð1þaÞ
4dLCð1þaÞ
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi d c2 2dlþc c2 4dl ta c2 4dl 6ldþc2 c2 4dl ta A pffiffiffiffiffiffiffiffiffiffiffi exp @ 2dLCð1þaÞ 4dLCð1þaÞ 2
db0 2da0 þcb0 þb0
b1 c2 2dlþc
c 4dl
ð3:38Þ
ffi 11 0 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2dl þ c 2 4dl ta 2 a 2 c d c 6ld þ c c 4dl t CC B B B CC þ b0 þ b1 exp B AA @@ 2dLCð1 þ aÞ 4dLCð1 þ aÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If we take a0 ¼ 1; b1 ¼ 1; b0 ¼ 2; c2 4dl ¼ 2d c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and 2d 2d þ 2c þ 2 c2 4dl ¼ c2 2dl þ c c2 4dl Eq. (3.38) becomes
ffi 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2dl þ c 2 4dl ta c d c B 1 1 u3;4 ðx; tÞ ¼ þ tanh B @ 2 2 4dLCð1 þ aÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c2 4dl ta A: 8dLCð1 þ aÞ
6ld þ c2
ð3:39Þ
4. Conclusions In this paper, we have used the fractional complex transform with the Exp-function method in order to reduce nonlinear FDEs to NODEs for finding exact analytic solutions for the nonlinear space–time fractional Telegraph equation and the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equation. As a result, some new exact solutions for them have been successfully found. These results are going to be very useful, very effective in different areas of applied mathematics, so this method can be applied to other nonlinear FDEs in the mathematical physics. We note that the exact solutions established
Bekir, A., Guner, O., Unsal, O., 2015a. The first integral method for exact solutions of nonlinear fractional differential equations. J. Computat. Nonlinear Dyn. 10, 463–470. Bekir, A., Aksoy, E., Cevikel, A.C., 2015b. Exact solutions of nonlinear time fractional partial differential equations by subequation method. Math. Methods Appl. Sci. 38 (13), 2779–2784. Bekir, A., Guner, O., Bhrawy, A.H., Biswas, A., 2015c. Solving nonlinear fractional differential equations using exp-function and ðG0 =GÞ -expansion methods. Rom. J. Phys. 60 (3–4), 360–378. Bekir, A., Guner, O., Aksoy, E., 2016. Periodic and hyperbolic solutions of nonlinear fractional differential equations. Appl. Comput. Math. 15 (1), 88–95. Bulut, H., Baskonus, H.M., Pandir, Y., 2013. The modified trial equation method for fractional wave equation and time fractional generalized burgers equation. Abstr. Appl. Anal. 2013, 636802. Caputo, M., 1967. Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. Royal Astronom. Soc. 13, 529–539. Daftardar-Gejji, V., Bhalekar, S., 2008. Solving multi-term linear and non-linear diffusion wave equations of fractional order by adomian decomposition method. Appl. Math. Comput. 202, 113–120. Ebaid, A., 2012. An improvement on the Exp-function method when balancing the highest order linear and nonlinear terms. J. Math. Anal. Appl. 3921, 1–5. Erturk, V.S., Momani, S., Odibat, Z., 2008. Application of generalized differential transform method to multi-order fractional differential equations. Commun. Nonlinear Sci. Numer. Simulat. 13, 1642– 1654.
Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002
6 Eslami, M., Vajargah, B.F., Mirzazadeh, M., Biswas, A., 2014. Application of first integral method to fractional partial differential equations. Indian J. Phys. 88, 177–184. Feng, J., Li, W., Wan, Q., 2011. Using ðG0 =GÞ-expansion method to seek the traveling wave solution of Kolmogorov–Petrovskii– Piskunov equation. Appl. Math. Computat. 217, 5860–5865. Gepreel, K.A., 2011. The homotopy perturbation method to the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations. Appl. Math. Lett. 24, 1428–1434. Gepreel, K.A., Mohamed, M.S., 2013. Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation. Chin. Phys. B 22, 010201–010211. Guner, O., 2015. Singular and non-topological soliton solutions for nonlinear fractional differential equations. Chin. Phys. B 24 (10), 100201. Guner, O., Bekir, A., Pandir, Y., 2015a. Solving nonlinear space–time fractional differential equations using the exp-function method. AIP Conf. Proc. 1648, 370005. Guner, O., Bekir, A., Cevikel, A.C., 2015b. A variety of exact solutions for the time fractional Cahn-Allen equation. Eur. Phys. J. Plus 130, 146. Guner, O., Bekir, A., 2016. Bright and dark soliton solutions for some nonlinear fractional differential equations. Chin. Phys. B 25 (3), 030203. Guner, O., Bekir, A., 2017. Exp-function method for nonlinear fractional differential equations. Nonlinear Sci. Lett. A 8 (1), 41–49. Hariharan, G., 2013. The homotopy analysis method applied to the Kolmogorov–Petrovskii–Piskunov (KPP) and fractional KPP equations. J. Math. Chem. 51, 992–1000. He, J.H., 2006. Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700–708. He, J.H., Abdou, M.A., 2007. New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos Solitons Fractals 34 (5), 1421–1429. He, J.H., 2011. A new fractal derivation. Thermal Sci. 15 (1), 145–147. He, J.H., Elegan, S.K., Li, Z.B., 2012. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 376, 257–259. He, J.H., 2013. Exp-function method for fractional differential equations. Int. J. Nonlinear Sci. Numer. Simul. 14 (6), 363–366. Hu, Y., He, J.H., 2016. On fractal space-time and fractional calculus. Thermal Sci. 20 (3), 773–777. Jafari, H., Tajadodi, H., Baleanu, D., 2014. Application of a homogeneous balance method to exact solutions of nonlinear fractional evolution equations. J. Computat. Nonlinear Dyn. 9. 021019-1. Jumarie, G., 2006. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 51, 1367–1376. Jumarie, G., 2009. Table of some basic fractional calculus formulae derived from a modified Riemann-Liouvillie derivative for nondifferentiable functions. Appl. Maths. Lett. 22, 378–385. Khan, K., Akbar, M.A., 2013. Exact and solitary wave solutions for the Tzitzeica–Dodd–Bullough and the modified KdV–Zakharov– Kuznetsov equations using the modified simple equation method. Ain Shams Eng. J. 4 (4), 903–909. Khan, K., Akbar, M.A., 2014. Exact solutions of the (2+1)-dimensional cubic Klein-Gordon equation and the (3+1)-dimensional Zakharov-Kuznetsov equation using the modified simple equation method. J. Assoc. Arab Univ. Basic Appl. Sci. 15, 74–81. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., 2006. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam. Li, Z., He, J.H., 2010. Fractional complex transform for fractional differential equations. Math. Comput. Appl 15 (5), 970–973. Liu, W., Chen, K., 2013. The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations. Pramana J. Phys. 81 (3), 377–384.
