Envelope And The Renormalization Group Method

In many usual problem of physics, engineering and mathematic ,we can not get their exact solutions ,so physicists, engineers and mathematicians only considered for the numerical solution or the approximate solution of the problem, after the joint efforts of the past century, they establish and develop of an important method for approximate solution - perturbative method. It uses the first several perturbative expansion to represent exact solution. By this way, the perturbative expansion may be divergent, and its effective range is usually restricted. The asymptotic expansion is non-uniformly valid because of infinite field, different types of partial differential equations, small parameter multiply highest derivative and singular point. In the case of infinite field, secular terms emergence is the essential problem of non-uniformity. However, we hope that the asymptotic expansion is valid all the time.In order to get uniformly valid asymptotic expansions, scientists have developed many effective methods, such as multiple scales, stretched coordinate, averaging, central manifold theory and otherwise. But in the use of these methods, to irregular part, such as the Boundary Layer, or by the location and thickness of surface may appear small fraction of the market parameters, such as the need to have some understanding. There is also a need for closer asymptotic matching necessary. This makes them much more restricted applications.Recently, a perturbative renormalization group method was developed by Chen, Goldenfeld, and Oono as a unified tool for asymptotic analysis. The main purpose of this paper is to introduce the renormalization group method and envelope theory, and interpret RG by classic envelope theory.This paper consist of three parts. In the first part, briefed the issue of singular perturbation of the research background and related research methods.In the second part is divided into two sections. In the first section, we gives the renormalization group method formulated in quantum field theory. The essence of the RG in quantum field theory (QFT) and statistical physics may be stated as follows: let (?) be the effective action (or thermodynamical potential) obtained by integration of the field variable with the energy scale down to A from infinity or a vary large cutoff (?). Here (?) is a collection of the coupling constants including the wave-function renormalization constant defined at the energy scale at A. Then the RG equation may be expressed as a simple fact that the effective action as a functional of the field variableφshould be the same, irrespective of how much the integration of the field variable is achieved, i.e.,If we take the limit (?)'→(?), we havewhich is the Wilson RG equation, or the flow equation in Wegner's terminology; Then this equation is rewritten asIf the number of the coupling constants is finite, the theory is called renormalizable. In this case, the functional space of the theory does not change in the flow given by the variation of A; one may say that the flow has an invariant manifold. And in this section also describes methods of renormalization group development process. In the second section, we consider the following system of differential equations:whereε> 0 is a small parameter, A is a complex matrix, assumed for simplicity to be diagonalizable, F is a polynomial vector function with respect to y.We obtain the first-order asymptotic expansion of this problem, In the third part is also divided into two sections. In the first section, we give a brief review of the theory of envelops, at here introduced for the family of curves or surface owners on how to seek its envelope curve or envelope surface. In the second section, we first introduce how to obtain the envelope equation of ordinary dynamics equation by envelope method. And explains the envelope equation is the same as the renormalization group equation. In fact,envelope method is equal to renormalization group method in the envelope theory. Moreover, we use the envelope method to obtain the uniformly valid asymptotic solution of initial value problem of ordinary differential equation same as the equation in the second chapterwe find that the results given by renormalization group method is as same as the one given by envelope method, which tell that envelope method is another interpretation of renormalization group method in the envelope theory.