The homotopy analysis method and the Lienard equation

International Journal of Computer Mathematics

Volumn 88, Issue 1, 2011, Pages 121-134

  Abbasbandy, S ^a,d   Lopez, J L ^b   Lopez Ruiz, R ^c  

^a ISLAMIC AZAD UNIVERSITY   (Iran)

^b PUBLIC UNIVERSITY OF NAVARRE   (Spain)

^c UNIVERSITY OF ZARAGOZA   (Spain)

^d IMAM KHOMEINI INTERNATIONAL UNIVERSITY   (Iran)

Author keywords

homotopy analysis method; Lienard equation; limit cycles; the van der Pol oscillator; perturbative techniques

Indexed keywords

HOMOTOPY ANALYSIS METHODS; LIENARD EQUATION; LIMIT-CYCLE; NONLINEAR DIFFERENTIAL EQUATION; PERTURBATIVE TECHNIQUES; VAN DER POL OSCILLATOR;

DIFFERENTIAL EQUATIONS;

CIRCUIT OSCILLATIONS;

EID: 78650583121     PISSN: 00207160     EISSN: 10290265     Source Type: Journal    
DOI: 10.1080/00207160903248659     Document Type: Article

References (44)

1. S. Abbasbandy, Homotopy analysis method for heat radiation equations, Int. Commun. Heat Mass Transf. 34(2007), pp. 380-387. (Pubitemid 46351900)
2. S. Abbasbandy, The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation, Phys. Lett. A 361(2007), pp. 478-483.
3. S. Abbasbandy, Y. Tan, and S. J. Liao, Newton-Homotopy analysis method for nonlinear equations, Appl. Math. Comput. 188(2007), pp. 1794-1800. (Pubitemid 46710069)
4. S. Abbasbandy, Solitary wave solutions to the Kuramoto-Sivashinsky equation by means of the homotopy analysis method, Nonlinear Dynam. 52(2008), pp. 35-40. (Pubitemid 351304393)
5. S. Abbasbandy, Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method, Appl. Math. Model 32(2008), pp. 2706-2714.
6. S. Abbasbandy and E. J. Parkes, Solitary smooth-hump solutions of the Camassa-Holm equation by means of the homotopy analysis method, Chaos Solitons Fractals 36(2008), pp. 581-591. (Pubitemid 50009117)
7. S. Abbasbandy and Zakaria F. Samadian, Soliton solutions for the fifth-order KdV equation with the homotopy analysis method, Nonlinear Dynam. 51(2008), pp. 83-87. (Pubitemid 47632348)
8. C. Anderson and J. Geer, Power series expansions for the frequency and period of the limit cycle of the van der Pol equation, SIAM J. Appl. Math. 42(1982) pp. 678-693.
9. A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators, Dover, New York, 1989.
10. A. S. Bataineh, M. S. M. Noorani, and I. Hashim, The homotopy analysis method for Cauchy reaction diffusion problems, Phys. Lett. A 371(2007) pp. 72-82.
11. A. S. Bataineh, M. S. M. Noorani, and I. Hashim, Approximate solutions of singular two-point BVPs by modified homotopy analysis method, Phys. Lett. A 372(2008), pp. 4062-4066.
12. M. C. Depassier and J. Mura, Variational approach to a class of nonlinear oscillators with several limit cycles, Phys. Rev. E 64(2001), 056217(6).
13. D. Dumortier, D. Panazzolo, and R. Roussarie, More limit cycles than expected in the Liénard equations, Proc. Am. Math. Soc. 135(2007) pp. 1895-1904.
14. H. Giacomini and S. Neukirch, Number of limit cycles of the Liénard equation, Phys. Rev. E 56(1997), pp. 3809-3813.
15. T. Hayat and M. Khan, Homotopy solutions for a generalized second-grade fluid past a porous plate, Nonlinear Dynam. 42(2005), pp. 395-405. (Pubitemid 41697996)
16. T. Hayat, M. Khan, and M. Ayub, On non-linear flows with slip boundary condition, Z. Angew. Math. Phys. (ZAMP) 56(2005), pp. 1012-1029.
17. M. Inc, Application of homotopy analysis method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Math. Comput. Simul. 79(2008), pp. 189-200.
18. H. Jafari and S. Seifi, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Commun. Nonlinear Sci. Numer. Simul. 14(2009), pp. 2006-2012.
19. S. J. Liao, The proposed homotopy analysis technique for the solutions of non-linear problems, Ph. D. thesis, Shanghai Jiao Tong University, 1992.
20. S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, FL, 2003.
21. S. J. Liao, An analytic approximate approach for free oscillations of self-excited systems, Int. J. Non-Linear Mech. 39(2004), pp. 271-280.
22. S. J. Liao, A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int. J. Heat Mass Transf. 48(2005), pp. 2529-2539.
23. S. J. Liao, Series solutions of unsteady boundary-layer flows over a stretching flat plate, Stud. Appl. Math. 117(2006), pp. 239-264.
24. S. J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul. 14(2009), pp. 983-997.
25. S. J. Liao and E. Magyari, Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones, Z. Angew. Math. Phys. (ZAMP) 57(2006), pp. 777-792.
26. S. J. Liao, J. Su, and A. T. Chwang, Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body, Int. J. Heat Mass Transf. 49(2006), pp. 2437-2445.
27. A. Lins, W. de Melo, and C. C. Pugh, On Liénard's Equation, Lectures Notes in Math Vol. 597, Springer-Verlag, New York, 1977, p. 355.
28. J. L. López and R. López-Ruiz, The limit cycles of Liénard equations in the strongly nonlinear regime, Chaos Solitons Fractals 11(2000), pp. 747-756.
29. J. L. López and R. López-Ruiz, The limit cycles of Liénard equations in the weakly nonlinear regime. ArXiv 2006;nlin/0605025.
30. J. L. López and R. López-Ruiz, Approximating the amplitude and form of limit cycles in the weakly nonlinear regime of Liénard systems, Chaos Solitons Fractals 34(2007), pp. 1307-1317.
31. R. López-Ruiz and J. L. López, Bifurcation curves of limit cycles in some Liénard systems, Int. J. Bifurcation Chaos 10(2000), pp. 971-980.
32. R. López-Ruiz and Y. Pomeau, Transition between two oscillation modes, Phys. Rev. E 55(1997), pp. R3820-R3823.
33. K. Odani, The limit cycle of the van der Pol equation is not algebraic, J. Differ. Equ. 115(1995), pp. 146-152.
34. 2i+1 is two, Differ. Equ. 11(1975), pp. 301-302.
35. M. Sajid, T. Hayat, and S. Asghar, On the analytic solution of the steady flow of a fourth grade fluid, Phys. Lett. A 355(2006), pp. 18-26. (Pubitemid 43737111)
36. L. Song and H. Q. Zhang, Solving the fractional BBM-Burgers equation using the homotopy analysis method, Chaos Solitons Fractals 40(2009), pp. 1616-1622.
37. Y Tan and S. Abbasbandy, Homotopy analysis method for quadratic Riccati differential equation, Commun. Nonlinear Sci. Numer. Simul. 13(2008) pp. 539-546. (Pubitemid 47497345)
38. Y Tan, H. Xu, and S. J. Liao, Explicit series solution of travelling waves with a front of Fisher equation, Chaos Solitons Fractals 31(2007), pp. 462-472.
39. F. Verhulst, Nonlinear Differential Equations and Dynamical Equations, Springer-Verlag, Berlin, 1990.
40. C. Wang, Analytic solutions for a liquid film on an unsteady stretching surface, Heat Mass Transf. 42(2006), pp. 759-766.
41. W. Wu and S. J. Liao, Solving solitary waves with discontinuity by means of the homotopy analysis method, Chaos Solitons Fractals 26(2005), pp. 177-185. (Pubitemid 40556381)
42. Y Ye, Theory of Limit Cycles, Transactions of the Mathematical Monographs Vol. 66. American Mathematical Society, Boston, MA, 1986.
43. T. T. Zhang, L. Jia, Z. C. Wang, and X. Li, The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels, Phys. Lett. A 372(2008), pp. 3223-3227.
44. Z. Ziabakhsh and G. Domairry, 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(2009), pp. 1284-1294.