- Email: [email protected]
G-expansion method for nonlinear diffusion equations with nonlinear source
Journal of the Franklin Institute 347 (2010) 1391–1398 www.elsevier.com/locate/jfranklin
Short communication 0
Application of the G /G-expansion method for nonlinear diffusion equations with nonlinear source Ghodrat Ebadia,, Anjan Biswasb a Faculty of Mathematical Sciences, University of Tabriz, Tabriz 51666–14766, Iran Applied Mathematics Research Center, Center for Research and Education in Optical Sciences and Applications, Department of Mathematical Sciences, Delaware State University, Dover, DE 19901–2277, USA
b
Received 20 April 2010; received in revised form 16 May 2010; accepted 25 May 2010
Abstract The generalized diffusion equations, with nonlinear source terms which encompasses the Fisher, Newell–Whitehead, FitzHugh–Nagumo and Allen–Cahn equations as particular forms are solved by 0 the G /G-expansion method. The exact solutions are in terms of hyperbolic, trigonometric and rational functions with external parameters. This paper concludes with the stationary topological soliton solution of the FitzHugh–Nagumo equation that is obtained by the Ansatz method. & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction In the recent decades, many effective methods in order to obtain the exact solutions of nonlinear diffusion equations have been presented, such as the tanh–coth function method [2], the exp-function method [3]. 0 Recently, Wang et al. [5] proposed the G /G-expansion method to find travelling wave solutions of nonlinear equations. More recently, this method were proposed to solve nonlinear equations, variable coefficient equations and high dimensional equations [1,6]. 0 In addition to the G /G-expansion method, the solitary wave Ansatz method will be utilized to carry out the integration of the FitzHugh–Nagumo equation. It will be seen that by using this method, it will be possible to obtain the stationary topological soliton solution to
Corresponding author. Fax: þ98 4113342102.
E-mail address: [email protected] (G. Ebadi). 0016-0032/$32.00 & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2010.05.013
G. Ebadi, A. Biswas / Journal of the Franklin Institute 347 (2010) 1391–1398
1392
this equation. The Ansatz method has been studied earlier to carry out the integration of several nonlinear evolution equations [7]. 0 In this paper, the G /G-expansion method is used to solve the nonlinear diffusion PDEs with nonlinear source terms given by ut ¼ uxx þ au þ bun ;
ð1Þ
ut ¼ uxx uð1uÞðauÞ;
ð2Þ
where a, b are constant numbers. In Eq. (1) when a=4, b=4 and n=3, it gives the Allen–Cahn equation. In Eq. (1) when n=3 and the coefficient b is replaced by b, then it becomes the Newell–Whitehead equation. However, when n=2 and b=a, Eq. (1) reduces to the well-known Fisher equation. Eq. (2) is the FitzHugh–Nagumo equation. The Allen–Cahn equation, the Newell–Whitehead equation, the FitzHugh–Nagumo equation and the Fisher equation are important for modelling a variety of physical, biological and chemical phenomenons [4,8,9]. 0
2. Description of the G /G-expansion method 0
We can summarize the G /G-expansion method as follows: Step 1. A given nonlinear partial differential equation Pðu;ux ;ut ;uxx ;uxt ;utt ; . . .Þ ¼ 0;
ð3Þ
can be converted to an ODE 00
QðU;U 0 ;U ; . . .Þ ¼ 0;
ð4Þ
where uðx; tÞ ¼ UðxÞ, x ¼ xct. Eq. (4) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. Step 2. Suppose the solution of Eq. (4) can be expressed by a polynomial in G 0 =G as follows: 0 i N X G ðxÞ ai ; ð5Þ UðxÞ ¼ GðxÞ i¼0 where ai are real constants with aN a0 and the positive integer N can be determined by considering the homogeneous balance the highest order derivatives and highest order nonlinear appearing in ODE (4). The function GðxÞ is the solution of the auxiliary linear ordinary differential equation 00
G ðxÞ þ lG0 ðxÞ þ mGðxÞ ¼ 0;
ð6Þ
where l and m are real constants to be determined. Step 3. Substituting the solution of Eq. (6) together with Eq. (5) into Eq. (4) yields an 0 algebraic equation involving powers of G /G. Equating the coefficients of each power of 0 G /G to zero gives a system of algebraic equations for ai, l, m and c. Then, we solve the system with Maple to determine these constants. Next, depending on the sign of D ¼ l2 4m, we get solutions of Eq. (4).
G. Ebadi, A. Biswas / Journal of the Franklin Institute 347 (2010) 1391–1398
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3. Applications 0
In this section, we apply the G /G-expansion method to solve the Allen–Cahn equation, the FitzHugh–Nagumo equation, the Newell–Whithead equation and the Fisher equation.