O. Guner, A. Bekir Lu, B., 2012. The first integral method for some time fractional differential equations. J. Math. Anal. Appl. 395, 684–693. Miller, K.S., Ross, B., 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York. Mohamed, M.S., Al-Malki, F., Talib, R., 2012. Jacobi elliptic numerical solutions for the time fractional dispersive long wave equation. International Journal of Pure and Applied Mathematics 80, 635–646. Naher, H., Abdullah, F.A., Akbar, M.A., 2013. Generalized and improved (G0 /G)-expansion method for (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation. PLoS One 8 (5), e64618. Navickas, Z., Ragulskis, M., Bikulciene, L., 2010. Be careful with the Exp-function method – additional remarks. Commun. Nonlinear Sci. Numer. Simul. 15, 3874–3886. Noor, M.A., Mohyud-Din, S.D., Waheed, A., 2008. Exp-function method for generalized traveling solutions of master partial differential equation. Acta Appl. Math. 104 (2), 131–137. Podlubny, I., 1999. Fractional Differential Equations. Academic Press, California. Sayevand, K., Pichaghchi, K., 2016. Analysis of nonlinear fractional KdV equation based on He’s fractional derivative. Nonlinear Sci. Lett. A 7 (3), 77–85. Song, L.N., Wang, W.G., 2012. Approximate solutions of nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations using an enhanced algorithm of the generalized two-dimensional differential transform method. Commun. Theor. Phys. 58 (2), 182–188. Sweilam, N.H., Khader, M.M., Al-Bar, R.F., 2007. Numerical studies for a multi order fractional differential equation. Phys. Lett. A 371, 26–33. Taghizadeh, N., Mirzazadeh, M., Rahimian, M., Akbari, M., 2013. Application of the simplest equation method to some timefractional partial differential equations. Ain Shams Eng. J. 4 (4), 897–902. Wang, D.S., Li, H., 2008. Single and multi-solitary wave solutions to a class of nonlinear evolution equations. J. Math. Anal. Appl. 343, 273–298. Wu, X.E., Liang, Y.S., 2017. Relationship between fractal dimensions and fractional calculus. Nonlinear Sci. Lett. A 8 (1), 77–89. Yan, L.M., Xu, F.S., 2015. Generalized exp-function method for nanlinear space-time fractional differential equations. Thermal Sci. 18 (5), 1573–1576. Zayed, E.M.E., Ibrahim, S.A.H., 2014. Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method. Acta Math. Appl. Sin. English Series 30 (3), 749– 754. Zayed, E.M.E., Al-Nowehy, A.G., 2016. Exact traveling wave solutions for nonlinear PDEs in mathematical physics using the generalized Kudryashov method. Serb. J. Electr. Eng. 13 (2), 203–227. Zayed, E.M.E., Amer, Y.A., 2015. Exact solutions for the nonlinear KPP equation by using the Riccati equation method combined with the ðG0 =GÞ-expansion method. Sci. Res. Essays 10 (3), 86–96. Zayed, E.M.E., Amer, Y.A., Shohib, R.M.A., 2015. The fractional ðDan G=GÞ-expansion method and its applications for solving four nonlinear space-time fractional PDEs in mathematial physics. Italian J. Pure Appl. Math. 34, 463–482. Zhang, S., Zong, Q.-A., Liu, D., Gao, Q., 2010. A generalized Expfunction method for fractional Riccati differential equations. Commun. Fract. Calculus 1, 48–51. Zhang, S., Zhang, H.-Q., 2011. Fractional sub-equation method and its applications to nonlinear fractional PDEs. Phys. Lett. A 375, 1069–1073. Zheng, B., 2012. ðG0 =GÞ-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun. Theor. Phys. 58, 623–630. Zheng, B., 2013. Exp-function method for solving fractional partial differential equations. Sci. World J. 2013, 465723.
Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002
University of Bahrain
Journal of the Association of Arab Universities for Basic and Applied Sciences www.elsevier.com/locate/jaaubas www.sciencedirect.com
The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics Ozkan Guner a,*, Ahmet Bekir b a Cankiri Karatekin University, Faculty of Economics and Administrative Sciences, Department of International Trade, Cankiri, Turkey b Eskisehir Osmangazi University, Art-Science Faculty, Department of Mathematics-Computer, Eskisehir, Turkey
Received 3 October 2016; revised 21 November 2016; accepted 9 December 2016
KEYWORDS Exact solution; Space–time fractional Telegraph equation; Space–time fractional KPP equation; Exp-function method
Abstract Using the Exp-function method, we derive exact solutions of the nonlinear space–time fractional Telegraph equation and space–time fractional KPP equation. As a result, we obtain many exact analytical solutions including hyperbolic function. The fractional derivative is described in Jumarie’s modified Riemann–Liouville sense. This method is very effective and convenient for solving nonlinear fractional differential equations. Ó 2016 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders. Fractional differential equations (FDEs) are used in various fields of physics, mathematics and engineering. So, they have gained much attention many mathematicians. Applications of FDEs for instance; signal processing, control theory, viscoelastic materials, systems identification, biomedical sciences, biology, finance, fractional dynamics as well as other disciplines (Miller and Ross, 1993;Podlubny, 1999; Kilbas et al., 2006). In the last two decades, a lot of attention is paid to finding appropriate solutions of FDEs. There are many effective methods to obtain numerical and analytical solutions of these kinds * Corresponding author. E-mail addresses: [email protected] (O. [email protected] (A. Bekir). Peer review under responsibility of University of Bahrain.
Guner),
of FDEs, such as homotopy analysis method, adomian decomposition method, variational iteration method, homotopy perturbation method (Daftardar-Gejji and Bhalekar, 2008; Erturk et al., 2008; Gepreel and Mohamed, 2013; Sweilam et al., 2007; Gepreel, 2011; Mohamed et al., 2012) and the tanh method, the Exp-function method, the first integral method, the functional variable method, the ðG0=GÞ-expansion method, the sub-equation method, the modified trial equation method, the simplest equation method, the generalized Kudryashov method, the ansatz method and so on (Naher et al., 2013; Zheng, 2012, 2013; Bekir and Guner, 2013, 2014; Bekir et al., 2015a,b,c; Bekir et al., 2016; Bulut et al., 2013; Yan and Xu, 2015; Zhang et al., 2010; Guner, 2015; Guner and Bekir, 2016; Guner et al., 2015a,b; Taghizadeh et al., 2013; Khan and Akbar, 2013, 2014; Lu, 2012; Eslami et al., 2014; Zhang and Zhang, 2011; Liu and Chen, 2013). The Exp-function method was first proposed by He and Wu in 2006 (He, 2006) and systematically studied in He and Abdou (2007), Noor et al. (2008), Ebaid (2012) and
http://dx.doi.org/10.1016/j.jaubas.2016.12.002 1815-3852 Ó 2016 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002
2
O. Guner, A. Bekir
Navickas et al. (2010). The Exp-function method was originally proposed to solve PDEs. However, its application to fractional calculus is rare and primary. In 2013, this method was successfully extended to fractional calculus, and it becomes an effective tool for fractional differential equations (see He, 2013; Guner and Bekir, 2017). There are different kinds of fractional derivative operators. The most famous one is the Caputo definition that the function should be differentiable (Caputo, 1967). Recently, Jumarie derived definitions for the fractional derivative called modified Riemann–Liouville, which are suitable for continuous and non-differentiable functions. This paper adopts the Jumarieı´ s derivative, which has been disadvantages, and has already been updated by some new ones, for example AtanganaBaleanu derivative and He derivative (Atangana and Koca, 2016;Atangana and Baleanu, 2016; Sayevand and Pichaghchi, 2016; He, 2011). The order a of Jumarie’s derivative is defined by Jumarie (2006, 2009): ( Daw fðwÞ ¼
1 d Cð1aÞ dw
Rw 0
;
h 6 a < h þ 1; h P 1: ð1:1Þ
where f : R ! R; w ! fðwÞ denotes a continuous (but not necessarily differentiable) function and satisfying above properties Daw wc ¼
Cð1 þ cÞ wca Cð1 þ c aÞ
c > 0;
ð1:2Þ
Daw ðCfðwÞÞ ¼ CDaw fðwÞ;
ð1:3Þ
Daw fafðwÞ þ bgðwÞg ¼ aDaw fðwÞ þ bDaw gðwÞ;
ð1:4Þ
Daw C ¼ 0;
ð1:5Þ
where a; b and C are constant. The outline of the present paper is as follows. In Section 2, we have a brief description of the Exp-function method for solving fractional partial differential equations. In Section 3, we apply the Exp-function method on the nonlinear space–time fractional Telegraph equation and the nonlinear fractional Kolmogorov Petrovskii-Piskunov (KPP) equation respectively. Finally, some conclusions are given in Section 4. 2. Description of the Exp-function method for solving fractional partial differential equations We take into consideration the following general nonlinear FDE of the type a b 2b Fðu; Dat u; Dbx u; D2a t u; Dt Dx u; Dx u; . . .Þ ¼ 0;
uðx; yÞ ¼ UðnÞ; n¼
kxb cta ; Cð1 þ bÞ Cð1 þ aÞ
0 < a; b 6 1 ð2:1Þ
where F is a polynomial of u; u is an unknown function and Da partial fractional derivative of u, in which the highest order derivatives and the nonlinear terms are involved. The fractional complex transform was proposed by He and Li to convert a fractal space (time) to its continuous partner, the physical basis was illustrated by He in his recent review articles (Li and He, 2010;He et al., 2012; Hu and He, 2016). Wu and Liang gave a very clear mathematic analysis of relationship between fractal dimensions and fractional calculus in Wu and Liang (2017).
