# Difference between revisions of "Math 547: Partial Differential Equations 1"

### Title

Partial Differential Equations 1. [Recommended change: Applied Partial Differential Equations]

3

### Prerequisite

Math 334, 342; or equivalents.

### Description

Methods of analysis for hyperbolic, elliptic, and parabolic equations, including characteristic manifolds, distributions, Green's functions, maximum principles and Fourier analysis.

## Desired Learning Outcomes

It is proposed that the focus of this course be applied partial differential equations. Thus, the students ought to be less concerned with knowing existence/uniqueness proofs than with understanding the properties and representation of solutions and with the modeling of important phenomena with PDEs.

### Minimal learning outcomes

1. General Cauchy problem
• Cauchy-Kowalevski Theorem
• Lewy Example
2. Method of characteristics for first-order equations
• Semilinear case
• Quasilinear case
• General case
3. Quasilinear systems of conservation laws on a line
• Riemann problem
• Rankine-Hugoniot jump condition
• Entropy condition
• Shocks
• Rarefaction waves
4. Classification of general second-order equations
5. Canonical forms for semilinear second-order equations

6. Hyperbolic equations
• The wave equation
• Cauchy problem
• Problems with boundary data
• Huygens' principle
• Applications
7. Elliptic equations
• Laplace's equation
• Poisson's equation
• Green's functions
• Maximum principles
• Applications
8. Parabolic equations
• The heat equation
• Green's functions
• The heat kernel
• Maximum principles
• Applications

### Textbooks

Possible textbooks for this course include (but are not limited to):

• John Ockendon, Sam Howison, Andrew Lacey, and Alexander Movchan, Applied Partial Differential Equations (Revised Edition), Oxford University Press, 1999.