3.1. The FitzHugh–Nagumo equation The FitzHugh–Nagumo equation is presented as ut ¼ uxx uð1uÞðauÞ;
ð7Þ
where a is a constant number. Through performing steps 1 and 2 of description of the 0 G /G-expansion method on Eq. (7) we get 00
cU 0 þ U Uð1UÞðaUÞ ¼ 0;
ð8Þ
where uðx; tÞ ¼ UðxÞ, x ¼ xct and 0 G ; a1 a0: UðxÞ ¼ a0 þ a1 G
ð9Þ 00
By using Eq. (9) and the solution of Eq. (6) we can find U 0 ðxÞ, U ðxÞ, U 2 ðxÞ and U 3 ðxÞ as 0 polynomials of G /G. Performing step 3 yields to a set of simultaneous algebraic equations for a0 ; a1 ; c; l and m as follows: a1 lmca1 ma0 a þ a20 þ a20 aa30 ¼ 0; a1 l2 þ 2a1 mca1 l þ 2a1 a0 3a1 a20 þ 2a1 a0 aa1 a ¼ 0; 3a1 lca1 3a21 a0 þ a21 þ a21 a ¼ 0; 2a1 a31 ¼ 0: Solving this system by Maple gives pffiffiffi pffiffiffi 2 1 a l þ ; a 1 ¼ 7 2; ðaÞ : a0 ¼ 7 2 2 2
pffiffiffi 2 ð1 þ aÞ; c¼8 2
pffiffiffi pffiffiffi 2 a l; a1 ¼ 7 2; ðbÞ : a0 ¼ 7 2 2
pffiffiffi 2 ða2Þ; c¼8 2
pffiffiffi pffiffiffi 2 1 ðcÞ : a0 ¼ 7 l; a1 ¼ 7 2; 2 2
pffiffiffi 2 ð2a1Þ; c¼7 2
1 1 1 1 m ¼ l2 þ a a2 : 4 8 4 8 ð10Þ
1 2 a2 : m¼ l 2 4
ð11Þ
1 2 1 l : 4 2
ð12Þ
m¼
Since in the above solution sets we have l2 4m40 so substituting the solution sets Eqs. (10)–(12) and the corresponding general solution of Eq. (6) into Eq. (9), we have the
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hyperbolic function travelling wave solutions of Eq. (8) respectively as 0 pffiffiffi ! pffiffiffi !1 ða1Þ 2x ða1Þ 2x þ c2 cosh Bc1 sinh C 4 4 C a þ 1 ða1ÞB B C; ! ! 7 U1 ðxÞ ¼ p ffiffi ffi p ffiffi ffi B C 2 2 @ ða1Þ 2x ða1Þ 2x A c1 cosh þ c2 sinh 4 4 pffiffiffi ð1 þ aÞ 2 t; x ¼ x8 2 0 pffiffiffi ! pffiffiffi !1 2ax 2ax c1 sinh þ c2 cosh pffiffiffi B C 4 4 C aB ða2Þ 2 U2 ðxÞ ¼ B 17 t; ; x ¼ x8 pffiffiffi ! pffiffiffi !C B C 2@ 2 2ax 2ax A þ c2 sinh c1 cosh 4 4 pffiffiffi ! pffiffiffi !1 2x 2x þ c2 cosh B c1 sinh 4 C 4 B C 1 ! ! U3 ðxÞ ¼ B 17 pffiffiffi pffiffiffi C B C; 2@ 2x 2x A þ c2 sinh c1 cosh 4 4 0
pffiffiffi ð2a1Þ 2 x ¼ x7 t: 2
In solutions Ui ðxÞ (i=1,2,3), c1 and c2 are arbitrary constants. In particular, if we take c1 a0 and c2 ¼ 0, then U1, U2 and U3 become, respectively, U11 ðxÞ ¼
a þ 1 a1 a1 aþ1 tanh pffiffiffi x pffiffiffi t ; 2 2 2 2 2
a a a2 ; U21 ðxÞ ¼ 1tanh pffiffiffi x pffiffiffi t 2 2 2 2 1 1 2a1 U31 ðxÞ ¼ : 1 þ tanh pffiffiffi x pffiffiffi t 2 2 2 2
ð13Þ
ð14Þ
ð15Þ
The solutions (13)–(15) are the same as Eqs. (55)–(57) in [4], respectively. If we take c1=0 and c2 a0, then U1, U2 and U3 become, respectively, U12 ðxÞ ¼
a þ 1 a1 a1 aþ1 coth pffiffiffi x pffiffiffi t ; 2 2 2 2 2
a a a2 1coth pffiffiffi x pffiffiffi t ; 2 2 2 2 1 1 2a1 U32 ðxÞ ¼ 1 þ coth pffiffiffi x pffiffiffi t ; 2 2 2 2
U22 ðxÞ ¼
ð16Þ
ð17Þ
ð18Þ
G. Ebadi, A. Biswas / Journal of the Franklin Institute 347 (2010) 1391–1398
or
1395
a þ 1 a1 a1 aþ1 a1 aþ1 tanh pffiffiffi x pffiffiffi t þ coth pffiffiffi x pffiffiffi t ; U13 ðxÞ ¼ 2 2 4 2 2 4 2 2 ð19Þ
