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

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## Revision as of 09:37, 31 May 2011

## Contents

## Catalog Information

### Title

Partial Differential Equations 1.

### Credit Hours

3

### Prerequisite

Math 334, 342; or equivalents.

### Recommended(?)

Math 314, 341; 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

### Prerequisites

### Minimal learning outcomes

- General Cauchy problem
- Cauchy-Kowalevski Theorem
- Lewy Example

- Method of characteristics for first-order equations
- Semilinear case
- Quasilinear case
- General case

- Quasilinear systems of conservation laws on a line
- Riemann problem
- Rankine-Hugoniot jump condition
- Entropy condition
- Shocks
- Rarefaction waves

- Classification of general second-order equations
- Canonical forms for semilinear second-order equations
- Hyperbolic equations
- The wave equation
- Cauchy problem
- Problems with boundary data
- Huygens' principle
- Applications

- Elliptic equations
- Laplace's equation
- Poisson's equation
- Green's functions
- Maximum principles
- Applications

- 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):