A review on some new recently developed nonlinear analytical techniques.

*(English)*Zbl 0966.65056Summary: This paper is a survey of some recent developments in nonlinear analytical techniques, which covers mainly the following 6 categories: 1) variational iteration method, 2) homotopy perturbation method, 3) linearized perturbation method, 4) parameterized perturbation method, 5) various modified Lindstedt-Poincaré methods, and 6) modified Adomian decomposition method. Each of those methods can be an effective procedure for analytical solutions of a wide class of both weakly and strongly nonlinear systems without small parameter assumption. Some of these new technologies have never appeared in any other literature. The emphasis is put upon the author’s recent work, and the references are not exhaustive.

##### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

##### Keywords:

survey paper; Lagrange multiplier; Duffing equation; variational iteration method; homotopy perturbation method; linearized perturbation method; parameterized perturbation method; Lindstedt-Poincaré methods; Adomian decomposition method; nonlinear systems
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\textit{J.-H. He}, Int. J. Nonlinear Sci. Numer. Simul. 1, No. 1, 51--70 (2000; Zbl 0966.65056)

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##### References:

[1] | Liu, Practice in Chinese Optimally weighted decomposition method in nonlinear applied mathematics th China Conf on Modern Math, Proc Mech 21 (1999) |

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