Advanced mathematical methods - 2021

Introduction

Approximate methods for LDEs

1. Classification of of singular points

2. Series expansion near a regular singular point

3. Behavior near finite irregular singular points

4. Irregular singular points at infinity and the asymptotic power series

5. Asymptotic relations

6. Asymptotic series

Boundary layer theory

1. Introduction

2. Inner, outer and intermediate regions of a boundary

3. Boundaries of width greater than epsilon and higher order boundary layer theory

4. Internal boundary layers

5. Boundary layer theory applied to NLDE

WKB theory

1. The WKB expansion

2. Turning points

3. Tunneling

4. Inhomogeneous equations

Asymptotic analysis of integrals

1. Intro

2. Laplace integrals

3. Stationary phase approximation

4. The method of steepest decent

Summation of series

1. Improving convergence and the Shanks transformation

2. Divergent sums and the Pade approximation

Introduction (10/3/2021): video, whiteboard

Behavior near ISPs #1 (17/3/2021): video, whiteboard

Behavior near ISPs #2 (7/4/2021): video, whiteboard

Asymptotic series & approximation (22/4/2021): video, whiteboard

Asymptotic relations for oscillatory and complex functions: video, whiteboard

Boundary layer theory (5/5/2021): video, whiteboard

Boundary layer theory 2 (12/5/2021): video, whiteboard

WKB 1 (19/5/21): video, whiteboard

WKB 2 (26/5/21): video, whiteboard

WKB 3 (2/6/21): video, whiteboard

Asymptotic expansion of int. - Laplace method (16/6/21): video, whiteboard

Asymptotic expansion of int. - stationary phase and steepest descent (23/6/21): video, whiteboard

Summation of series (30/6/21): video, whiteboard