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Title of Journal: Nonlinear Dyn

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Abbravation: Nonlinear Dynamics

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Springer Netherlands

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DOI

10.1002/anie.201007638

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1573-269X

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The renormalization method based on the Taylor exp

Authors: Chengshi Liu
Publish Date: 2017/01/09
Volume: 88, Issue: 2, Pages: 1099-1124
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Abstract

Based on the Taylor expansion we propose a renormalization method TR method for simplicity for asymptotic analysis The standard renormalization group RG method for asymptotic analysis can be derived out from this new method and hence the mathematical essence of the RG method is also recovered The biggest advantage of the proposed method is that the secular terms in perturbation series are automatically eliminated but in usual perturbation theory we need more efforts and tricks to eliminate these terms At the same time the mathematical foundation of the method is simple and the logic of the method is very clear Therefore it is very easy to use in practice As application we obtain the uniform valid asymptotic solutions to some typical problems including vector field boundary layer boundary value problems of nonlinear wave equations the normal form theory and reduction equations of dynamical systems Further by combining the topological deformation with the TR method a modified method namely the homotopy renormalization method for simplicity HTR is proposed to overcome some weaknesses of the standard RG method and TR method In this HTR method since there is a freedom to choose the firstorder approximate solution in perturbation expansion we can improve the global solution In particular for those equations not including a small parameter the HTR method can also be applied Some concrete applications including multisolution problems the forced Duffing equation and the Blasius equation are givenThanks to anonymous referees for their valuable comments on the first version of the paper I would like to thank Prof Nayfeh for his helpful suggestions I also appreciate the Editor’s kindly help Special thanks to the referees of the second version for their detailed comments and suggestions so that I can improve the paper This project was supported by Scientific Research Fund of Education Department of Heilongjiang Province of China under Grant No 12541083

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