# 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 using PDEs to model important phenomena.

### Prerequisites

Since ODEs appear in certain approaches to PDEs (in particular, the method of characteristics), Math 334 is a prerequisite. Math 342 is a prerequisite to ensure that students are reasonably comfortable with analysis in several dimensions.

### Minimal learning outcomes

Outlined below are topics that all successful Math 547 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, and ability to make direct application of those results to related problems, including calculations.

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.