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 Operator 11 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 General Formulation 17.2 Lorentz Model Revisited 18 Relativistic Causal Fluid dynamic Equation 19 Numerical Analysis of Transport Coefficients and Relaxation Times 20 Reactive-Multi-component Systems 21 Non-relativistic Case and Application to Cold Atoms PART IV Summary and Future Prospect 22 Summary and Future Prospects.