Application of series method with Padé and Laplace-Padé resummation methods to solve a model for the evolution of smoking habit in Spain

We obtain approximated analytical solutions of a mathematical model of the evolution of smoking habit in Spain (Guerrero et al. in Int J Drug Policy 22:247–251, 2011) using the series method. To enlarge the domain of convergence, we apply the Padé and Laplace-Padé resummation methods to the series solution. We present a comparison of our results and a solution obtained using homotopy analysis method (Guerrero et al. in Nonlinear Anal Real World Appl 14:549–558, 2013b), resulting that the combination of series method with Laplace-Padé resummation method generates the best results for the complete domain.

[1]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[2]  A. R. Forsyth.  Theory of Differential Equations , 1961 .

[3]  T. Nakayama,et al.  Ordinary Differential Equations , 2009 .

[4]  Keith O. Geddes.  Convergence behavior of the Newton iteration for first order differential equations , 1979, EUROSAM.

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

[6]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

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

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

[9]  Ji-Huan He.  Variational iteration method—Some recent results and new interpretations , 2007 .

[10]  Ji-Huan He,et al.  Variational iteration method: New development and applications , 2007, Comput. Math. Appl..

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

[12]  S. Abbasbandy,et al.  Homotopy analysis method for quadratic Riccati differential equation , 2008 .

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

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

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

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

[17]  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..

[18]  Syed Tauseef Mohyud-Din,et al.  The Comparative Boubaker Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) for solving a standard nonlinear second-order boundary value problem , 2011, Math. Comput. Model..

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

[20]  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.

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

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

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

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

[25]  Ji-Huan He,et al.  Notes on the optimal variational iteration method , 2012, Appl. Math. Lett..

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

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

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

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

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

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

[32]  Luis Hernández-Martínez,et al.  Removal of Noise Oscillation Term Appearing in the Nonlinear Equation Solution , 2012, J. Appl. Math..

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

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

[35]  U. Filobello-Niño,et al.  A general solution for troesch's problem , 2012 .

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

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

[38]  Abraham J. Arenas,et al.  A nonstandard finite difference numerical scheme applied to a mathematical model of the prevalence of smoking in Spain: a case study , 2014 .