|
週次 |
日期 |
單元主題 |
|
第1週 |
2/15 |
Introduction:
1. Dimensional Analysis 2. Perturbation Methods 3. Calculus of Variations |
|
第2週 |
2/22 |
1 Dimensional Analysis and Scaling:
a) model, variables and parameters, equations and relations
b) dimensions and units
c) basic properties of units, examples
d) pendulum
e) statement of Pi Theorem: dimensionless, unit-free |
|
第3週 |
3/01 |
Homework and applications of Pi Theorem |
|
第4週 |
3/08 |
Proof of the Buckingham Pi Theorem
Pendulum |
|
第5週 |
3/15 |
Diffusion process and heat kernel |
|
第6週 |
3/22 |
Poincare's inequality, Sobolev's inequality, interpolation inequality;
Gauss-Bonnet Theorem |
|
第7週 |
3/29 |
2. Perturbation Methods
2-1. The regular perturbation method for algebraic equations
Implicit function theorem and regular perturbation |
|
第9週 |
4/12 |
2-2 Singular perturbation methods for algebraic equations
2-3 The regular perturbation method for ODE
Motion in a resistive medium
|
|
第10週 |
4/19 |
Nonlinear oscillation
Poincare-Lindstedt method |
|
第11週 |
4/26 |
Midterm examination |
|
第12週 |
5/03 |
2-4 Singular perturbation methods for ODE (PDE)
2-4-1 Problems with a small parameter
(a) small diffusion
(b) Navier-Stokes equations and Euler equations
(c) Newton's laws and Boltzmann equation
2-4-2 An example of ODE/PDE with singular behavior: small diffusion |
|
第13週 |
5/10 |
2-4-3 Boundary layers
(a) Outer layer and outer approximation
(b) Inner layer and inner approximation
(c) Matching condition
(d) Location of the boundary layer
(e) An example with no boundary layer
(f) Importance of the small parameter in boundary layers |
|
第14週 |
5/17 |
2-4-4 Initial layers
2-4-4-1 Damped spring-mass system with a small mass
(a) outer approximation
(b) inner approximation
(c) matching condition
2-4-4-2 Enzyme kinetics
(a) enzyme reaction
(b) system of rate equations and its reduction
(c) outer (slow time) and inner (fast time) approximations |
|
第15週 |
5/24 |
2-5 WKB method
2-5-1 Introduction
2-5-2 Non-oscillatory case
2-5-3 Oscillatory case
Chapter 3 Calculus of variations
3-1 Some history
a). Examples: isoperimetric problem, Newton's minimal resistance problem, brachistochrone problem
b). Johann Bernoulli's solution to the brachistochrone problem
|
|
第16週 |
5/31 |
Chapter 3 Calculus of variations
3-2 Examples with mathematical formulation
3-3 Normed linear spaces and Banach spaces
3-4 Linear functionals and linear operators
3-5 Variation of functionals:
(a) Gateaux's derivatives and Frechet's derivatives;
(b) necessary conditions for extremals |
|
第17週 |
6/07 |
3-5 Variation of functionals:
(c) the Euler-Lagrange equation
3-6 Applications
(a) examples
(b) first integrals |
|
第18週 |
6/14 |
3-6 Applications
(c) electrostatics (PDE)
3-7 Lagrangian with higher derivatives/several functions
3-8 Lagrangian mechanics, Hamilton's principle
3-9 Variational problems with constraints |