Improved (G′/G)-expansion and extended tanh methods for (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation

Abstract Traveling wave solutions of nonlinear (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation are constructed by using improved (G′/G)-expansion and extended tanh-function methods. We demonstrate that some of the obtained exact solutions of these two methods are analogous, but they are not one and the same. The solutions obtained in this article may be imperative and significant for the explanation of some practical physical phenomena. It is observed that improved (G′/G)-expansion method gives a variety of exact solutions.

[1]  Xiangzheng Li,et al.  Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations , 2005 .

[2]  A. A. Soliman,et al.  New application of Exp-function method for improved Boussinesq equation , 2007 .

[3]  Norhashidah Hj. Mohd. Ali,et al.  A Generalized and Improved -Expansion Method for Nonlinear Evolution Equations , 2012 .

[4]  J. A. Tenreiro Machado,et al.  Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators , 2014 .

[5]  Abdul-Majid Wazwaz,et al.  New solutions of distinct physical structures to high-dimensional nonlinear evolution equations , 2008, Appl. Math. Comput..

[6]  M. Noor,et al.  Variational iteration method for solving twelfth-order boundary-value problems using He’s polynomials , 2010 .

[7]  Nikolai A. Kudryashov,et al.  Seven common errors in finding exact solutions of nonlinear differential equations , 2009, 1011.4268.

[8]  Ahmet Bekir,et al.  Exact solutions of coupled nonlinear evolution equations , 2008 .

[9]  Mingliang Wang,et al.  Periodic wave solutions to a coupled KdV equations with variable coefficients , 2003 .

[10]  Ji-Huan He,et al.  Exp-function method for nonlinear wave equations , 2006 .

[11]  Maria Luz Gandarias,et al.  The Calogero–Bogoyavlenskii–Schiff Equation in 2+1 Dimensions , 2003 .

[12]  Nikolai A. Kudryashov,et al.  Exact solutions of the generalized Kuramoto-Sivashinsky equation , 1990 .

[13]  M. A. Abdou The extended tanh method and its applications for solving nonlinear physical models , 2007, Appl. Math. Comput..

[14]  Zuntao Fu,et al.  JACOBI ELLIPTIC FUNCTION EXPANSION METHOD AND PERIODIC WAVE SOLUTIONS OF NONLINEAR WAVE EQUATIONS , 2001 .

[15]  Xiao‐Jun Yang,et al.  Fractal heat conduction problem solved by local fractional variation iteration method , 2013 .

[16]  Xiaoling Zhang,et al.  A new generalized Riccati equation rational expansion method to a class of nonlinear evolution equations with nonlinear terms of any order , 2007, Appl. Math. Comput..

[17]  Nasir Taghizadeh,et al.  The first integral method to some complex nonlinear partial differential equations , 2011, J. Comput. Appl. Math..

[18]  Ahmad T. Ali New generalized Jacobi elliptic function rational expansion method , 2011, J. Comput. Appl. Math..

[19]  E. Zayed,et al.  A further improved (G′/G)- expansion method and the extended tanh- method for finding exact solutions of nonlinear PDEs , 2011 .

[20]  Mingliang Wang,et al.  The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics , 2008 .

[21]  Mingliang Wang Exact solutions for a compound KdV-Burgers equation , 1996 .

[22]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[23]  Syed Tauseef Mohyud-Din,et al.  Assessment of the further improved (G'/G)-expansion method and the extended tanh-method in probing exact solutions of nonlinear PDEs , 2013, SpringerPlus.

[24]  M. Ali Akbar,et al.  New Solitary and Periodic Solutions of Nonlinear Evolution Equation by Exp-function Method , 2012 .

[25]  C. S. Gardner,et al.  Method for solving the Korteweg-deVries equation , 1967 .

[26]  W. Malfliet Solitary wave solutions of nonlinear wave equations , 1992 .

[27]  Jiao-Ling Zhang,et al.  A generalized (G′G)-expansion method and its applications , 2008 .

[28]  Muhammad Aslam Noor,et al.  Exp-Function Method for Generalized Travelling Solutions of Calogero-Degasperis-Fokas Equation , 2010 .

[29]  M. A. Abdou,et al.  RETRACTION (G' / G)-expansion method equivalent to the extended tanh-function method , 2010 .

[30]  Norhashidah Hj. Mohd. Ali,et al.  Abundant Exact Traveling Wave Solutions of Generalized Bretherton Equation via Improved (G′/G)-Expansion Method , 2012 .

[31]  Nikolai A. Kudryashov,et al.  On types of nonlinear nonintegrable equations with exact solutions , 1991 .

[32]  Kouichi Toda,et al.  The Painlevé Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients , 2006 .

[33]  Norhashidah Hj. Mohd. Ali,et al.  The modified alternative -expansion method for finding the exact solutions of nonlinear PDEs in mathematical physics , 2011 .

[34]  J. A. E. R. Institute,et al.  N soliton solutions to the Bogoyavlenskii-Schiff equation and a quest for the soliton solution in (3 1) dimensions , 1998, solv-int/9801003.

[35]  R. Hirota Exact solution of the Korteweg-deVries equation for multiple collision of solitons , 1971 .

[36]  T. Xia,et al.  A generalized new auxiliary equation method and its applications to nonlinear partial differential equations , 2007 .

[37]  Abdul-Majid Wazwaz,et al.  The extended tanh method for new compact and noncompact solutions for the KP–BBM and the ZK–BBM equations , 2008 .

[38]  Sheng Zhang,et al.  A further improved tanh function method exactly solving the -dimensional dispersive long wave equations. , 2008 .

[39]  Emmanuel Yomba,et al.  A generalized auxiliary equation method and its application to nonlinear Klein–Gordon and generalized nonlinear Camassa–Holm equations , 2008 .

[40]  Saudi Arabia,et al.  APPROXIMATE SOLUTIONS FOR DIFFUSION EQUATIONS ON CANTOR SPACE-TIME , 2013 .

[41]  Zhen Wang,et al.  Many new kinds exact solutions to (2+1)-dimensional Burgers equation and Klein-Gordon equation used a new method with symbolic computation , 2007, Appl. Math. Comput..

[42]  Muhammad Aslam Noor,et al.  Homotopy Perturbation Method for Solving Fourth-Order Boundary Value Problems , 2007 .

[43]  E. Fan,et al.  Extended tanh-function method and its applications to nonlinear equations , 2000 .

[44]  Qi Wang,et al.  Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1 + 1)-dimensional dispersive long wave equation , 2005 .

[45]  Mingliang Wang,et al.  The periodic wave solutions for the Klein–Gordon–Schrödinger equations , 2003 .

[46]  A. Wazwaz Partial Differential Equations and Solitary Waves Theory , 2009 .

[47]  Mingliang Wang,et al.  Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation , 2005 .

[48]  M. Noor,et al.  Variational Iteration Method for Fifth-Order Boundary Value Problems Using He's Polynomials , 2008 .

[49]  E. J. Parkes,et al.  Observations on the basic (G'/G)-expansion method for finding solutions to nonlinear evolution equations , 2010, Appl. Math. Comput..

[50]  Farah Aini Abdullah,et al.  The exp-function method for new exact solutions of the nonlinear partial differential equations , 2011 .

[51]  Yan-ze Peng,et al.  New Types of Localized Coherent Structures in the Bogoyavlenskii-Schiff Equation , 2006 .

[52]  Syed Tauseef Mohyud-Din,et al.  Solutions of Nonlinear Differential Equations by Exp-function Method , 2009 .