ð2:2Þ ð2:3Þ
where k – 0 and c – 0 are constants. The fractional chain rule is defined as dU a Dn dn t dU a D n Dax u ¼ rx dn x Dat u ¼ rt
ð2:4Þ
where rt and rx are named as sigma indexes (Aksoy et al., 2016). We can choose rt ¼ rx ¼ L, where L is a constant. When we substitute, (2.2) with (1.2) and (2.4) into (2.1), we can rewrite Eq. (2.1) in the following NODE; QðU; U0 ; U00 ; U000 ; . . .Þ ¼ 0:
ðw nÞa ðfðnÞ fð0ÞÞdn; 0 < a < 1
ðahÞ
ðfðhÞ ðwÞÞ
Using the fractional complex transform
ð2:5Þ
where UðnÞ is the nth derivative of U with respect to n. Suppose that the solution of NODE (2.5) can be expressed in the following form Pd an exp ½nn UðnÞ ¼ Pqn¼c ð2:6Þ b m¼p m exp ½mn where p; q; c and d are positive integers which are known to be further determined, an and bm are unknown constants. To determine the value of c and p, we balance the linear term of the lowest order of equation Eq. (2.5) with the lowest order nonlinear term. Similarly, the value of d and q can be determined. 3. Applications of the proposed method Example 1: We consider the nonlinear space–time fractional Telegraph equation (Jafari et al., 2014) 2a a 3 D2a tt u Dxx u þ Dt u þ cu þ bu ¼ 0;
ð3:1Þ
a is a parameter describing the order of the fractional space and time derivative. When a ¼ 1, Eq. (3.1) is called the nonlinear Telegraph equation. Wang and Li, presented to construct the traveling wave solutions of this equation (Wang and Li, 2008). Also, Jafari et al. applied the sub equation method Eq. (3.1) and obtained the many exact solutions. Zayed and Al-Nowehy solved the equation by the generalized Kudryashov method and obtained the exact solutions of nonlinear PDEs in mathematical physics (Zayed and Al-Nowehy, 2016). Using the fractional complex transform, uðx; tÞ ¼ UðnÞ; n¼
kxa cta ; Cð1 þ aÞ Cð1 þ aÞ
ð3:2Þ ð3:3Þ
where k and c are constants. When we substitute (3.3) with (1.2) and (2.4) into (3.1), this Eq. (3.1) can reduced to an ODE ðc2 k2 ÞU00 cU0 þ cU þ bU3 ¼ 0; 0
where U ¼
ð3:4Þ
dU . dn
Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002
The Exp-function method for in mathematical physics
3
Balancing the order of U00 and U3 in Eq. (3.4), we obtain c1 exp½ðc þ 3pÞn þ ; U ¼ c2 exp½4pn þ 00
ð3:5Þ
If we take b1 ¼ 1 and b0 ¼ 1 Eq. (3.16) becomes u3;4 ðx; tÞ ¼
pffiffiffi pffiffiffi b þ b cosh
pffiffiffiffiffiffiffiffiffiffi
pffiffiffi i c
9c2 2cxa 3cta 2Cð1þaÞ
pffiffiffi b sinh
pffiffiffiffiffiffiffiffiffiffi
9c2 2cxa 3cta 2Cð1þaÞ
ð3:17Þ
and U3 ¼
:
c3 exp½3cn þ ; c4 exp½3pn þ
ð3:6Þ
where ci is determined coefficient only for simplicity. Balancing the lowest order of Exp-function in Eqs. (3.5) and (3.6) we obtain ðc þ 3pÞ ¼ ð3c þ pÞ;
ð3:7Þ
which leads to the result p ¼ c:
ð3:8Þ
In the same way also we have 3q þ d ¼ 4d;
ð3:9Þ
and this gives q ¼ d:
ð3:10Þ
If we set p ¼ c ¼ 1 and q ¼ d ¼ 1, Eq. (2.5) reduces to 0 þa1 expðnÞ UðnÞ ¼ ab11 expðnÞþa : expðnÞþb0 þb1 expðnÞ
ð3:11Þ
When we substitute Eq. (3.11) into Eq. (3.1), by the help of Maple, we get the following results, Case 1: qffiffiffiffiffiffiffi a1 ¼ 0; a1 ¼ b1 bc ; a0 ¼ 0; b1 ¼ b1 ;
b0 ¼ 0; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 4 :
c ¼ 3c4 ;
ð3:12Þ
b1 ¼ b1 ;
where b1 is free parameter. When substitute these results into (3.11), we get the following exact solution pffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffi 9c2 2cxa 3cta bc exp 4Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi : 9c2 2cxa 3cta 9c2 2cxa 3cta þ b1 exp 4Cð1þaÞ 4Cð1þaÞ
b1 pffiffiffiffiffiffiffiffiffiffi
u1;2 ðx;tÞ ¼ b1 exp
ð3:13Þ
If we take b1 ¼ 1 and b1 ¼ 1 Eq. (3.13) becomes pffiffiffi i c cosh u1;2 ðx;tÞ ¼
pffiffiffiffiffiffiffiffiffiffi
4Cð1þaÞ
pffiffiffi 2 b cosh
pffiffiffiffiffiffiffiffiffiffi
pffiffiffi þ i c sinh pffiffiffiffiffiffiffiffiffiffi
9c2 2cxa 3cta
9c2 2cxa 3cta 4Cð1þaÞ
9c2 2cxa 3cta 4Cð1þaÞ
:
b1 ¼ 0; c ¼ 3c2 ;
b0 ¼ b0 ; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 2 :
a1 ¼ 0; b1 ¼ b1 ;
a0 ¼ 0;
a1 ¼ b1
b1 ¼ 0;
b0 ¼ b0 ; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 2 :
b1 ¼ b1 ;
c¼
3c2 ;
qffiffiffiffiffiffiffi bc ; ð3:18Þ
where b1 is free parameter. Similarly, we obtain pffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi 9c2 2cxa þ3cta b1 bc exp 2Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi : u5;6 ðx; tÞ ¼ 2 9c 2cxa þ3cta b0 þ b1 exp 2Cð1þaÞ