a a a2 a a2 ; 1tanh pffiffiffi x pffiffiffi t coth pffiffiffi x pffiffiffi t 4 2 2 4 2 4 2 1 1 2a1 1 2a1 þ coth pffiffiffi x pffiffiffi t : U33 ðxÞ ¼ 2 þ tanh pffiffiffi x pffiffiffi t 4 2 2 4 2 4 2 U23 ðxÞ ¼
ð20Þ
ð21Þ
The solutions (16)–(21) are the same as Eqs. (58)–(63) in [4], respectively. So our solutions logically contain the solutions in [4]. 3.2. The Allen–Cahn equation, the Newell–Whithead equation and the Fisher equation 0
Through performing step 1 of description of the (G /G)-expansion method on Eq. (1) we get the ODE 00
cU 0 ðxÞ þ U ðxÞ þ aUðxÞ þ bU n ðxÞ ¼ 0; or 00
ðn1Þcvv0 þ ðn1Þvv þ ð2nÞðv0 Þ2 þ aðn1Þ2 v2 þ bðn1Þ2 v3 ¼ 0; 00
1=n1
where UðxÞ ¼ v
ð22Þ
3
ðxÞ. Balancing vv with v gives N=2. Therefore, we can write 0 0 2 G G þ a2 ; a2 a0: vðxÞ ¼ a0 þ a1 G G
ð23Þ
00
By using Eq. (23) and the solution of Eq. (6) we can find v0 ðxÞ; v ðxÞ; v2 ðxÞ and v3 ðxÞ as 0 0 polynomials of G /G. Performing step 3 of description of the G /G-expansion method yields to a system of algebraic equation and solving it by Maple gives ðaÞ : a0 ¼
c ¼ 0;
ðbÞ : a0 ¼
a1 ¼
c¼
aðn1Þ2 ðn þ 1Þ þ l2 ðn þ 1Þ ; 2bðn1Þ2
a1 ¼
2lðn þ 1Þ 2ðn þ 1Þ ; a2 ¼ ; 2 bðn1Þ bðn1Þ2
1 1 1 1 m ¼ an2 an þ a þ l2 : 4 2 4 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðn1Þ2 þ 2l2 ðn þ 1Þ82l 2aðn1Þ2 ðn þ 1Þ 4bðn1Þ2
2lðn þ 1Þ7
7ðn þ 3Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2aðn1Þ2 ðn þ 1Þ
bðn1Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2aðn1Þ2 ðn þ 1Þ 2ðn2 1Þ
; a2 ¼
; m¼
ð24Þ
;
2ðn þ 1Þ ; bðn1Þ2
an2 2l2 n2an2l2 þ a : 8ð1 þ nÞ
ð25Þ
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By substituting the solution set (24) and the corresponding solutions of Eq. (6) into Eq. (23), we obtain the trigonometric and hyperbolic travelling wave solutions of Eq. (22) as follows: Case 1: When a40 then D ¼ l2 4m ¼ aðn1Þ2 o0, so we have v1 ðxÞ ¼
aðn þ 1Þðc21 þ c22 Þ ; 2bðc2 sinðbÞ þ c1 cosðbÞÞ2
b¼
n1 pffiffiffi n1 pffiffiffi ax ¼ ax: 2 2
ð26Þ
Case 2: When ao0 then D ¼ l2 4m ¼ aðn1Þ2 40, so we can get v2 ðxÞ ¼
aðn þ 1Þðc22 c21 Þ n1 pffiffiffiffiffiffiffi n1 pffiffiffiffiffiffiffi ; b¼ ax ¼ ax: 2 2 2 2bðc2 sinhðbÞ þ c1 coshðbÞÞ
ð27Þ
Also substituting the solution set (25) and the corresponding solutions of Eq. (6) into Eq. (23), we obtain the hyperbolic travelling wave solutions of Eq. (22) as follows: Case 1: When a40 then D ¼ l2 4m ¼ aðn1Þ2 =2ð1 þ nÞ40, so we have pffiffiffiffiffi aðc1 c2 Þ2 ðcoshðbÞsinhðbÞÞ2 ðn1Þ 2a v3 ðxÞ ¼ ; b ¼ pffiffiffiffiffiffiffiffiffiffiffi x ð28Þ 4 nþ1 4bðc2 sinhðbÞ þ c1 coshðbÞÞ2 and aðc1 þ c2 Þ2 ðcoshðbÞ þ sinhðbÞÞ2 v4 ðxÞ ¼ ; 4bðc2 sinhðbÞ þ c1 coshðbÞÞ2
pffiffiffiffiffi ðn1Þ 2a p ffiffiffiffiffiffiffiffiffiffiffi x: b¼ 4 nþ1
ð29Þ
