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Section B

The homotopy analysis method and the Liénard equation

, &
Pages 121-134 | Received 17 Jan 2009, Accepted 30 Jul 2009, Published online: 22 Oct 2010

References

  • Abbasbandy , S. 2007 . Homotopy analysis method for heat radiation equations . Int. Commun. Heat Mass Transf. , 34 : 380 – 387 .
  • Abbasbandy , S. 2007 . The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation . Phys. Lett. A , 361 : 478 – 483 .
  • Abbasbandy , S. , Tan , Y. and Liao , S. J. 2007 . Newton-Homotopy analysis method for nonlinear equations . Appl. Math. Comput. , 188 : 1794 – 1800 .
  • Abbasbandy , S. 2008 . Solitary wave solutions to the Kuramoto–Sivashinsky equation by means of the homotopy analysis method . Nonlinear Dynam. , 52 : 35 – 40 .
  • Abbasbandy , S. 2008 . Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method . Appl. Math. Model , 32 : 2706 – 2714 .
  • Abbasbandy , S. and Parkes , E. J. 2008 . Solitary smooth-hump solutions of the Camassa–Holm equation by means of the homotopy analysis method . Chaos Solitons Fractals , 36 : 581 – 591 .
  • Abbasbandy , S. and Samadian , Zakaria F. 2008 . Soliton solutions for the fifth-order KdV equation with the homotopy analysis method . Nonlinear Dynam. , 51 : 83 – 87 .
  • Anderson , C. and Geer , J. 1982 . Power series expansions for the frequency and period of the limit cycle of the van der Pol equation . SIAM J. Appl. Math. , 42 : 678 – 693 .
  • Andronov , A. A. , Vitt , A. A. and Khaikin , S. E. 1989 . Theory of Oscillators , New York : Dover .
  • Bataineh , A. S. , Noorani , M. S.M. and Hashim , I. 2007 . The homotopy analysis method for Cauchy reaction diffusion problems . Phys. Lett. A , 371 : 72 – 82 .
  • Bataineh , A. S. , Noorani , M. S.M. and Hashim , I. 2008 . Approximate solutions of singular two-point BVPs by modified homotopy analysis method . Phys. Lett. A , 372 : 4062 – 4066 .
  • Depassier , M. C. and Mura , J. 2001 . Variational approach to a class of nonlinear oscillators with several limit cycles . Phys. Rev. E , 64 : 056217(6)
  • Dumortier , D. , Panazzolo , D. and Roussarie , R. 2007 . More limit cycles than expected in the Liénard equations . Proc. Am. Math. Soc. , 135 : 1895 – 1904 .
  • Giacomini , H. and Neukirch , S. 1997 . Number of limit cycles of the Liénard equation . Phys. Rev. E , 56 : 3809 – 3813 .
  • Hayat , T. and Khan , M. 2005 . Homotopy solutions for a generalized second-grade fluid past a porous plate . Nonlinear Dynam. , 42 : 395 – 405 .
  • Hayat , T. , Khan , M. and Ayub , M. 2005 . On non-linear flows with slip boundary condition . Z. Angew. Math. Phys. (ZAMP) , 56 : 1012 – 1029 .
  • Inc , M. 2008 . Application of homotopy analysis method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity . Math. Comput. Simul. , 79 : 189 – 200 .
  • Jafari , H. and Seifi , S. 2009 . Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation . Commun. Nonlinear Sci. Numer. Simul. , 14 : 2006 – 2012 .
  • Liao , S. J. 1992 . “ The proposed homotopy analysis technique for the solutions of non-linear problems ” . Shanghai Jiao Tong University . Ph.D. thesis
  • Liao , S. J. 2003 . Beyond Perturbation: Introduction to the Homotopy Analysis Method , Boca Raton, FL : Chapman & Hall/CRC Press .
  • Liao , S. J. 2004 . An analytic approximate approach for free oscillations of self-excited systems . Int. J. Non-Linear Mech. , 39 : 271 – 280 .
