Unit · year 3
MU-305 · Partial Differential Equations
Threads change · space24 lectures5 theorems
The heat, wave and Laplace equations and the methods that solve them.
Lectures
| L01 | What a PDE Is: Order, Linearity, Well-Posedness — |
| L02 | First-Order PDEs and Transport |
| L03 | The Method of Characteristics |
| L04 | Quasilinear Equations and Shock Formation |
| L05 | Classification of Second-Order PDEs — |
| L06 | The Wave Equation in One Dimension |
| L07 | d'Alembert's Solution |
| L08 | Domains of Dependence and Influence |
| L09 | The Wave Equation on a Finite Interval |
| L10 | Separation of Variables |
| L11 | Sturm–Liouville Problems |
| L12 | Fourier Series: Coefficients and Orthogonality |
| L13 | Convergence of Fourier Series |
| L14 | The Gibbs Phenomenon and Uniform Convergence |
| L15 | The Heat Equation — |
| L16 | The Fundamental Solution and Smoothing — |
| L17 | The Maximum Principle for the Heat Equation |
| L18 | Laplace's Equation and Harmonic Functions |
| L19 | The Mean Value Property |
| L20 | The Maximum Principle and Uniqueness |
| L21 | Green's Functions — |
| L22 | The Fourier Transform and PDEs — |
| L23 | Energy Methods and Uniqueness — |
| L24 | Synthesis: Three Equations, Three Behaviours |
Theorems in this unit
T-100
Separation of variables
Reducing a PDE to ODEs via product solutions.
T-101
d'Alembert's solution
The 1D wave equation solved by travelling waves.
T-102
The maximum principle
A harmonic function attains its extrema on the boundary.
T-103
Convergence of Fourier series
When and how a periodic function equals its Fourier series.
T-104
The method of characteristics
First-order PDEs solved along characteristic curves.