apm346:PDE
apm346:PDE
What's a PDE?
Distinguish Equation Types: Algebraic, ODE, PDE, Other
What does it mean to "solve" a PDE?
Idea of Wellposedness; Examples
Different kinds of problems for different PDE
First Order PDE
Geometric Interpretation
Method of Characteristics
Do Several Examples!
Wave Equation in 1d
3d discussion
D'Alembert's Formula
Causality
Domain of Dependence
Domain of Influence
Finite Speed of Propagation
Energy
Solutions do not get smoother
Diffusion/Heat Equation in 1d
Fundamental Solution
Duhamel's Principle
Instantaneous Smoothing
Infinite Speed of Propagation
Energy Decreases
Separation of Variables
Separated Form Ansatz
Emergence of Eignevalue Parameter
Do Several Examples!
Fourier Series
Emergence from Separation of Variables
Coefficient Formulae
Why so useful?
Relation to Fourier Transform
Decay and Smoothness
Convolution/Multiplication
Translation/Modulation
Dilation/Dilation
Fourier Transform
Relation to Fourier Series
Basic Properties
Decay and Smoothness
Convolution/Multiplication
Translation/Modulation
Dilation/Dilation
Laplace's Equation
Dirichlet Problem
Dirichlet Energy Minimizer
Neumann Problem
Dirichlet Problem on Disk in R2
Poisson's Formula
Smoothness of Solutions
Green's Functions
Unifying Themes
Energy
Integral Solution Formulae
Fourier Methods
Convolution
Relevance in models of natural phenomena