ð3:19Þ
Eq. (3.19) becomes, if we take b1 ¼ 1 and b0 ¼ 1 pffiffiffiffiffiffiffiffiffiffi pffiffiffi 9c2 2cxa þ3cta i c sinh 2Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi : u5;6 ðx;tÞ ¼ pffiffiffi pffiffiffi pffiffiffi 9c2 2cxa þ3cta 9c2 2cxa þ3cta b sinh b þ b cosh 2Cð1þaÞ 2Cð1þaÞ pffiffiffi i c cosh
pffiffiffiffiffiffiffiffiffiffi
9c2 2cxa þ3cta 2Cð1þaÞ
ð3:20Þ
where i ¼ 1. Case 4: qffiffiffiffiffiffiffi a1 ¼ b1 bc ; 2
b1 ¼ b1 ; c ¼ 3c2 ;
a0 ¼ 0;
a1 ¼ 0;
b0 ¼ b0 ; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 2 :
b1 ¼ 0;
where b1 is free parameter. In the same way we get pffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi 9c2 2cxa 3cta b1 bc exp 2Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi : u7;8 ðx; tÞ ¼ 9c2 2cxa 3cta b0 þ b1 exp 2Cð1þaÞ
ð3:21Þ
ð3:22Þ
Eq. (3.22) becomes, when we take b1 ¼ 1 and b0 ¼ 1 pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi pffiffiffi 9c2 2cxa 3cta þ i c sinh 2Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi : u7;8 ðx;tÞ ¼ pffiffiffi pffiffiffi pffiffiffi 9c2 2cxa 3cta 9c2 2cxa 3cta þ b sinh b þ b cosh 2Cð1þaÞ 2Cð1þaÞ
9c2 2cxa 3cta 2Cð1þaÞ
ð3:23Þ
where i2 ¼ 1. Case 2: a1 ¼ 0;
a1 ¼ 0;
pffiffiffi i c cosh
ð3:14Þ
qffiffiffiffiffiffiffi a0 ¼ b0 bc ;
where i2 ¼ 1. Case 3:
ð3:15Þ
where b0 is free parameter. Substituting these results into (3.11), we get qffiffiffiffiffiffiffi b0 bc pffiffiffiffiffiffiffiffiffiffi : u3;4 ðx; tÞ ¼ ð3:16Þ 9c2 2cxa 3cta b0 þ b1 exp 2Cð1þaÞ
where i ¼ 1. Case 5: 2
a1 ¼ 0;
a0 ¼ 0;
a1 ¼ b1
b1 ¼ b1 ;
b0 ¼ 0; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 4 :
b1 ¼ b1 ;
c¼
3c4 ;
qffiffiffiffiffiffiffi bc ; ð3:24Þ
where b1 is free parameter. These results give pffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffi 9c2 2cxa þ3cta b1 bc exp 4Cð1þaÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi : u9;10 ðx;tÞ ¼ 9c2 2cxa þ3cta 9c2 2cxa þ3cta b1 exp þ b1 exp 4Cð1þaÞ 4Cð1þaÞ ð3:25Þ
Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002
4
O. Guner, A. Bekir If we take b1 ¼ 1 and b1 ¼ 1 Eq. (3.25) becomes pffiffiffi i c cosh
pffiffiffiffiffiffiffiffiffiffi
u9;10 ðx;tÞ ¼
pffiffiffi 2 b cosh
pffiffiffiffiffiffiffiffiffiffi
pffiffiffi i c sinh pffiffiffiffiffiffiffiffiffiffi
9c2 2cxa þ3cta 4Cð1þaÞ
9c2 2cxa þ3cta 4Cð1þaÞ
9c2 2cxa þ3cta 4Cð1þaÞ
2 3 Dat u D2a x u þ lu þ cu þ du ¼ 0;
: ð3:26Þ
where i ¼ 1. Case 6: 2
qffiffiffiffiffiffiffi bc ;
a1 ¼ 0;
a0 ¼ b0
b1 ¼ b1 ;
b0 ¼ b0 ; pffiffiffiffiffiffiffiffiffiffi 9c2 2c k¼ 2 :
c ¼ 3c2 ;
a1 ¼ 0; ð3:27Þ
b1 ¼ 0;
where b0 is free parameter. Substituting these results into (3.11), we have qffiffiffiffiffiffiffi b0 bc pffiffiffiffiffiffiffiffiffiffi : ð3:28Þ u11;12 ðx; tÞ ¼ 9c2 2cxa þ3cta b0 þ b1 exp 2Cð1þaÞ If we take b1 ¼ 1 and b0 ¼ 1 Eq. (3.28) becomes u3;4 ðx; tÞ ¼
pffiffiffi pffiffiffi b þ b cosh
pffiffiffiffiffiffiffiffiffiffi
pffiffiffi i c
9c2 2cxa þ3cta 2Cð1þaÞ
pffiffiffi þ b sinh
pffiffiffiffiffiffiffiffiffiffi
9c2 2cxa þ3cta 2Cð1þaÞ
:
ð3:29Þ
where i2 ¼ 1. Case 7: a1 ¼ a1 ; b1 ¼ ia1
qffiffi
b ; c
c ¼ c;
a0 ¼ a0 ; b0 ¼ ia0
qffiffi
b ; c
a1 ¼ b1
qffiffiffiffiffiffiffi bc ; ð3:30Þ
b1 ¼ b1 ;
k ¼ k:
where a1 ; a0 and b1 are free parameters. Substituting these results into (3.11) we get a a qffiffi cta cta þ a0 ib1 bc exp kx a1 exp kx Cð1þaÞ Cð1þaÞ qffiffi qffiffi a a : u13;14 ðx; tÞ ¼ cta cta ia0 bc þ b1 exp kx ia1 bc exp kx Cð1þaÞ Cð1þaÞ ð3:31Þ where i2 ¼ 1 and from ((3.14)), ((3.17)), ((3.24)), ((3.23)), ((3.25)), ((3.31)) it is possible to see that the solution will exist provided c – 0 and c – 29. Example 2: Nonlinear fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation has the form (Zayed and Amer, 2015):
u1;2 ðx; tÞ ¼
pffiffiffiffiffiffiffiffiffiffiffi
c2 4dlc b1 2d
ð3:32Þ
