| The renormalization group(RG)method was first proposed in the study of quantum field theory.It was primarily used for research the problem of"renormalization charge"in electrodynamics.Since the group properties be-tween these charge transformations are involved in the study,they are called renormalized groups.Early 1990s,Goldenfeld applied renormalization group method to solve large-scale asymptotic behavior of perturbed nonlinear dif?ferential equations.In 1999,ziane constructed the approximate solution of a class of equation by means of renormalization group method,and gave the error estimate between the approximate solution and the real solution.Constructive renormalization group is a method to study the long-time asymptotic solutions of nonlinear parabolic partial differential equations,which was proposed by Bricmont in 1993.It is widely used in coupled equations,high-order differential equations,fractional differential equations and so on.The error between the approximate solution and the exact solution is smaller.In this paper,we will take a two-dimensional ordinary differential equation system(?)as an example.First,we will introduce the main ideas and results of the con-structive renormalization group methods.Then,we will do further analysis by scaling transformation and singular perturbation renormalization group(RG)method.Our results show that the corresponding long time asymptotic be-havior can also be obtained by the singularly perturbed RG method,and the results obtained are more specific than the former.This paper is divided into three chapters.In the first chapter,we intro-duce the research background and results of perturbed renormalization group method and constructive renormalization group method.In the second chap-ter,we introduce the singular perturbed renorrmalization group method and how to use it to solve the initial value problem of differential equation.The third chapter is the main work of our paper.In the first section,we introduce the constructive renormalization group method and its important results with the example of above equation.In the second section,we reanalyze the above equations by means of singular perturbation renormalization group method and obtain new and more specific results. |