Case 2: When ao0 then we have some complex travelling wave solutions. In particular, if we take c1=1 and c2=0, then v3 and v4 become a ð17tanhðbÞÞ2 ; v5 ðxÞ ¼ 4b that these solutions are the same as Eq. (22) in [4]. If we take c1=0 and c2=1, then v3 and v4 become a ð17cothðbÞÞ2 ; v6 ðxÞ ¼ 4b that these solutions are the same as Eqs. (23) in [4]. So our solutions logically contain the solutions in [4] and our solutions are more general. Since the Allen–Cahn, the Newell–Whithead and the Fisher equations are special cases of the Eq. (1), So setting a=4, b=4, n=3 and n=2, b=a in solutions (26)–(29), they become the solutions of the Allen–Cahn and the Fisher equations. Also, when n=3 and the coefficient b is replaced by b in the solutions (26)–(29), they become the solutions of the Newell–Whitehead equation. 4. Ansatz method In this section the solitary wave ansatz method will be used to integrate the FitzHugh–Nagumo equation. The form of this equation that will be studied in this
G. Ebadi, A. Biswas / Journal of the Franklin Institute 347 (2010) 1391–1398
1397
section is given by qt ¼ aqxx qð1qÞðbqÞ:
ð30Þ
The focus will be on obtaining topological soliton solutions of this equation. The starting hypothesis of this is therefore going to be qðx;tÞ ¼ Atanhp t;
t ¼ BðxvtÞ;
ð31Þ
where A and B are free parameters and v is the velocity of the soliton. Thus from Eq. (31), it is possible to obtain qt ¼ pvABðtanhpþ1 ttanhp1 tÞ qxx ¼ pðp1ÞAB2 tanhp2 t2p2 AB2 tanhp t þ pðp þ 1Þtanhpþ2 t: Thus, Eq. (30) reduces to pvABðtanhpþ1 ttanhp1 tÞ ¼ apðp1ÞAB2 tanhp2 t2ap2 AB2 tanhp t þ apðp þ 1Þtanhpþ2 t bAtanhp t þ A2 tanh2p t þ bA2 tanh2p tA3 tanh3p t:
ð32Þ
Now, from Eq. (32), equating the exponents 2p and pþ1 gives 2p=pþ1 which gives p=1. The same value of p is obtained when the exponents 3p and pþ2 are equated. Thus, the linearly independent functions in Eq. (32) are tanhpþj t for j=1, 0, 1, 2. Therefore setting pffiffiffiffiffi their respective coefficients to zero yields A=1, B ¼ 1= 2a, v=0, b=1. Therefore the FitzHugh–Nagumo equation modifies to qt ¼ aqxx þ qð1q2 Þ whose stationary 1-soliton pffiffiffiffiffi solution is therefore given by qðx; tÞ ¼ tanhðx= 2aÞ. 5. Conclusions In this paper, the exact travelling wave solutions to the nonlinear FitzHugh–Nagumo, Allen–Cahn, Newell–Whitehead and Fisher equations was obtained by the aid of 0 G /G-expansion method. The solutions contains free parameters. These solutions will be very useful in various physical situations where these equations arise. Finally, using the Ansatz method, the stationary topological soliton solution of the FitzHugh–Nagumo equation has been obtained. Acknowledgement The research work of the second author (AB) was fully supported by NSF-CREST Grant No. HRD-0630388 and this support is genuinely and sincerely appreciated. References 0
[1] A. Bekir, Application of the (G /G)-expansion method for nonlinear evolution equations, Phys. Lett. A 372 (19) (2008) 3400–3406. [2] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (4–5) (2000) 212–218. [3] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fract. 30 (3) (2006) 700–708. [4] A.M. Wazwaz, The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations, Appl. Math. Comput. 188 (2) (2007) 1467–1475.
