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

Abstract

Based on the Taylor expansion, we propose a renormalization method 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 in practice. As application, we obtain the uniform valid asymptotic solutions to some problems including vector field, boundary layer and boundary value problems of nonlinear wave equations. Moreover, we discuss the normal form theory and reduction equations of dynamical systems. Furthermore, by combining the topological deformation and the RG method, a modified method namely the homotopy renormalization method (for simplicity, HTR) was proposed to overcome the weaknesses of the standard RG 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 including no a small parameter, the HTR method can also be applied. Some concrete applications including multi-solutions problems, the forced Duffing equation and the Blasius equation are given.

The renormalization method from continuous to discrete dynamical systems: asymptotic solutions, reductions and invariant manifolds

“…This is the most important formula for our theory from which we can give every thing of the standard RG method and more. In fact, the main results on our renormalization method in [1] can be we summarized as follows: Theorem 2.1. The exact solution of the Eq.…”

“…The renormalization methods is not omnipotent, and has some weaknesses [1]. In order to overcome these weaknesses, we propose an iteration method as done in [1]. In fact, what we need is only to have a freedom to choose the first approximate solution.…”

“…which cannot be dealt with by the TR method to improve the asymptotic solution by the similar reason in [1], where η is a parameter. Here we find its nontrivial asymptotic solution by the homotopy renormalization method.…”

Efficient Computation of Periodic Orbits of Forced Rayleigh Equation in the Framework of Novel Asymptotic Structures

“…Over past few decades, various non-perturbative modifications of naive perturbation method such as Method of Multiple Scales, Method of Boundary Layer [2], WKB Method [4], Homotopy Averaging Method [18], Homotopy Analysis Method [11], Variational Iteration Method [19], homotopy RG method [20] etc., have been investigated and advocated widely for faster and efficient computation of periodic oscillations of strongly nonlinear systems as well as to singularly perturbed problems, that should yield reasonable fits with experimental values for any value of the control parameter ε > 0. But over time it has become evident that none of the these asymptotic methods could yield uniformly valid approximate solutions to the system variables concerned, both for a large control parameter space as well as for sufficiently large time [16], unless special care and methods are invented and considered.…”

Exact dynamical behavior for a dual Kaup–Boussinesq system by symmetry reduction and coupled trial equations method

“…These methods have been extensively developed and applied to a lot of nonlinear problems [26][27][28][29][30][31][32][33][34][35][36][37]. Liu's renormalization method and its applications can be found in [38][39][40][41][42][43][44][45]. Some new methods and results on fractional differential equations can be seen in [46][47][48][49][50][51][52][53][54][55][56][57] and the references therein.…”