Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation.

*(English)*Zbl 1221.65281Summary: In this paper, the homotopy analysis method (HAM) proposed by Liao in 1992 and the homotopy perturbation method (HPM) proposed by He in 1998 are compared through an evolution equation used as the second example in a recent paper by D. D. Ganji, H. Tari and M. B. Jooybari [Comput. Math. Appl. 54, No. 7–8, 1018–1027 (2007; Zbl 1141.65384)]. It is found that the HPM is a special case of the HAM when \(\hbar=-1\). However, the HPM solution is divergent for all \(x\) and \(t\) except \(t=0\). It is also found that the solution given by the variational iteration method (VIM) is divergent too. On the other hand, using the HAM, one obtains convergent series solutions which agree well with the exact solution. This example illustrates that it is very important to investigate the convergence of approximation series. Otherwise, one might get useless results.

##### MSC:

65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

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\textit{S. Liang} and \textit{D. J. Jeffrey}, Commun. Nonlinear Sci. Numer. Simul. 14, No. 12, 4057--4064 (2009; Zbl 1221.65281)

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##### References:

[1] | Nayfeh, A.H., Perturbation methods, (2000), Wiley New York |

[2] | Abbasbandy, S., The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys lett A, 360, 109-113, (2006) · Zbl 1236.80010 |

[3] | Lyapunov, A.M., General problem on stability of motion, (1992), Taylor & Francis London, (English translation) · Zbl 0786.70001 |

[4] | Karmishin, A.V.; Zhukov, A.I.; Kolosov, V.G., Methods of dynamics calculation and testing for thin-walled structures, (1990), Mashinostroyenie Moscow |

[5] | Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Dordrecht · Zbl 0802.65122 |

[6] | He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput meth appl mech eng, 167, 57-68, (1998) · Zbl 0942.76077 |

[7] | He, J.H., Variational iteration method: a kind of nonlinear analytical technique: some examples, Int J non-linear mech, 34, 699-708, (1999) · Zbl 1342.34005 |

[8] | Ganji, D.D.; Tari, H.; Jooybari, M.B., Variational iteration method and homotopy perturbation method for nonlinear evolution equations, Comput math appl, 54, 1018-1027, (2007) · Zbl 1141.65384 |

[9] | Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis. Shanghai: Shanghai Jiao Tong University; 1992. |

[10] | Liao, S.J., An approximate solution technique not depending on small parameters: a special example, Int J non-linear mech, 30, 371-380, (1995) · Zbl 0837.76073 |

[11] | Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton |

[12] | Liao, S.J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, Stud appl math, 119, 297-354, (2007) |

[13] | Liao, S.J., Notes on the homotopy analysis method: some definitions and theorems, Comm nonlinear sci numer simul, 14, 983-997, (2009) · Zbl 1221.65126 |

[14] | Bataineh, A.S.; Noorani, M.S.M.; Hashim, I., Solutions of time-dependent emden – fowler type equations by homotopy analysis method, Phys lett A, 371, 72-82, (2007) · Zbl 1209.65104 |

[15] | Van*Gorder, R.A.; Vajravelu, K., Analytic and numerical solutions to the lane – emden equation, Phys lett A, 372, 6060-6065, (2008) · Zbl 1223.85004 |

[16] | Hayat, T.; Sajid, M., On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder, Phys lett A, 361, 316-322, (2007) · Zbl 1170.76307 |

[17] | Sajid, M.; Hayat, T., Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations, Nonlinear anal (B), 9, 2290-2295, (2008) · Zbl 1156.76436 |

[18] | Song, L.; Zhang, H., Application of homotopy analysis method to fractional KdV-burgers – kuramoto equation, Phys lett A, 367, 88-94, (2007) · Zbl 1209.65115 |

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