where 0 < a 6 1 and l; c; d are non zero constants. Nonlinear fractional KPP equation is important in the physical fields, and it includes the fractional Klein–Gordon equation, fractional Fitzhugh–Nagumo equation, fractional Newell–Whitehead equation, fractional Burgers equation, fractional Fisher equation, fractional Cahn–Allen equation, fractional Chaffee–Infanfe equation and fractional Huxlay equation. Zayed and Amer applied the improved Riccati equation method and found more general exact solutions which include the solitary wave solutions, the periodic solutions and the rational function solutions. Zayed et al. applied the fractional ðDan G=GÞ-expansion method Eq. (3.32) and obtained three types of solutions via the solitary, trigonometric and rational solutions (Zayed et al., 2015). Song and Wang have implemented the differential transform method for the approximate solution of this equation (Song and Wang, 2012). When a ¼ 1 Eq. (3.32) is called the nonlinear KPP equation and it has been discussed in (Feng et al., 2011) using the ðG0=GÞ-expansion method. In Zayed and Ibrahim (2014), Zayed and Ibrahim have obtained the solitary wave solutions via the modified simple equation method. Hariharan has obtained the analytical/numerical solutions of this equation with the Homotopy analysis method (HAM) for linear and nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) and fractional KPP equations (Hariharan, 2013). Now, we solve Eq. (3.32) using the proposed method of Section 2. To reduce this equation to the following ODE with integer order, we use the expression (2.3): cLU0 k2 L2 U00 þ lu þ cu2 þ du3 ¼ 0;
ð3:33Þ
Same approach in the previous example, we have p ¼ c and q ¼ d. Substituting Eq. (3.11) into Eq. (3.33), and collecting all the terms with the same power of e j ; j ¼ 3; . . . ; 3 together and then equating each coefficient equal to zero, yields a set of algebraic equations. By solving this algebraic equations with help of symbolic computation, we get. Case 1: a1 ¼ 0;
a0 ¼ 0;
b1 ¼ b1 ; c¼
pffiffiffiffiffiffiffiffiffiffiffi
6ldþc2 c2 4dl ; 8Ld
a1 ¼
pffiffiffiffiffiffiffiffiffiffiffi
c2 4dlc b1 ; 2d
b0 ¼ 0; b1 ¼ b1 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi k¼
d c2 2dlþc 4Ld
c2 4dl
: ð3:34Þ
where a1 and b1 are free parameters. By substituting these results into (3.11), we get
0 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi d c2 2dlþc c2 4dl ta 6ldþc2 c2 4dl ta AA exp @@ 4LdCð1þaÞ
8LdCð1þaÞ
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi d c2 2dlþc c2 4dl ta 6ldþc2 c2 4dl ta A b1 exp @ 4LdCð1þaÞ 8LdCð1þaÞ
ð3:35Þ
ffi 0 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2dl þ c 2 4dl ta 2 a 2 c d c 6ld þ c c 4dl t CC B B B CC þ b1 exp B @@ AA 4LdCð1 þ aÞ 8LdCð1 þ aÞ
Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002
The Exp-function method for in mathematical physics
5
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If we take b1 ¼ 2; b1 ¼ 1 and c2 4dl ¼ 2d c Eq. (3.35) becomes 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d c2 2dl þ c 2d c ta 1 1 u1;2 ðx; tÞ ¼ tanh @ 2 2 4dLCð1 þ aÞ ! p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6ld þ c2 2d c ta : ð3:36Þ 8dLCð1 þ aÞ Case 2: a1 ¼ 0;
a0 ¼ a0 ;
a1 ¼
pffiffiffiffiffiffiffiffiffiffiffi
c2 4dlc b1 ; 2d
pffiffiffiffiffiffiffiffiffiffiffi db0 2da0 þcb0 þb0 c2 4dl pffiffiffiffiffiffiffiffiffiffiffi ; b0 ¼ b0 ; b1 ¼ b1 ¼ b1 ; b1 c2 2dlþc c2 4dl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffi d c2 2dlþc c2 4dl 6ldþc2 c2 4dl c¼ ; k¼ : 4Ld 2Ld
ð3:37Þ
where a0 ; b0 and b1 are free parameters. Substituting these results into (3.11), we get
a0 þ u3;4 ðx; tÞ ¼
pffiffiffiffiffiffiffiffiffiffiffi
c2 4dlc b1 2d
in (3.14), (3.17), (3.20), (3.23), (3.26), (3.31), (3.36) and (3.39) are new exact solutions to the these equations. References Aksoy, E., Kaplan, M., Bekir, A., 2016. Exponential rational functionmethod for space–time fractional differential equations. Waves Random Complex Media 26 (2), 142–151. Atangana, A., Koca, I., 2016. Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos Solitons Fractals 89, 447–454. Atangana, A., Baleanu, D., 2016. New fractional derivatives with nonlocal and non-singular kernel. Theory Appl. Heat Transfer Model Thermal Sci. 20 (2), 763–769. Bekir, A., Guner, O., 2013. Exact solutions of nonlinear fractional differential equations by ðG0 =GÞ-expansion method. Chin. Phys. B 22 (11), 110202. Bekir, A., Guner, O., 2014. Exact solutions of distinct physical structures to the fractional potential Kadomtsev-Petviashvili equation. Computat. Methods Differ. Eq. 2 (1), 26–36.
