Approximate solution for system of differential-difference equations by means of the homotopy analysis method

Abstract In this paper, the homotopy analysis method is successfully applied to solve approximate solutions of the differential-difference system. The solution of another relativistic Toda lattice system is considered. Comparisons made between the results of the proposed method and exact solutions reveal that the homotopy analysis method is very effective in solving differential-difference system.

[1]  S. Liao.  On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet , 2003, Journal of Fluid Mechanics.

[2]  M. Wadati,et al.  Bäcklund Transformation for the Exponential Lattice , 1975 .

[3]  Abdul-Majid Wazwaz,et al.  Partial differential equations : methods and applications , 2002 .

[4]  Thiab R. Taha,et al.  Analytical and numerical aspects of certain nonlinear evolution equations. 1V. numerical modified Korteweg-de Vries equation , 1988 .

[5]  S. I. Svinolupov,et al.  The multi-field Schrödinger lattices , 1991 .

[6]  S. Abbasbandy.  THE APPLICATION OF HOMOTOPY ANALYSIS METHOD TO NONLINEAR EQUATIONS ARISING IN HEAT TRANSFER , 2006 .

[7]  R. Yamilov.  CONSTRUCTION SCHEME FOR DISCRETE MIURA TRANSFORMATIONS , 1994 .

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

[9]  S. Liao.  An explicit, totally analytic approximate solution for Blasius’ viscous flow problems , 1999 .

[10]  Qi Wang,et al.  An extended Jacobi elliptic function rational expansion method and its application to (2 + 1)-dimensional dispersive long wave equation , 2005 .

[11]  S. Liao.  An approximate solution technique not depending on small parameters: A special example , 1995 .

[12]  R. Conte,et al.  Painleve analysis and Backlund transformation in the Kuramoto-Sivashinsky equation , 1989 .

[13]  S. Abbasbandy.  The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation , 2007 .

[14]  Qi Wang,et al.  A new Jacobi elliptic function rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation , 2005 .

[15]  S. Liao.  A kind of approximate solution technique which does not depend upon small parameters — II. An application in fluid mechanics , 1997 .

[16]  Qi Wang,et al.  New rational formal solutions for (1 + 1)-dimensional Toda equation and another Toda equation , 2006 .

[17]  Willy Hereman,et al.  Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations , 2003, Comput. Phys. Commun..

[18]  M. Wadati,et al.  Conservation Laws of a Volterra System and Nonlinear Self-Dual Network Equation , 1977 .

[19]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

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

[21]  S. Liao,et al.  Beyond Perturbation: Introduction to the Homotopy Analysis Method , 2003 .

[22]  S. Lou,et al.  Special solutions from the variable separation approach: the Davey - Stewartson equation , 1996 .

[23]  D. Levi,et al.  Extension of the spectral-transform method for solving nonlinear differential difference equations , 1978 .

[24]  Shijun Liao,et al.  A new branch of solutions of boundary-layer flows over a permeable stretching plate , 2007 .

[25]  S. I. Svinolupov,et al.  Multi-component Volterra and Toda type integrable equations , 1999 .

[26]  S. Liao.  A new branch of solutions of boundary-layer flows over an impermeable stretched plate , 2005 .

[27]  S. Liao.  AN EXPLICIT TOTALLY ANALYTIC APPROXIMATION OF BLASIUS VISCOUS FLOW PROBLEMS , 1999 .

[28]  Y. Hon,et al.  Quasi-periodic solutions for modified Toda lattice equation , 2009 .

[29]  M. Kruskal,et al.  New similarity reductions of the Boussinesq equation , 1989 .

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

[31]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .