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Advanced Partial Differential Equations with Applications >> Content Detail



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SES #TOPICSKEY DATES
1Introduction, Theme for the Course, Initial and Boundary Conditions, Well-posed and Ill-posed Problems
2Conservation Laws in (1 + 1) Dimensions

Introduction to 1st-order PDEs: Linear and Homogeneous, and Linear, Non-Homogeneous PDEs
3Theory of 1st-order PDEs (cont.): Quasilinear PDEs, and General Case, Charpit's EquationsHomework 1 out
4Theory of 1st-order PDEs (cont.): Examples, The Eikonal Equation, and the Monge Cone

Introduction to Traffic Flow
5Solutions for the Traffic-flow Problem, Hyperbolic Waves

Breaking of Waves, Introduction to Shocks, Shock Velocity

Weak Solutions
6Shock Structure (with a Foretaste of Boundary Layers), Introduction to Burgers' Equation

Introduction to PDE Systems, The Wave Equation
7Systematic Theory, and Classification of PDE SystemsHomework 1 due

Homework 2 out
8PDE Systems (cont.): Example from Elementary Gas Dynamics, Riemann Invariants

More on the Wave Equation, The D'Alembert Solution
9Remarks on the D'Alembert Solution

The Wave Equation in a Semi-infinite Interval

The Diffusion (or Heat) Equation in an Infinite Interval, Fourier Transform and Green's Function
10Properties of Solutions to the Diffusion Equation (with a Foretaste of Similarity Solutions)

Conversion of Nonlinear PDEs to Linear PDEs: Simple Transformations, Parabolic PDE with Quadratic Nonlinearity, Viscous Burgers' Equation and the Cole-Hopf Transformation
11The Laplace Equation in a Finite Region, Separation of Variables in a Circular Disc

Conversion of Nonlinear PDEs to Linear PDEs: Potential Functions
Homework 2 due
12Generalities on Separation of Variables for Solving Linear PDEs, The Principle of Linear Superposition

Conversion of PDEs to ODEs, Traveling Waves, Fisher's Equation

Conversion of Nonlinear PDEs to linear PDEs: The Hodograph Transform

Quiz 1
13Conversion of Nonlinear PDEs to Linear PDEs: The Legendre Transform

Natural Frequencies and Separation of Variables: Linear PDEs, Fourier Series, Example: Vibrating String

The Sturm-Liouville Problem

About the Question: Can One Hear the Shape of a Drum?
Homework 3 out
14Natural Frequencies for Linear PDEs (cont.): Vibrating Circular Membrane, Bessel's Functions, Linear Schrödinger's Equation
15Vibrating Circular Membrane (cont.)

Natural Frequencies and Separation of Variables: Nonlinear PDEs, Example: Nonlinear Schrödinger's Equation, Elliptic Integrals and Functions
16Remarks on the Nonlinear Schrödinger Equation

General Eigenvalue Problem for Linear PDEs with Self-adjoint Operators

Classification of 2nd-order Quasilinear PDEs, Initial and Boundary Data
Homework 3 due

Homework 4 out
17Introduction to Green's Functions, The Poisson Equation in 3D, Integral Equation for the "Nonlinear Poisson Equation"

Green's Functions for Nonlinear Problems
18Green's Functions for Nonlinear PDEs: Example: Infinite Vibrating String with Forcing, The Issue of (Classical) Causality, Formulation of the Integral Equation, Analytical Solution by Regular Perturbation
19Conversion of Self-adjoint Problems to Integral Equations

Introduction to Dispersive Waves, Dispersion Relations, Uniform Klein-Gordon Equation, Linear Superposition and the Fourier Transform, The Stationary-phase Method for Linear Dispersive Waves
Homework 4 due

Homework 5 out
20Extra Lecture

Linear Dispersive Waves (cont.): Phase and Group Velocities, Energy Propagation, Theory of Caustics, Airy Function

Generalizations: Local Wave Number and Frequency, Slowly Varying Wave Amplitudes
21Asymptotic Expansions for Non-uniform PDEs, Example: Non-uniform Klein-Gordon Equation

Kinematic Derivation of Group Velocity
22Dimensional Analysis for Stationary-phase Method (Linear Dispersive Waves), Characteristic Length and Time of a Dispersive System

Introduction to Dimensional Analysis and Similarity for PDEs, Example: The Diffusion Equation
23Dimensional Analysis and Similarity (cont.): Idea of Stretching Transformations, Example: Nonlinear DiffusionHomework 5 due

Homework 6 out
24Extra Lecture

Dimensional Analysis and Similarity (cont.): More on Nonlinear Diffusion, Solutions of Compact Support
25Comments on the Blasius Problem

Introduction to Perturbation Methods for PDEs: Regular Perturbation, Example
26Regular Perturbation for Linear Schrödinger Equation with a Potential

Perturbation Methods for PDEs: Singular Perturbation, Boundary Layers, Elementary Example
27Singular Perturbation for PDEs (cont.), More Advanced Examples

Quiz 2
Homework 6 due
28Boundary Layers (cont.): Anatomy of Inner and Outer Solutions

Introduction to Solitary Waves and Solitons, Water Waves, Solitary Waves for the KdV Equation, The Sine-Gordon Equation: Kink and Anti-kink Solutions
29Extra Lecture

(Heuristic) Definition of Soliton, Some Nonlinear Evolution PDEs with Soliton Solutions, Solutions to the Sine-Gordon Equation via Separation of Variables, Outline of the Inverse Scattering Transform Idea and Technique

Special Topics: The Painlevé Conjecture, The Painlevé Property, The Painlevé Equations

 








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