The Application of the Homotopy Analysis Method and the Homotopy Perturbation Method to the Davey-Stewartson Equations and Comparison between Them and Exact Solutions

We introduce two powerful methods to solve the Davey-Stewartson equations: one is the homotopy perturbation method (HPM) and the other is the homotopy analysis method (HAM). HAM is a strong and easy to use analytic tool for nonlinear problems. Comparison of the HPM results with the HAM results, and compute the absolute errors between the exact solutions of the DS equations with the HPM solutions and HAM solutions are obtained.

[1]  Hassan A. Zedan,et al.  Symmetry analysis of an integrable Ito coupled system , 2010, Comput. Math. Appl..

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

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

[4]  Ji-Huan He,et al.  Asymptotology by homotopy perturbation method , 2004, Appl. Math. Comput..

[5]  Ji-Huan He.  Application of homotopy perturbation method to nonlinear wave equations , 2005 .

[6]  Mohammad Mehdi Rashidi,et al.  Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method , 2009 .

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

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

[9]  Shijun Liao,et al.  On the homotopy analysis method for nonlinear problems , 2004, Appl. Math. Comput..

[10]  H. Zedan,et al.  Solution of Davey-Stewartson equations by homotopy perturbation method , 2009 .

[11]  Saeid Abbasbandy,et al.  Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method , 2006, Appl. Math. Comput..

[12]  Shijun Liao,et al.  Comparison between the homotopy analysis method and homotopy perturbation method , 2005, Appl. Math. Comput..

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

[14]  Ji-Huan He,et al.  Comparison of homotopy perturbation method and homotopy analysis method , 2004, Appl. Math. Comput..

[15]  Ji-Huan He.  Homotopy Perturbation Method for Bifurcation of Nonlinear Problems , 2005 .

[16]  Saeid Abbasbandy,et al.  Numerical solutions of the integral equations: Homotopy perturbation method and Adomian's decomposition method , 2006, Appl. Math. Comput..

[17]  Davood Domiri Ganji,et al.  Explicit Solutions of Helmholtz Equation and Fifth-order KdV Equation using Homotopy Perturbation Method , 2006 .

[18]  K. Stewartson,et al.  On three-dimensional packets of surface waves , 1974, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[19]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[20]  T. Hayat,et al.  Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations , 2008 .

[21]  Hassan A. Zedan,et al.  Exact solutions for the generalized KdV equation by using Backlund transformations , 2011, J. Frankl. Inst..

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

[23]  Ji-Huan He,et al.  The homotopy perturbation method for nonlinear oscillators with discontinuities , 2004, Appl. Math. Comput..