0 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi d c2 2dlþc c2 4dl ta 6ldþc2 c2 4dl ta AA exp @@ 2dLCð1þaÞ
4dLCð1þaÞ
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi d c2 2dlþc c2 4dl ta c2 4dl 6ldþc2 c2 4dl ta A pffiffiffiffiffiffiffiffiffiffiffi exp @ 2dLCð1þaÞ 4dLCð1þaÞ 2
db0 2da0 þcb0 þb0
b1 c2 2dlþc
c 4dl
ð3:38Þ
ffi 11 0 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2dl þ c 2 4dl ta 2 a 2 c d c 6ld þ c c 4dl t CC B B B CC þ b0 þ b1 exp B AA @@ 2dLCð1 þ aÞ 4dLCð1 þ aÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If we take a0 ¼ 1; b1 ¼ 1; b0 ¼ 2; c2 4dl ¼ 2d c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and 2d 2d þ 2c þ 2 c2 4dl ¼ c2 2dl þ c c2 4dl Eq. (3.38) becomes
ffi 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2dl þ c 2 4dl ta c d c B 1 1 u3;4 ðx; tÞ ¼ þ tanh B @ 2 2 4dLCð1 þ aÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c2 4dl ta A: 8dLCð1 þ aÞ
6ld þ c2
ð3:39Þ
4. Conclusions In this paper, we have used the fractional complex transform with the Exp-function method in order to reduce nonlinear FDEs to NODEs for finding exact analytic solutions for the nonlinear space–time fractional Telegraph equation and the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equation. As a result, some new exact solutions for them have been successfully found. These results are going to be very useful, very effective in different areas of applied mathematics, so this method can be applied to other nonlinear FDEs in the mathematical physics. We note that the exact solutions established
Bekir, A., Guner, O., Unsal, O., 2015a. The first integral method for exact solutions of nonlinear fractional differential equations. J. Computat. Nonlinear Dyn. 10, 463–470. Bekir, A., Aksoy, E., Cevikel, A.C., 2015b. Exact solutions of nonlinear time fractional partial differential equations by subequation method. Math. Methods Appl. Sci. 38 (13), 2779–2784. Bekir, A., Guner, O., Bhrawy, A.H., Biswas, A., 2015c. Solving nonlinear fractional differential equations using exp-function and ðG0 =GÞ -expansion methods. Rom. J. Phys. 60 (3–4), 360–378. Bekir, A., Guner, O., Aksoy, E., 2016. Periodic and hyperbolic solutions of nonlinear fractional differential equations. Appl. Comput. Math. 15 (1), 88–95. Bulut, H., Baskonus, H.M., Pandir, Y., 2013. The modified trial equation method for fractional wave equation and time fractional generalized burgers equation. Abstr. Appl. Anal. 2013, 636802. Caputo, M., 1967. Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. Royal Astronom. Soc. 13, 529–539. Daftardar-Gejji, V., Bhalekar, S., 2008. Solving multi-term linear and non-linear diffusion wave equations of fractional order by adomian decomposition method. Appl. Math. Comput. 202, 113–120. Ebaid, A., 2012. An improvement on the Exp-function method when balancing the highest order linear and nonlinear terms. J. Math. Anal. Appl. 3921, 1–5. Erturk, V.S., Momani, S., Odibat, Z., 2008. Application of generalized differential transform method to multi-order fractional differential equations. Commun. Nonlinear Sci. Numer. Simulat. 13, 1642– 1654.
Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002
6 Eslami, M., Vajargah, B.F., Mirzazadeh, M., Biswas, A., 2014. Application of first integral method to fractional partial differential equations. Indian J. Phys. 88, 177–184. Feng, J., Li, W., Wan, Q., 2011. Using ðG0 =GÞ-expansion method to seek the traveling wave solution of Kolmogorov–Petrovskii– Piskunov equation. Appl. Math. Computat. 217, 5860–5865. Gepreel, K.A., 2011. The homotopy perturbation method to the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations. Appl. Math. Lett. 24, 1428–1434. Gepreel, K.A., Mohamed, M.S., 2013. Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation. Chin. Phys. B 22, 010201–010211. Guner, O., 2015. Singular and non-topological soliton solutions for nonlinear fractional differential equations. Chin. Phys. B 24 (10), 100201. Guner, O., Bekir, A., Pandir, Y., 2015a. Solving nonlinear space–time fractional differential equations using the exp-function method. AIP Conf. Proc. 1648, 370005. Guner, O., Bekir, A., Cevikel, A.C., 2015b. A variety of exact solutions for the time fractional Cahn-Allen equation. Eur. Phys. J. Plus 130, 146. Guner, O., Bekir, A., 2016. Bright and dark soliton solutions for some nonlinear fractional differential equations. Chin. Phys. B 25 (3), 030203. Guner, O., Bekir, A., 2017. Exp-function method for nonlinear fractional differential equations. Nonlinear Sci. Lett. A 8 (1), 41–49. Hariharan, G., 2013. The homotopy analysis method applied to the Kolmogorov–Petrovskii–Piskunov (KPP) and fractional KPP equations. J. Math. Chem. 51, 992–1000. He, J.H., 2006. Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700–708. He, J.H., Abdou, M.A., 2007. New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos Solitons Fractals 34 (5), 1421–1429. He, J.H., 2011. A new fractal derivation. Thermal Sci. 15 (1), 145–147. He, J.H., Elegan, S.K., Li, Z.B., 2012. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 376, 257–259. He, J.H., 2013. Exp-function method for fractional differential equations. Int. J. Nonlinear Sci. Numer. Simul. 