Exact traveling wave solution of nonlinear variants of the RLW and the PHI-four equations

By means of the modified extended tanh-function (METF) method the multiple traveling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system. The solutions for the nonlinear equations such as variants of the RLW and variant of the PHI-four equations are exactly obtained and so the efficiency of the method can be demonstrated.

[1]  L. Gardner,et al.  Solitary waves of the regularized long-wave equation , 1990 .

[2]  A. A. Soliman Numerical scheme based on similarity reductions for the regularized long wave equation , 2004, Int. J. Comput. Math..

[3]  A. A. Soliman,et al.  New applications of variational iteration method , 2005 .

[4]  L. R. Scott,et al.  Numerical schemes for a model for nonlinear dispersive waves , 1985 .

[5]  A. A. Soliman,et al.  Collocation method using quadratic B-spline for the RLW equation , 2001, Int. J. Comput. Math..

[6]  J. Bona,et al.  Model equations for long waves in nonlinear dispersive systems , 1972, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[7]  Zhenya Yan,et al.  New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water , 2001 .

[8]  B. Duffy,et al.  An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations , 1996 .

[9]  S. A. El-Wakil,et al.  Modified extended tanh-function method and its applications to nonlinear equations , 2005, Appl. Math. Comput..

[10]  A. A. Soliman,et al.  Variational iteration method for solving Burger's and coupled Burger's equations , 2005 .

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

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

[13]  A. A. Soliman,et al.  The numerical simulation for stiff systems of ordinary differential equations , 2007, Comput. Math. Appl..

[14]  Abdulkadir Dogan,et al.  A least‐squares finite element scheme for the RLW equation , 1996 .

[15]  D. Peregrine Calculations of the development of an undular bore , 1966, Journal of Fluid Mechanics.

[16]  J. Dye,et al.  An inverse scattering scheme for the regularized long-wave equation , 2000 .

[17]  A. A. Soliman,et al.  Numerical solutions of nonlinear evolution equations using variational iteration method , 2007 .

[18]  M. Wadati,et al.  Relationships among Inverse Method, Bäcklund Transformation and an Infinite Number of Conservation Laws , 1975 .

[19]  M. E. Alexander,et al.  Galerkin methods applied to some model equations for non-linear dispersive waves , 1979 .

[20]  A. A. Soliman,et al.  A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations , 2006 .

[21]  G. R. McGuire,et al.  Numerical Study of the Regularized Long-Wave Equation. II: Interaction of Solitary Waves , 1977 .

[22]  A. A. Soliman,et al.  Collocation solution for RLW equation with septic spline , 2005, Appl. Math. Comput..

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

[24]  S. A. El-Wakil,et al.  Modified extended tanh-function method for solving nonlinear partial differential equations , 2002 .

[25]  A. A. Soliman,et al.  The modified extended tanh-function method for solving Burgers-type equations , 2006 .

[26]  Zhenya Yan New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations , 2001 .

[27]  A. A. Soliman,et al.  Modified extended tanh-function method and its application on nonlinear physical equations , 2006 .

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

[29]  V. G. Makhankov,et al.  One more example of inelastic soliton interaction , 1976 .

[30]  A. A. Soliman,et al.  Numerical simulation of the generalized regularized long wave equation by He's variational iteration method , 2005, Math. Comput. Simul..