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[5] M.L. Wang, X.Z. Li, J.L. Zhang, The (G /G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A 372 (4) (2008) 417–423. 0 [6] S. Zhang, J.L. Tong, W. Wang, A generalized (G /G)-expansion method for the mKdV equation with variable coefficients, Phys. Lett. A 372 (13) (2008) 2254–2257. [7] A. Biswas, D. Milovic, Bright and dark solitons of the generalized nonlinear Schrodinger’s equation, Commun. Nonlinear Sci. Numer. Simulation 15 (6) (2010) 1473–1484. [8] J.G. Alford, Bifurcation structure of rotating wave solutions of the FitzHugh–Nagumo equations, Commun. in Nonlinear Sci. Numer. Simulation 14 (8) (2009) 3282–3291. [9] P.W. Bates, S. Brown, J. Han, Numerical analysis for a nonlocal Allen–Cahn equation, Int. J. Numer. Anal. Modeling 6 (1) (2009) 33–49.
Short communication 0
Application of the G /G-expansion method for nonlinear diffusion equations with nonlinear source Ghodrat Ebadia,, Anjan Biswasb a Faculty of Mathematical Sciences, University of Tabriz, Tabriz 51666–14766, Iran Applied Mathematics Research Center, Center for Research and Education in Optical Sciences and Applications, Department of Mathematical Sciences, Delaware State University, Dover, DE 19901–2277, USA
b
Received 20 April 2010; received in revised form 16 May 2010; accepted 25 May 2010
Abstract The generalized diffusion equations, with nonlinear source terms which encompasses the Fisher, Newell–Whitehead, FitzHugh–Nagumo and Allen–Cahn equations as particular forms are solved by 0 the G /G-expansion method. The exact solutions are in terms of hyperbolic, trigonometric and rational functions with external parameters. This paper concludes with the stationary topological soliton solution of the FitzHugh–Nagumo equation that is obtained by the Ansatz method. & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction In the recent decades, many effective methods in order to obtain the exact solutions of nonlinear diffusion equations have been presented, such as the tanh–coth function method [2], the exp-function method [3]. 0 Recently, Wang et al. [5] proposed the G /G-expansion method to find travelling wave solutions of nonlinear equations. More recently, this method were proposed to solve nonlinear equations, variable coefficient equations and high dimensional equations [1,6]. 0 In addition to the G /G-expansion method, the solitary wave Ansatz method will be utilized to carry out the integration of the FitzHugh–Nagumo equation. It will be seen that by using this method, it will be possible to obtain the stationary topological soliton solution to
Corresponding author. Fax: þ98 4113342102.
E-mail address: [email protected] (G. Ebadi). 0016-0032/$32.00 & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2010.05.013
G. Ebadi, A. Biswas / Journal of the Franklin Institute 347 (2010) 1391–1398
1392
this equation. The Ansatz method has been studied earlier to carry out the integration of several nonlinear evolution equations [7]. 0 In this paper, the G /G-expansion method is used to solve the nonlinear diffusion PDEs with nonlinear source terms given by ut ¼ uxx þ au þ bun ;
ð1Þ
ut ¼ uxx uð1uÞðauÞ;
ð2Þ
where a, b are constant numbers. In Eq. (1) when a=4, b=4 and n=3, it gives the Allen–Cahn equation. In Eq. (1) when n=3 and the coefficient b is replaced by b, then it becomes the Newell–Whitehead equation. However, when n=2 and b=a, Eq. (1) reduces to the well-known Fisher equation. Eq. (2) is the FitzHugh–Nagumo equation. The Allen–Cahn equation, the Newell–Whitehead equation, the FitzHugh–Nagumo equation and the Fisher equation are important for modelling a variety of physical, biological and chemical phenomenons [4,8,9]. 0
2. Description of the G /G-expansion method 0
We can summarize the G /G-expansion method as follows: Step 1. A given nonlinear partial differential equation Pðu;ux ;ut ;uxx ;uxt ;utt ; . . .Þ ¼ 0;
ð3Þ
can be converted to an ODE 00
QðU;U 0 ;U ; . . .Þ ¼ 0;
ð4Þ
where uðx; tÞ ¼ UðxÞ, x ¼ xct. Eq. (4) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. Step 2. Suppose the solution of Eq. (4) can be expressed by a polynomial in G 0 =G as follows: 0 i N X G ðxÞ ai ; ð5Þ UðxÞ ¼ GðxÞ i¼0 where ai are real constants with aN a0 and the positive integer N can be determined by considering the homogeneous balance the highest order derivatives and highest order nonlinear appearing in ODE (4). The function GðxÞ is the solution of the auxiliary linear ordinary differential equation 00
G ðxÞ þ lG0 ðxÞ þ mGðxÞ ¼ 0;
ð6Þ
where l and m are real constants to be determined. Step 3. Substituting the solution of Eq. (6) together with Eq. (5) into Eq. (4) yields an 0 algebraic equation involving powers of G /G. Equating the coefficients of each power of 0 G /G to zero gives a system of algebraic equations for ai, l, m and c. Then, we solve the system with Maple to determine these constants. Next, depending on the sign of D ¼ l2 4m, we get solutions of Eq. (4).
