An improvement on the exp-function method when balancing the highest order linear and nonlinear terms. (English) Zbl 1239.35008

Ebaid, Abdelhalim

J. Math. Anal. Appl. 392, No. 1, 1-5 (2012).

Summary: The exp-function method has been widely used to solve different kinds of nonlinear partial differential equations. These nonlinear partial differential equations are transformed first into nonlinear ordinary differential equations and then the ansatz of the exp-function method: \(u=\frac{\sum^p_{n=-c}a_n\exp(n\eta)}{\sum^q_{m=-d}b_n\exp(m\eta)}\) is applied to obtain the solution. However, a part of the solution using this method is to construct the relations between \(c\), \(p\), \(d\) and \(q\) by balancing the highest order linear term with the highest order nonlinear term. It is proved in the present paper that \(c=d\) and \(p=q\) are the only relations that can be obtained by applying this method to any nonlinear ordinary differential equation. Therefore, the additional calculations of balancing the highest order linear term with the highest order nonlinear term are not longer required in the future. Hence, the method becomes more straightforward.

MSC:

35A24	Methods of ordinary differential equations applied to PDEs
35G20	Nonlinear higher-order PDEs
34A34	Nonlinear ordinary differential equations and systems

Keywords:

exp-function method; nonlinear wave equations; soliton

References:

[1]	He, Ji-Huan; Wu, Xu-Hong, Exp-function method for nonlinear wave equations, Chaos solitons fractals, 30, 3, 700-708, (2006) · Zbl 1141.35448
[2]	Wu, Xu-Hong (Benn); He, Ji-Huan, Solitary solutions, periodic solutions and compacton-like solutions using the exp-function method, Comput. math. appl., 54, 966-986, (2007) · Zbl 1143.35360
[3]	Ebaid, A., Exact solitary wave solutions for some nonlinear evolution equations via exp-function method, Phys. lett. A, 365, 213-219, (2007) · Zbl 1203.35213
[4]	Wu, Xu-Hong (Benn); He, Ji-Huan, Exp-function method and its application to nonlinear equations, Chaos solitons fractals, 38, 903-910, (2008) · Zbl 1153.35384
[5]	He, Ji-Huan; Zhang, Li-Na, Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the exp-function method, Phys. lett. A, 372, 1044-1047, (2008) · Zbl 1217.35152
[6]	Ebaid, A., Exact solutions for the generalized Klein-Gordon equation via a transformation and exp-function method and comparison with adomianʼs method, Comput. appl. math., 223, 278-290, (2009) · Zbl 1155.65079
[7]	Ebaid, A., Application of the exp-function method for solving some evolution equations with nonlinear terms of any orders, Z. naturforschung A, 65, 1039-1044, (2010)
[8]	Zhang, Sheng, Exp-function method for Riccati equation and new exact solutions with two arbitrary functions of (\(2 + 1\))-dimensional konopelchenko-dubrovsky equations, Appl. math. comput., 216, 5, 1546-1552, (2010) · Zbl 1189.35299
[9]	Aslan, Ismail, The exp-function approach to the Schwarzian Korteweg-de Vries equation, Comput. math. appl., 59, 8, 2896-2900, (2010) · Zbl 1193.35180
[10]	Wazwaz, Abdul-Majid; Mehanna, Mona S., Application of exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer-Kaup-Kupershmidt equations, Appl. math. comput., 217, 1484-1490, (2010) · Zbl 1203.35247
[11]	Bekir, Ahmet; Aksoy, Esin, Exact solutions of nonlinear evolution equations with variable coefficients using exp-function method, Appl. math. comput., 217, 1, 430-436, (2010) · Zbl 1197.35213
[12]	Inan, Ibrahim E.; Ugurlu, Yavuz, Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation, Phys. lett. A, 217, 4, 1294-1299, (2010) · Zbl 1202.35228
[13]	Yildirim, Ahmet; Pinar, Zehra, Application of the exp-function method for solving nonlinear reaction-diffusion equations arising in mathematical biology, Comput. math. appl., 60, 1873-1880, (2010) · Zbl 1205.35325
[14]	Chun, Changbum, New solitary wave solutions to nonlinear evolution equations by the exp-function method, Comput. math. appl., 61, 8, 2107-2110, (2011) · Zbl 1219.35224
[15]	Shuimeng Yu, N-soliton solutions of the KP equation by Exp-function method, Appl. Math. Comput., doi:10.1016/j.amc.2010.12.095, in press. · Zbl 1311.35267
[16]	Chang, Jin-Rong, The exp-function method and generalized solitary solutions, Comput. math. appl., 61, 8, 2081-2084, (2011) · Zbl 1219.35001