  • Liao , S. J. 2005 . A new branch of solutions of boundary-layer flows over an impermeable stretched plate . Int. J. Heat Mass Transf. , 48 : 2529 – 2539 .
  • Liao , S. J. 2006 . Series solutions of unsteady boundary-layer flows over a stretching flat plate . Stud. Appl. Math. , 117 : 239 – 264 .
  • Liao , S. J. 2009 . Notes on the homotopy analysis method: Some definitions and theorems . Commun. Nonlinear Sci. Numer. Simul. , 14 : 983 – 997 .
  • Liao , S. J. and Magyari , E. 2006 . Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones . Z. Angew. Math. Phys. (ZAMP) , 57 : 777 – 792 .
  • Liao , S. J. , Su , J. and Chwang , A. T. 2006 . Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body . Int. J. Heat Mass Transf. , 49 : 2437 – 2445 .
  • Lins , A. , de Melo , W. and Pugh , C. C. 1977 . On Liénard's Equation , Lectures Notes in Math Vol. 597 355 New York : Springer-Verlag .
  • López , J. L. and López-Ruiz , R. 2000 . The limit cycles of Liénard equations in the strongly nonlinear regime . Chaos Solitons Fractals , 11 : 747 – 756 .
  • J.L. López and R. López-Ruiz, The limit cycles of Liénard equations in the weakly nonlinear regime. ArXiv 2006;nlin/0605025
  • López , J. L. and López-Ruiz , R. 2007 . Approximating the amplitude and form of limit cycles in the weakly nonlinear regime of Liénard systems . Chaos Solitons Fractals , 34 : 1307 – 1317 .
  • López-Ruiz , R. and López , J. L. 2000 . Bifurcation curves of limit cycles in some Liénard systems . Int. J. Bifurcation Chaos , 10 : 971 – 980 .
  • López-Ruiz , R. and Pomeau , Y. 1997 . Transition between two oscillation modes . Phys. Rev. E , 55 : R3820 – R3823 .
  • Odani , K. 1995 . The limit cycle of the van der Pol equation is not algebraic . J. Differ. Equ. , 115 : 146 – 152 .
  • Rychkov , G. S. 1975 . The maximum number of limit cycles of the system is two . Differ. Equ. , 11 : 301 – 302 .
  • Sajid , M. , Hayat , T. and Asghar , S. 2006 . On the analytic solution of the steady flow of a fourth grade fluid . Phys. Lett. A , 355 : 18 – 26 .
  • Song , L. and Zhang , H. Q. 2009 . Solving the fractional BBM-Burgers equation using the homotopy analysis method . Chaos Solitons Fractals , 40 : 1616 – 1622 .
  • Tan , Y. and Abbasbandy , S. 2008 . Homotopy analysis method for quadratic Riccati differential equation . Commun. Nonlinear Sci. Numer. Simul. , 13 : 539 – 546 .
  • Tan , Y. , Xu , H. and Liao , S. J. 2007 . Explicit series solution of travelling waves with a front of Fisher equation . Chaos Solitons Fractals , 31 : 462 – 472 .
  • Verhulst , F. 1990 . Nonlinear Differential Equations and Dynamical Equations , Berlin : Springer-Verlag .
  • Wang , C. 2006 . Analytic solutions for a liquid film on an unsteady stretching surface . Heat Mass Transf. , 42 : 759 – 766 .
  • Wu , W. and Liao , S. J. 2005 . Solving solitary waves with discontinuity by means of the homotopy analysis method . Chaos Solitons Fractals , 26 : 177 – 185 .
  • Ye , Y. 1986 . Theory of Limit Cycles , Boston, MA : American Mathematical Society . Transactions of the Mathematical Monographs Vol. 66
  • Zhang , T. T. , Jia , L. , Wang , Z. C. and Li , X. 2008 . The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels . Phys. Lett. A , 372 : 3223 – 3227 .
  • Ziabakhsh , Z. and Domairry , G. 2009 . Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method . Commun. Nonlinear Sci. Numer. Simul. , 14 : 1284 – 1294 .

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