The renormalization method based on the Taylor expansion and applications for asymptotic analysis

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 recover...

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Bibliographic Details
Published in:Nonlinear dynamics 2017-04, Vol.88 (2), p.1099-1124
Main Author: Liu, Cheng-shi
Format: Article
Language:English
Subjects:
Asymptotic methods
Asymptotic properties
Asymptotic series
Automotive Engineering
Blasius equation
Boundary layer
Boundary layers
Boundary value problems
Canonical forms
Classical Mechanics
Control
Deformation
Duffing equation
Dynamical Systems
Engineering
Fields (mathematics)
Mathematical analysis
Mechanical Engineering
Nonlinear equations
Original Paper
Perturbation methods
Perturbation theory
Taylor series
Vibration
Wave equations
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Summary: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 first-order 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 multi-solution problems, the forced Duffing equation and the Blasius equation are given.
ISSN:0924-090X
1573-269X
DOI:10.1007/s11071-016-3298-8