Renormalization Group Method

Abstract

Renormalization Group (RG) method is a general method whose aim is to globally approximate solutions to differential equations involving a small parameter. In this thesis, we will give an algorithm for the RG method to generate the RG equation needed in the process of finding an approximate solution for ODEs. In chapter 1, we have some introduction to perturbation theory and introducing some traditional methods in perturbation theory. In chapter 2 we compare the results of RG and other conventional methods using numerical or explicit methods. Thereafter, in chapter 3, we rigorously compare the approximate solution obtained using the RG method and the true solution using two classes of system of ordinary differential equations. In chapter 4, we present a simplified RG method and apply it to the second order RG. In chapter 5 we briefly explain the first order Normal Form (NF) theory and then its relation to the RG method. Also a similar geometric interpretation for the RG equation and NF's outcome has been provided. In the Appendix, we have added definitions and proofs used in this thesis. The RG method is much more straightforward than other traditional methods and does not require prior information about the solutions. One begins with a naive perturbative expansion which already contains all the necessary information that we need to construct a solution. Using RG, there is no need to asymptotically match the solutions in the overlapping regions, which is a key point in some other methods. In addition, the RG method is applicable to most of perturbed differential equations and will produce a closed form solution which is, most of the times, as accurate as or even more accurate than the solutions obtained by other conventional methods.ii