| 1 |
Introduction, Theme for the Course, Initial and Boundary Conditions, Well-posed and Ill-posed Problems |
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| 2 |
Conservation Laws in (1 + 1) Dimensions
Introduction to 1st-order PDEs: Linear and Homogeneous, and Linear, Non-Homogeneous PDEs |
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| 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 |
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| 5 |
Solutions for the Traffic-flow Problem, Hyperbolic Waves
Breaking of Waves, Introduction to Shocks, Shock Velocity
Weak Solutions |
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| 6 |
Shock Structure (with a Foretaste of Boundary Layers), Introduction to Burgers' Equation
Introduction to PDE Systems, The Wave Equation |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 15 |
Vibrating Circular Membrane (cont.)
Natural Frequencies and Separation of Variables: Nonlinear PDEs, Example: Nonlinear Schrödinger's Equation, Elliptic Integrals and Functions |
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| 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 |
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| 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 |
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| 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 |
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| 21 |
Asymptotic Expansions for Non-uniform PDEs, Example: Non-uniform Klein-Gordon Equation
Kinematic Derivation of Group Velocity |
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| 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 |
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| 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 |
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| 25 |
Comments on the Blasius Problem
Introduction to Perturbation Methods for PDEs: Regular Perturbation, Example |
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| 26 |
Regular Perturbation for Linear Schrödinger Equation with a Potential
Perturbation Methods for PDEs: Singular Perturbation, Boundary Layers, Elementary Example |
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| 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 |
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| 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 |
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