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

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# Canonical forms for semilinear second-order equations | # Canonical forms for semilinear second-order equations | ||

# Hyperbolic equations | # Hyperbolic equations | ||

+ | #* The wave equation | ||

#* Cauchy problem | #* Cauchy problem | ||

#* Problems with boundary data | #* Problems with boundary data | ||

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#* Applications | #* Applications | ||

# Elliptic equations | # Elliptic equations | ||

+ | #* Laplace's equation | ||

+ | #* Poisson's equation | ||

+ | #* Green's functions | ||

+ | #* Maximum principles | ||

+ | #* Applications | ||

# Parabolic equations | # Parabolic equations | ||

# Classical theory for the canonical second-order linear equations on '''R'''<sup>''n''</sup> | # Classical theory for the canonical second-order linear equations on '''R'''<sup>''n''</sup> | ||

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#** Weak and strong maximum principles | #** Weak and strong maximum principles | ||

#** Uniqueness for the Dirichlet problem | #** Uniqueness for the Dirichlet problem | ||

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#* Heat equation | #* Heat equation | ||

#** Fourier transforms | #** Fourier transforms | ||

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#** Uniqueness for the IBVP | #** Uniqueness for the IBVP | ||

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=== Textbooks === | === Textbooks === | ||

## Revision as of 10:35, 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
- Classical theory for the canonical second-order linear equations on
**R**^{n}- Laplace's equation
- Green's first and second identities
- Mean Value Principle and its converse
- Weak and strong maximum principles
- Uniqueness for the Dirichlet problem

- Heat equation
- Fourier transforms
- The heat kernel
- Existence for the IVP
- Weak and strong maximum principles
- Uniqueness for the IBVP

- Laplace's equation

### Textbooks

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