Introduction into Partial Differential Equations
HWS 2024
Ross Ogilvie
Contents
1
First Order PDEs
1.1
Homogeneous Transport Equation
1.2
Inhomogeneous Transport Equation
1.3
Scalar Conservation Laws
1.4
Noncharacteristic Hypersurfaces
1.5
Method of Characteristics
1.6
Weak Solutions
2
General Concepts
2.1
Types of Second Order PDEs
2.2
The Questions
2.3
Divergence Theorem
2.4
Distributions
3
Laplace Equation
3.1
Fundamental Solution
3.2
Mean Value Property
3.3
Maximum Principle
3.4
Green’s Function
3.5
A PDE with no solutions
4
Heat Equation
4.1
Spectral Theory and the Fourier Transform
4.2
Fundamental Solution
4.3
Maximum Principle
4.4
Heat Kernels
4.5
Heat Kernel of
(
0
,
1
)
5
Wave Equation
5.1
D’Alembert’s Formula
5.2
Solution on the half-line
5.3
Spherical Means of the Wave Equation
5.4
Solution in Dimension 3
5.5
Solution in Dimension 2
5.6
Inhomogeneous Wave Equation
5.7
Energy Methods
Appendix
Literature
Changes for 2024
Changes for 2023