G. Ebadi, A. Biswas / Journal of the Franklin Institute 347 (2010) 1391–1398
1393
3. Applications 0
In this section, we apply the G /G-expansion method to solve the Allen–Cahn equation, the FitzHugh–Nagumo equation, the Newell–Whithead equation and the Fisher equation.
3.1. The FitzHugh–Nagumo equation The FitzHugh–Nagumo equation is presented as ut ¼ uxx uð1uÞðauÞ;
ð7Þ
where a is a constant number. Through performing steps 1 and 2 of description of the 0 G /G-expansion method on Eq. (7) we get 00
cU 0 þ U Uð1UÞðaUÞ ¼ 0;
ð8Þ
where uðx; tÞ ¼ UðxÞ, x ¼ xct and 0 G ; a1 a0: UðxÞ ¼ a0 þ a1 G
ð9Þ 00
By using Eq. (9) and the solution of Eq. (6) we can find U 0 ðxÞ, U ðxÞ, U 2 ðxÞ and U 3 ðxÞ as 0 polynomials of G /G. Performing step 3 yields to a set of simultaneous algebraic equations for a0 ; a1 ; c; l and m as follows: a1 lmca1 ma0 a þ a20 þ a20 aa30 ¼ 0; a1 l2 þ 2a1 mca1 l þ 2a1 a0 3a1 a20 þ 2a1 a0 aa1 a ¼ 0; 3a1 lca1 3a21 a0 þ a21 þ a21 a ¼ 0; 2a1 a31 ¼ 0: Solving this system by Maple gives pffiffiffi pffiffiffi 2 1 a l þ ; a 1 ¼ 7 2; ðaÞ : a0 ¼ 7 2 2 2
pffiffiffi 2 ð1 þ aÞ; c¼8 2
pffiffiffi pffiffiffi 2 a l; a1 ¼ 7 2; ðbÞ : a0 ¼ 7 2 2
pffiffiffi 2 ða2Þ; c¼8 2
pffiffiffi pffiffiffi 2 1 ðcÞ : a0 ¼ 7 l; a1 ¼ 7 2; 2 2
pffiffiffi 2 ð2a1Þ; c¼7 2
1 1 1 1 m ¼ l2 þ a a2 : 4 8 4 8 ð10Þ
1 2 a2 : m¼ l 2 4
ð11Þ
1 2 1 l : 4 2
ð12Þ
m¼
Since in the above solution sets we have l2 4m40 so substituting the solution sets Eqs. (10)–(12) and the corresponding general solution of Eq. (6) into Eq. (9), we have the
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hyperbolic function travelling wave solutions of Eq. (8) respectively as 0 pffiffiffi ! pffiffiffi !1 ða1Þ 2x ða1Þ 2x þ c2 cosh Bc1 sinh C 4 4 C a þ 1 ða1ÞB B C; ! ! 7 U1 ðxÞ ¼ p ffiffi ffi p ffiffi ffi B C 2 2 @ ða1Þ 2x ða1Þ 2x A c1 cosh þ c2 sinh 4 4 pffiffiffi ð1 þ aÞ 2 t; x ¼ x8 2 0 pffiffiffi ! pffiffiffi !1 2ax 2ax c1 sinh þ c2 cosh pffiffiffi B C 4 4 C aB ða2Þ 2 U2 ðxÞ ¼ B 17 t; ; x ¼ x8 pffiffiffi ! pffiffiffi !C B C 2@ 2 2ax 2ax A þ c2 sinh c1 cosh 4 4 pffiffiffi ! pffiffiffi !1 2x 2x þ c2 cosh B c1 sinh 4 C 4 B C 1 ! ! U3 ðxÞ ¼ B 17 pffiffiffi pffiffiffi C B C; 2@ 2x 2x A þ c2 sinh c1 cosh 4 4 0
pffiffiffi ð2a1Þ 2 x ¼ x7 t: 2
In solutions Ui ðxÞ (i=1,2,3), c1 and c2 are arbitrary constants. In particular, if we take c1 a0 and c2 ¼ 0, then U1, U2 and U3 become, respectively, U11 ðxÞ ¼
a þ 1 a1 a1 aþ1 tanh pffiffiffi x pffiffiffi t ; 2 2 2 2 2
a a a2 ; U21 ðxÞ ¼ 1tanh pffiffiffi x pffiffiffi t 2 2 2 2 1 1 2a1 U31 ðxÞ ¼ : 1 þ tanh pffiffiffi x pffiffiffi t 2 2 2 2
ð13Þ
ð14Þ
ð15Þ
The solutions (13)–(15) are the same as Eqs. (55)–(57) in [4], respectively. If we take c1=0 and c2 a0, then U1, U2 and U3 become, respectively, U12 ðxÞ ¼
a þ 1 a1 a1 aþ1 coth pffiffiffi x pffiffiffi t ; 2 2 2 2 2
a a a2 1coth pffiffiffi x pffiffiffi t ; 2 2 2 2 1 1 2a1 U32 ðxÞ ¼ 1 þ coth pffiffiffi x pffiffiffi t ; 2 2 2 2
U22 ðxÞ ¼
ð16Þ
ð17Þ
ð18Þ