14 (6), 363–366. Hu, Y., He, J.H., 2016. On fractal space-time and fractional calculus. Thermal Sci. 20 (3), 773–777. Jafari, H., Tajadodi, H., Baleanu, D., 2014. Application of a homogeneous balance method to exact solutions of nonlinear fractional evolution equations. J. Computat. Nonlinear Dyn. 9. 021019-1. Jumarie, G., 2006. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 51, 1367–1376. Jumarie, G., 2009. Table of some basic fractional calculus formulae derived from a modified Riemann-Liouvillie derivative for nondifferentiable functions. Appl. Maths. Lett. 22, 378–385. Khan, K., Akbar, M.A., 2013. Exact and solitary wave solutions for the Tzitzeica–Dodd–Bullough and the modified KdV–Zakharov– Kuznetsov equations using the modified simple equation method. Ain Shams Eng. J. 4 (4), 903–909. Khan, K., Akbar, M.A., 2014. Exact solutions of the (2+1)-dimensional cubic Klein-Gordon equation and the (3+1)-dimensional Zakharov-Kuznetsov equation using the modified simple equation method. J. Assoc. Arab Univ. Basic Appl. Sci. 15, 74–81. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., 2006. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam. Li, Z., He, J.H., 2010. Fractional complex transform for fractional differential equations. Math. Comput. Appl 15 (5), 970–973. Liu, W., Chen, K., 2013. The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations. Pramana J. Phys. 81 (3), 377–384.
O. Guner, A. Bekir Lu, B., 2012. The first integral method for some time fractional differential equations. J. Math. Anal. Appl. 395, 684–693. Miller, K.S., Ross, B., 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York. Mohamed, M.S., Al-Malki, F., Talib, R., 2012. Jacobi elliptic numerical solutions for the time fractional dispersive long wave equation. International Journal of Pure and Applied Mathematics 80, 635–646. Naher, H., Abdullah, F.A., Akbar, M.A., 2013. Generalized and improved (G0 /G)-expansion method for (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation. PLoS One 8 (5), e64618. Navickas, Z., Ragulskis, M., Bikulciene, L., 2010. Be careful with the Exp-function method – additional remarks. Commun. Nonlinear Sci. Numer. Simul. 15, 3874–3886. Noor, M.A., Mohyud-Din, S.D., Waheed, A., 2008. Exp-function method for generalized traveling solutions of master partial differential equation. Acta Appl. Math. 104 (2), 131–137. Podlubny, I., 1999. Fractional Differential Equations. Academic Press, California. Sayevand, K., Pichaghchi, K., 2016. Analysis of nonlinear fractional KdV equation based on He’s fractional derivative. Nonlinear Sci. Lett. A 7 (3), 77–85. Song, L.N., Wang, W.G., 2012. Approximate solutions of nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations using an enhanced algorithm of the generalized two-dimensional differential transform method. Commun. Theor. Phys. 58 (2), 182–188. Sweilam, N.H., Khader, M.M., Al-Bar, R.F., 2007. Numerical studies for a multi order fractional differential equation. Phys. Lett. A 371, 26–33. Taghizadeh, N., Mirzazadeh, M., Rahimian, M., Akbari, M., 2013. Application of the simplest equation method to some timefractional partial differential equations. Ain Shams Eng. J. 4 (4), 897–902. Wang, D.S., Li, H., 2008. Single and multi-solitary wave solutions to a class of nonlinear evolution equations. J. Math. Anal. Appl. 343, 273–298. Wu, X.E., Liang, Y.S., 2017. Relationship between fractal dimensions and fractional calculus. Nonlinear Sci. Lett. A 8 (1), 77–89. Yan, L.M., Xu, F.S., 2015. Generalized exp-function method for nanlinear space-time fractional differential equations. Thermal Sci. 18 (5), 1573–1576. Zayed, E.M.E., Ibrahim, S.A.H., 2014. Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method. Acta Math. Appl. Sin. English Series 30 (3), 749– 754. Zayed, E.M.E., Al-Nowehy, A.G., 2016. Exact traveling wave solutions for nonlinear PDEs in mathematical physics using the generalized Kudryashov method. Serb. J. Electr. Eng. 13 (2), 203–227. Zayed, E.M.E., Amer, Y.A., 2015. Exact solutions for the nonlinear KPP equation by using the Riccati equation method combined with the ðG0 =GÞ-expansion method. Sci. Res. Essays 10 (3), 86–96. Zayed, E.M.E., Amer, Y.A., Shohib, R.M.A., 2015. The fractional ðDan G=GÞ-expansion method and its applications for solving four nonlinear space-time fractional PDEs in mathematial physics. Italian J. Pure Appl. Math. 34, 463–482. Zhang, S., Zong, Q.-A., Liu, D., Gao, Q., 2010. A generalized Expfunction method for fractional Riccati differential equations. Commun. Fract. Calculus 1, 48–51. Zhang, S., Zhang, H.-Q., 2011. Fractional sub-equation method and its applications to nonlinear fractional PDEs. Phys. Lett. A 375, 1069–1073. Zheng, B., 2012. ðG0 =GÞ-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun. Theor. Phys. 58, 623–630. Zheng, B., 2013. Exp-function method for solving fractional partial differential equations. Sci. World J. 2013, 465723.
Please cite this article in press as: Guner, O., Bekir, A. The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences (2017), http://dx.doi.org/10.1016/j.jaubas.2016.12.002