GHM method for obtaining rationalsolutions of nonlinear differential equations

AbstractIn this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30

[1]  Yasir Khan,et al.  An effective modification of the homotopy perturbation method for MHD viscous flow over a stretching sheet , 2013 .

[2]  D. Bahuguna,et al.  A comparative study of numerical methods for solving an integro-differential equation , 2009, Comput. Math. Appl..

[3]  H. Vázquez-Leal.  Generalized homotopy method for solving nonlinear differential equations , 2014 .

[4]  Jafar Biazar,et al.  A new homotopy perturbation method for solving systems of partial differential equations , 2011, Comput. Math. Appl..

[5]  M. Matinfar,et al.  Exact and Numerical Solution of Lienard's Equation by the Variational Homotopy Perturbation Method , 2011 .

[6]  Davood Domiri Ganji,et al.  ASSESSMENT OF HOMOTOPY-PERTURBATION AND PERTURBATION METHODS IN HEAT RADIATION EQUATIONS , 2006 .

[7]  Y. Khan,et al.  Application of modified Laplace decomposition method for solving boundary layer equation , 2011 .

[8]  Cha'o-Kuang Chen,et al.  An approximate analytic solution of the nonlinear Riccati differential equation , 2010, J. Frankl. Inst..

[9]  U. Filobello-Niño,et al.  High Accurate Simple Approximation of Normal Distribution Integral , 2012 .

[10]  Yasir Khan,et al.  The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet , 2011, Comput. Math. Appl..

[11]  Yasir Khan,et al.  An efficient iterated method for mathematical biology model , 2012, Neural Computing and Applications.

[12]  L. Erro,et al.  HPM Applied to Solve Nonlinear Circuits: A Study Case , 2012 .

[13]  Ahmet Yildirim,et al.  On the numerical solution of the model for HIV infection of CD4+ T cells , 2011, Comput. Math. Appl..

[14]  A. Ebaid.  A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities , 2011 .

[15]  F. Santonja,et al.  Analysing the Spanish smoke-free legislation of 2006: a new method to quantify its impact using a dynamic model. , 2011, The International journal on drug policy.

[16]  Habibolla Latifizadeh.  Solution of the Falkner–Skan wedge flow by HPM–Pade’ method , 2011 .

[17]  Erwin Fehlberg,et al.  Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme , 1970, Computing.

[18]  Ji-Huan He,et al.  An elementary introduction to the homotopy perturbation method , 2009, Comput. Math. Appl..

[19]  J. Biazar,et al.  The homotopy perturbation method for solving neutral functional–differential equations with proportional delays , 2012 .

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

[21]  Wen-Hui Lin,et al.  A Homotopy Perturbation-Based Method for Large Deflection of a Cantilever Beam Under a Terminal Follower Force , 2012 .

[22]  Héctor Vázquez-Leal,et al.  Rational Homotopy Perturbation Method , 2012, J. Appl. Math..

[23]  Hossein Aminikhah,et al.  The combined Laplace transform and new homotopy perturbation methods for stiff systems of ODEs , 2012 .

[24]  F. Santonja,et al.  Solving a model for the evolution of smoking habit in Spain with homotopy analysis method , 2013 .

[25]  U. Filobello,et al.  An Approximate Solution of Blasius Equation by using HPM Method , 2012 .

[26]  Alejandro Díaz-Sánchez,et al.  Fixed-Term Homotopy , 2013, J. Appl. Math..

[27]  Wayne H. Enright,et al.  Interpolants for Runge-Kutta formulas , 1986, TOMS.

[28]  Nasser Hassan Sweilam,et al.  Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method , 2009, Comput. Math. Appl..

[29]  Alejandro Díaz-Sánchez,et al.  Rational Biparameter Homotopy Perturbation Method and Laplace-Padé Coupled Version , 2012, J. Appl. Math..

[30]  Francisco J. Campa,et al.  Characterization and stability analysis of a multivariable milling tool by the enhanced multistage homotopy perturbation method , 2012 .

[31]  C. Dang,et al.  An aftertreatment technique for improving the accuracy of Adomian's decomposition method , 2002 .

[32]  Ahmet Yildirim,et al.  Series solution of a nonlinear ODE arising in magnetohydrodynamic by HPM-Padé technique , 2011, Comput. Math. Appl..

[33]  Yasir Khan,et al.  Analytical solution of electrically conducted rotating flow of a second grade fluid over a shrinking surface , 2011 .

[34]  A. Yildirim,et al.  The modified algorithm for the differential transform method to solution of Genesio systems , 2012 .

[35]  Davood Domiri Ganji,et al.  Solution of the Falkner-Skan wedge flow by HPM-Pade' method , 2012, Adv. Eng. Softw..

[36]  Arturo Sarmiento-Reyes,et al.  Power Series Extender Method for the Solution of Nonlinear Differential Equations , 2015 .

[37]  Vedat Suat Ertürk,et al.  Solutions of non-linear oscillators by the modified differential transform method , 2008, Comput. Math. Appl..

[38]  Muhammad Aslam Noor,et al.  Variational Homotopy Perturbation Method for Solving Higher Dimensional Initial Boundary Value Problems , 2008 .

[39]  Syed Tauseef Mohyud-Din,et al.  Coupling of homotopy perturbation and modified Lindstedt–Poincaré methods for traveling wave solutions of the nonlinear Klein–Gordon equation , 2012 .

[40]  U. Filobello-Niño,et al.  Modified HPMs Inspired by Homotopy Continuation Methods , 2012 .

[41]  Davood Domiri Ganji,et al.  Application of Homotopy Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations , 2008 .

[42]  Ahmed M. A. El-Sayed,et al.  A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains , 2012, Appl. Math. Comput..

[43]  Yasir Khan,et al.  A new fractional analytical approach via a modified Riemann-Liouville derivative , 2012, Appl. Math. Lett..

[44]  G. H. Erjaee,et al.  The modified homotopy perturbation method for solving strongly nonlinear oscillators , 2009, Comput. Math. Appl..

[45]  H. Vázquez-Leal,et al.  Application of series method with Padé and Laplace-Padé resummation methods to solve a model for the evolution of smoking habit in Spain , 2014 .

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

[47]  Hessameddin Yaghoobi,et al.  Novel solution for acceleration motion of a vertically falling spherical particle by HPM–Padé approximant , 2011 .