G. Ebadi, A. Biswas / Journal of the Franklin Institute 347 (2010) 1391–1398
or
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a þ 1 a1 a1 aþ1 a1 aþ1 tanh pffiffiffi x pffiffiffi t þ coth pffiffiffi x pffiffiffi t ; U13 ðxÞ ¼ 2 2 4 2 2 4 2 2 ð19Þ
a a a2 a a2 ; 1tanh pffiffiffi x pffiffiffi t coth pffiffiffi x pffiffiffi t 4 2 2 4 2 4 2 1 1 2a1 1 2a1 þ coth pffiffiffi x pffiffiffi t : U33 ðxÞ ¼ 2 þ tanh pffiffiffi x pffiffiffi t 4 2 2 4 2 4 2 U23 ðxÞ ¼
ð20Þ
ð21Þ
The solutions (16)–(21) are the same as Eqs. (58)–(63) in [4], respectively. So our solutions logically contain the solutions in [4]. 3.2. The Allen–Cahn equation, the Newell–Whithead equation and the Fisher equation 0
Through performing step 1 of description of the (G /G)-expansion method on Eq. (1) we get the ODE 00
cU 0 ðxÞ þ U ðxÞ þ aUðxÞ þ bU n ðxÞ ¼ 0; or 00
ðn1Þcvv0 þ ðn1Þvv þ ð2nÞðv0 Þ2 þ aðn1Þ2 v2 þ bðn1Þ2 v3 ¼ 0; 00
1=n1
where UðxÞ ¼ v
ð22Þ
3
ðxÞ. Balancing vv with v gives N=2. Therefore, we can write 0 0 2 G G þ a2 ; a2 a0: vðxÞ ¼ a0 þ a1 G G
ð23Þ
00
By using Eq. (23) and the solution of Eq. (6) we can find v0 ðxÞ; v ðxÞ; v2 ðxÞ and v3 ðxÞ as 0 0 polynomials of G /G. Performing step 3 of description of the G /G-expansion method yields to a system of algebraic equation and solving it by Maple gives ðaÞ : a0 ¼
c ¼ 0;
ðbÞ : a0 ¼
a1 ¼
c¼
aðn1Þ2 ðn þ 1Þ þ l2 ðn þ 1Þ ; 2bðn1Þ2
a1 ¼
2lðn þ 1Þ 2ðn þ 1Þ ; a2 ¼ ; 2 bðn1Þ bðn1Þ2
1 1 1 1 m ¼ an2 an þ a þ l2 : 4 2 4 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðn1Þ2 þ 2l2 ðn þ 1Þ82l 2aðn1Þ2 ðn þ 1Þ 4bðn1Þ2
2lðn þ 1Þ7
7ðn þ 3Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2aðn1Þ2 ðn þ 1Þ
bðn1Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2aðn1Þ2 ðn þ 1Þ 2ðn2 1Þ
; a2 ¼
; m¼
ð24Þ
;
2ðn þ 1Þ ; bðn1Þ2
an2 2l2 n2an2l2 þ a : 8ð1 þ nÞ
ð25Þ
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G. Ebadi, A. Biswas / Journal of the Franklin Institute 347 (2010) 1391–1398
By substituting the solution set (24) and the corresponding solutions of Eq. (6) into Eq. (23), we obtain the trigonometric and hyperbolic travelling wave solutions of Eq. (22) as follows: Case 1: When a40 then D ¼ l2 4m ¼ aðn1Þ2 o0, so we have v1 ðxÞ ¼
aðn þ 1Þðc21 þ c22 Þ ; 2bðc2 sinðbÞ þ c1 cosðbÞÞ2
b¼
n1 pffiffiffi n1 pffiffiffi ax ¼ ax: 2 2
ð26Þ
Case 2: When ao0 then D ¼ l2 4m ¼ aðn1Þ2 40, so we can get v2 ðxÞ ¼
aðn þ 1Þðc22 c21 Þ n1 pffiffiffiffiffiffiffi n1 pffiffiffiffiffiffiffi ; b¼ ax ¼ ax: 2 2 2 2bðc2 sinhðbÞ þ c1 coshðbÞÞ
ð27Þ
Also substituting the solution set (25) and the corresponding solutions of Eq. (6) into Eq. (23), we obtain the hyperbolic travelling wave solutions of Eq. (22) as follows: Case 1: When a40 then D ¼ l2 4m ¼ aðn1Þ2 =2ð1 þ nÞ40, so we have pffiffiffiffiffi aðc1 c2 Þ2 ðcoshðbÞsinhðbÞÞ2 ðn1Þ 2a v3 ðxÞ ¼ ; b ¼ pffiffiffiffiffiffiffiffiffiffiffi x ð28Þ 4 nþ1 4bðc2 sinhðbÞ þ c1 coshðbÞÞ2 and aðc1 þ c2 Þ2 ðcoshðbÞ þ sinhðbÞÞ2 v4 ðxÞ ¼ ; 4bðc2 sinhðbÞ þ c1 coshðbÞÞ2
