##
**Homotopy perturbation method: a new nonlinear analytical technique.**
*(English)*
Zbl 1030.34013

Summary: In this paper, a new perturbation method is proposed. In contrast to the traditional perturbation methods, this technique does not require a small parameter in an equation. In this method, according to the homotopy technique, a homotopy with an imbedding parameter \(p\in [0,1]\) is constructed, and the imbedding parameter is considered as a “small parameter”, so the method is called the homotopy perturbation method, which can take the full advantages of the traditional perturbation methods and homotopy techniques. To illustrate its effectiveness and its convenience, a Duffing equation with high order of nonlinearity is used; the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter, and is more accurate than the perturbation solutions.

### MSC:

34A45 | Theoretical approximation of solutions to ordinary differential equations |

65L99 | Numerical methods for ordinary differential equations |

34E99 | Asymptotic theory for ordinary differential equations |

### Keywords:

Duffing equation
Full Text:
DOI

### References:

[1] | G.L. Liu, New research directions in singular perturbation theory: artificial parameter approach and inverse-perturbation technique, in: Conference of 7th Modern Mathematics and Mechanics, Shanghai, 1997; G.L. Liu, New research directions in singular perturbation theory: artificial parameter approach and inverse-perturbation technique, in: Conference of 7th Modern Mathematics and Mechanics, Shanghai, 1997 |

[2] | Liao, S. J., An approximate solution technique not depending on small parameters: a special example, Int. J. Non-Linear Mech., 30, 3, 371-380 (1995) · Zbl 0837.76073 |

[3] | Liao, S. J., Boundary element method for general nonlinear differential operators, Eng. Anal. Boundary Elem., 20, 2, 91-99 (1997) |

[4] | 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 |

[5] | He, J. H., Approximate solution for nonlinear differential equations with convolution product nonlinearities, Comput. Meth. Appl. Mech. Eng., 167, 69-73 (1998) · Zbl 0932.65143 |

[6] | He, J. H., Variational iteration method: a kind of nonlinear analytical technique: some examples, Int. J. Non-Linear Mech., 34, 4, 699-708 (1999) · Zbl 1342.34005 |

[7] | He, J. H., A review on some new recently developed nonlinear analytical techniques, Int. J. Nonlinear Sci. Numer. Simul., 1, 1, 51-70 (2000) · Zbl 0966.65056 |

[8] | He, J. H., Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng., 178, 257-262 (1999) · Zbl 0956.70017 |

[9] | He, J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Non-Linear Mech., 35, 1, 37-43 (2000) · Zbl 1068.74618 |

[10] | Nayfeh, A. H., Problems in Perturbation (1985), Wiley: Wiley New York · Zbl 0573.34001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.