PART I Introduction to Renormalization Group (RG) Method
1 Introduction: Notion of Effective Theories in Physical Sciences
2 Divergence and Secular Term in the Perturbation Series of Ordinary Differential Equations
3 Traditional Resummation Methods
3.1 Reductive Perturbation Theory
3.2 Lindstedt's Method
3.3 Krylov-Bogoliubov-Mitropolsky's Method for Nonlinear Oscillators
4 Elementary Introduction of the RG method in Terms of the Notion of Envelopes
4.1 Notion of Envelopes of Family of Curves Adapted for a Geometrical Formulation of the RG Method
4.2 Elementary Examples: Damped Oscillator and Boundary-Layer Problem
5 General Formulation and Foundation of the RG Method: Ei-Fujii-Kunihiro
Formulation and Relation to Kuramoto's reduction scheme
6 Relation to the RG Theory in Quantum Field Theory
7 Resummation of the Perturbation Series in Quantum Methods
PART II Extraction of Slow Dynamics Described by Differential and Difference Equations
8 Illustrative Examples
8.1 Rayleigh/Van der Pol equation and jumping phenomena
8.2 Lotka-Volterra Equation
8.3 Lorents Model
9 Slow Dynamics Around Critical Point in Bifurcation Phenomena
10 Dynamical Reduction of A Generic Non-linear Evolution Equation with Semi-simple Linear Operator11 A Generic Case when the Linear Operator Has a Jordan-cell Structure
12 Dynamical Reduction of Difference Equations (Maps)
13 Slow Dynamics in Some Partial Differential Equations
13.1 Dissipative One-Dimensional Hyperbolic Equation
13.2 Swift-Hohenberg Equation
13.3 Damped Kuramoto-Shivashinsky Equation
13.4 Diffusion in Porus Medium --- Barrenblatt Equation
14 Appendix: Some Mathematical Formulae
PART III Application to Extracting Slow Dynamics of Non-equilibrium Phenomena
15 Dynamical Reduction of Kinetic Equations
15.1 Derivation of Boltzmann Equation from Liouville Equation
15.2 Derivation of the Fokker-Planck (FP) Equation from Langevin Equation
15.3 Adiabatic Elimination of Fast Variables in FP Equation: Derivation of Generalized Kramers Equations
16 Relativistic First-Order Fluid Dynamic Equation
17 Doublet Scheme and its Applications
17.1
