| 1 | Introduction, Theme for the Course, Initial and Boundary Conditions, Well-posed and Ill-posed Problems | |
| 2 | Conservation Laws in (1 + 1) Dimensions
Introduction to 1st-order PDEs: Linear and Homogeneous, and Linear, Non-Homogeneous PDEs | |
| 3 | Theory of 1st-order PDEs (cont.): Quasilinear PDEs, and General Case, Charpit's Equations | Homework 1 out |
| 4 | Theory of 1st-order PDEs (cont.): Examples, The Eikonal Equation, and the Monge Cone
Introduction to Traffic Flow | |
| 5 | Solutions for the Traffic-flow Problem, Hyperbolic Waves
Breaking of Waves, Introduction to Shocks, Shock Velocity
Weak Solutions | |
| 6 | Shock Structure (with a Foretaste of Boundary Layers), Introduction to Burgers' Equation
Introduction to PDE Systems, The Wave Equation | |
| 7 | Systematic Theory, and Classification of PDE Systems | Homework 1 due
Homework 2 out |
| 8 | PDE Systems (cont.): Example from Elementary Gas Dynamics, Riemann Invariants
More on the Wave Equation, The D'Alembert Solution | |
| 9 | Remarks 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 | |
| 10 | Properties 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 | |
| 11 | The 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 |
| 12 | Generalities 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 | |
| 13 | Conversion 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 |
| 14 | Natural Frequencies for Linear PDEs (cont.): Vibrating Circular Membrane, Bessel's Functions, Linear Schrödinger's Equation | |
| 15 | Vibrating Circular Membrane (cont.)
Natural Frequencies and Separation of Variables: Nonlinear PDEs, Example: Nonlinear Schrödinger's Equation, Elliptic Integrals and Functions | |
| 16 | Remarks 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 |
| 17 | Introduction to Green's Functions, The Poisson Equation in 3D, Integral Equation for the "Nonlinear Poisson Equation"
Green's Functions for Nonlinear Problems | |
| 18 | Green'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 | |
| 19 | Conversion 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 |
| 20 | Extra 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 | |
| 21 | Asymptotic Expansions for Non-uniform PDEs, Example: Non-uniform Klein-Gordon Equation
Kinematic Derivation of Group Velocity | |
| 22 | Dimensional 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 | |
| 23 | Dimensional Analysis and Similarity (cont.): Idea of Stretching Transformations, Example: Nonlinear Diffusion | Homework 5 due
Homework 6 out |
| 24 | Extra Lecture
Dimensional Analysis and Similarity (cont.): More on Nonlinear Diffusion, Solutions of Compact Support | |
| 25 | Comments on the Blasius Problem
Introduction to Perturbation Methods for PDEs: Regular Perturbation, Example | |
| 26 | Regular Perturbation for Linear Schrödinger Equation with a Potential
Perturbation Methods for PDEs: Singular Perturbation, Boundary Layers, Elementary Example | |
| 27 | Singular Perturbation for PDEs (cont.), More Advanced Examples
Quiz 2 | Homework 6 due |
| 28 | Boundary 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 | |
| 29 | Extra 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 | |