A new technique of using homotopy analysis method for second order nonlinear differential equations

Abstract In this paper, a new technique of homotopy analysis method (nHAM) is proposed for solving second order nonlinear differential equations. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM). The proposed provides an approximate solution by rewriting the second order nonlinear differential equation in the form of two first order differential equations. The solution of these two differential equations is obtained as a power series solution. This scheme is tested on four non-linear exactly solvable differential equations. Three of the examples are initial value problems and the fourth is boundary value problem. The results demonstrate reliability and efficiency of the algorithm developed.

[1]  Saeid Abbasbandy,et al.  Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems , 2010, Numerical Algorithms.

[2]  S. Abbasbandy,et al.  Solution of the MHD Falkner-Skan flow by homotopy analysis method , 2009 .

[3]  T. Hayat,et al.  The influence of thermal radiation on MHD flow of a second grade fluid , 2007 .

[4]  Saeid Abbasbandy,et al.  Prediction of multiplicity of solutions of nonlinear boundary value problems: Novel application of homotopy analysis method , 2010 .

[5]  Zhi Zong,et al.  Solving the discrete KdV equation with homotopy analysis method , 2007 .

[6]  Naveed Ahmed,et al.  On the MHD flow of a second grade fluid in a porous channel , 2007, Comput. Math. Appl..

[7]  H. N. Hassan,et al.  A new technique of using homotopy analysis method for solving high‐order nonlinear differential equations , 2011 .

[8]  Dogan Kaya.  A numerical solution of the sine-Gordon equation using the modified decomposition method , 2003, Appl. Math. Comput..

[9]  S. Abbasbandy,et al.  The homotopy analysis method for multiple solutions of nonlinear boundary value problems , 2009 .

[10]  S. Abbasbandy,et al.  Analytic Solution of the Sharma-Tasso-Olver Equation by Homotopy Analysis Method , 2010 .

[11]  L. Tao,et al.  Homotopy analysis of 1D unsteady, nonlinear groundwater flow through porous media , 2007 .

[12]  Tasawar Hayat,et al.  Non-similar solution for the axisymmetric flow of a third-grade fluid over a radially stretching sheet , 2007 .

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

[14]  Tasawar Hayat,et al.  Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid , 2007 .

[15]  H. N. Hassan,et al.  An efficient analytic approach for solving two‐point nonlinear boundary value problems by homotopy analysis method , 2011 .

[16]  Mohd. Salmi Md. Noorani,et al.  Solutions of time-dependent Emden–Fowler type equations by homotopy analysis method , 2007 .

[17]  M. El-Tawil,et al.  SOLVING CUBIC AND COUPLED NONLINEAR SCHRÖ DINGER EQUATIONS USING THE HOMOTOPY ANALYSIS METHOD , 2011 .

[18]  Hong-qing Zhang,et al.  Applying homotopy analysis method for solving differential-difference equation , 2007 .

[19]  S. Abbasbandy.  Homotopy analysis method for heat radiation equations , 2007 .

[20]  S. Abbasbandy.  Soliton solutions for the Fitzhugh–Nagumo equation with the homotopy analysis method , 2008 .

[21]  T. Hayat,et al.  Non-similar analytic solution for MHD flow and heat transfer in a third-order fluid over a stretching sheet , 2007 .

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

[23]  T. Hayat,et al.  Rotating flow of a third grade fluid in a porous space with Hall current , 2007 .

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

[25]  H. P. McKean,et al.  Boussinesq's equation on the circle , 1981 .

[26]  Cheng-shi Liu,et al.  Comparison of a General Series Expansion Method and the Homotopy Analysis Method , 2010, 1105.6189.

[27]  Subrata K. Chakrabarti,et al.  Nonlinear progressive waves in water of finite depth — an analytic approximation , 2007 .

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

[29]  S. Abbasbandy,et al.  Soliton solutions for the fifth-order KdV equation with the homotopy analysis method , 2007 .

[30]  Cheng-shi Liu,et al.  The essence of the homotopy analysis method , 2010, Appl. Math. Comput..

[31]  S. Liao.  Notes on the homotopy analysis method: Some definitions and theorems , 2009 .

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

[33]  A. Molabahrami,et al.  THE HOMOTOPY ANALYSIS METHOD TO SOLVE THE BURGERS–HUXLEY EQUATION , 2009 .

[34]  S. Ha.  A nonlinear shooting method for two-point boundary value problems , 2001 .

[35]  T. Hayat,et al.  Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid , 2007 .

[36]  L. Tao,et al.  Homotopy analysis of 1 D unsteady , nonlinear groundwater flow through porous media , 2008 .

[37]  T. Hayat,et al.  On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder , 2007 .

[38]  Cheng-shi Liu.  The essence of the generalized Newton binomial theorem , 2010 .

[39]  T. Hayat,et al.  Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet , 2007 .

[40]  T. Hayat,et al.  Wire coating analysis using MHD Oldroyd 8-constant fluid , 2007 .

[41]  Zhi-bin Li,et al.  The homotopy analysis method for approximating the solution of the modified Korteweg-de Vries equation , 2009 .

[42]  S. Abbasbandy.  Homotopy analysis method for the Kawahara equation , 2010 .

[43]  Shijun Liao,et al.  An analytic approach to solve multiple solutions of a strongly nonlinear problem , 2005, Appl. Math. Comput..

[44]  Elçin Yusufoglu,et al.  The variational iteration method for studying the Klein-Gordon equation , 2008, Appl. Math. Lett..

[45]  Lina Song,et al.  Application of homotopy analysis method to fractional KdV–Burgers–Kuramoto equation , 2007 .

[46]  Cheng-shi Liu.  The essence of the generalized Taylor theorem as the foundation of the homotopy analysis method , 2011 .

[47]  Tasawar Hayat,et al.  The effect of the slip condition on flows of an Oldroyd 6-constant fluid , 2007 .