| 1 |
Introduction, Theme for the Course, Initial and Boundary Conditions, Well-posed and Ill-posed Problems (PDF) |
| 2 |
Conservation Laws in (1 + 1) Dimensions (PDF)
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 (PDF) |
| 4 |
Theory of 1st-order PDEs (cont.): Examples, The Eikonal Equation, and the Monge Cone (PDF)
Introduction to Traffic Flow |
| 5 |
Solutions for the Traffic-flow Problem, Hyperbolic Waves (PDF)
Breaking of Waves, Introduction to Shocks, Shock Velocity
Weak Solutions |
| 6 |
Shock Structure (with a Foretaste of Boundary Layers), Introduction to Burgers' Equation (PDF)
Introduction to PDE Systems, The Wave Equation |
| 7 |
Systematic Theory, and Classification of PDE Systems (PDF) |
| 8 |
PDE Systems (cont.): Example from Elementary Gas Dynamics, Riemann Invariants (PDF)
More on the Wave Equation, The D'Alembert Solution |
| 9 |
Remarks on the D'Alembert Solution (PDF)
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) (PDF)
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 (PDF)
Conversion of Nonlinear PDEs to Linear PDEs: Potential Functions |
| 12 |
Generalities on Separation of Variables for Solving Linear PDEs, The Principle of Linear Superposition (PDF)
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 (PDF)
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? |
| 14 |
Natural Frequencies for Linear PDEs (cont.): Vibrating Circular Membrane, Bessel's Functions, Linear Schrödinger's Equation (PDF) |
| 15 |
Vibrating Circular Membrane (cont.) (PDF)
Natural Frequencies and Separation of Variables: Nonlinear PDEs, Example: Nonlinear Schrödinger's Equation, Elliptic Integrals and Functions |
| 16 |
Remarks on the Nonlinear Schrödinger Equation (PDF)
General Eigenvalue Problem for Linear PDEs with Self-adjoint Operators
Classification of 2nd-order Quasilinear PDEs, Initial and Boundary Data |
| 17 |
Introduction to Green's Functions, The Poisson Equation in 3D, Integral Equation for the "Nonlinear Poisson Equation" (PDF)
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 (PDF) |
| 19 |
Conversion of Self-adjoint Problems to Integral Equations (PDF)
Introduction to Dispersive Waves, Dispersion Relations, Uniform Klein-Gordon Equation, Linear Superposition and the Fourier Transform, The Stationary-phase Method for Linear Dispersive Waves |
| 20 |
Extra Lecture (PDF)
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 (PDF)
Kinematic Derivation of Group Velocity |
| 22 |
Dimensional Analysis for Stationary-phase Method (Linear Dispersive Waves), Characteristic Length and Time of a Dispersive System (PDF)
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 (PDF) |
| 24 |
Extra Lecture (PDF)
Dimensional Analysis and Similarity (cont.): More on Nonlinear Diffusion, Solutions of Compact Support |
| 25 |
Comments on the Blasius Problem (PDF)
Introduction to Perturbation Methods for PDEs: Regular Perturbation, Example |
| 26 |
Regular Perturbation for Linear Schrödinger Equation with a Potential (PDF)
Perturbation Methods for PDEs: Singular Perturbation, Boundary Layers, Elementary Example |
| 27 |
Singular Perturbation for PDEs (cont.), More Advanced Examples (PDF)
Quiz 2 |
| 28 |
Boundary Layers (cont.): Anatomy of Inner and Outer Solutions (PDF)
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 (PDF)
(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 |