pffiffiffiffiffi ðn1Þ 2a p ffiffiffiffiffiffiffiffiffiffiffi x: b¼ 4 nþ1
ð29Þ
Case 2: When ao0 then we have some complex travelling wave solutions. In particular, if we take c1=1 and c2=0, then v3 and v4 become a ð17tanhðbÞÞ2 ; v5 ðxÞ ¼ 4b that these solutions are the same as Eq. (22) in [4]. If we take c1=0 and c2=1, then v3 and v4 become a ð17cothðbÞÞ2 ; v6 ðxÞ ¼ 4b that these solutions are the same as Eqs. (23) in [4]. So our solutions logically contain the solutions in [4] and our solutions are more general. Since the Allen–Cahn, the Newell–Whithead and the Fisher equations are special cases of the Eq. (1), So setting a=4, b=4, n=3 and n=2, b=a in solutions (26)–(29), they become the solutions of the Allen–Cahn and the Fisher equations. Also, when n=3 and the coefficient b is replaced by b in the solutions (26)–(29), they become the solutions of the Newell–Whitehead equation. 4. Ansatz method In this section the solitary wave ansatz method will be used to integrate the FitzHugh–Nagumo equation. The form of this equation that will be studied in this
G. Ebadi, A. Biswas / Journal of the Franklin Institute 347 (2010) 1391–1398
1397
section is given by qt ¼ aqxx qð1qÞðbqÞ:
ð30Þ
The focus will be on obtaining topological soliton solutions of this equation. The starting hypothesis of this is therefore going to be qðx;tÞ ¼ Atanhp t;
t ¼ BðxvtÞ;
ð31Þ
where A and B are free parameters and v is the velocity of the soliton. Thus from Eq. (31), it is possible to obtain qt ¼ pvABðtanhpþ1 ttanhp1 tÞ qxx ¼ pðp1ÞAB2 tanhp2 t2p2 AB2 tanhp t þ pðp þ 1Þtanhpþ2 t: Thus, Eq. (30) reduces to pvABðtanhpþ1 ttanhp1 tÞ ¼ apðp1ÞAB2 tanhp2 t2ap2 AB2 tanhp t þ apðp þ 1Þtanhpþ2 t bAtanhp t þ A2 tanh2p t þ bA2 tanh2p tA3 tanh3p t:
ð32Þ
Now, from Eq. (32), equating the exponents 2p and pþ1 gives 2p=pþ1 which gives p=1. The same value of p is obtained when the exponents 3p and pþ2 are equated. Thus, the linearly independent functions in Eq. (32) are tanhpþj t for j=1, 0, 1, 2. Therefore setting pffiffiffiffiffi their respective coefficients to zero yields A=1, B ¼ 1= 2a, v=0, b=1. Therefore the FitzHugh–Nagumo equation modifies to qt ¼ aqxx þ qð1q2 Þ whose stationary 1-soliton pffiffiffiffiffi solution is therefore given by qðx; tÞ ¼ tanhðx= 2aÞ. 5. Conclusions In this paper, the exact travelling wave solutions to the nonlinear FitzHugh–Nagumo, Allen–Cahn, Newell–Whitehead and Fisher equations was obtained by the aid of 0 G /G-expansion method. The solutions contains free parameters. These solutions will be very useful in various physical situations where these equations arise. Finally, using the Ansatz method, the stationary topological soliton solution of the FitzHugh–Nagumo equation has been obtained. Acknowledgement The research work of the second author (AB) was fully supported by NSF-CREST Grant No. HRD-0630388 and this support is genuinely and sincerely appreciated. References 0
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