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

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=== Description === | === Description === | ||

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

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== Desired Learning Outcomes == | == Desired Learning Outcomes == | ||

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

# Hyperbolic equations | # Hyperbolic equations | ||

+ | #* Cauchy problem | ||

+ | #* Problems with boundary data | ||

+ | #* Huygens' principle | ||

+ | #* Applications | ||

# Elliptic equations | # Elliptic equations | ||

# Parabolic equations | # Parabolic equations | ||

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#** Existence for the Dirichlet Problem on domains with regular boundaries and for continuous boundary data | #** Existence for the Dirichlet Problem on domains with regular boundaries and for continuous boundary data | ||

#** Interior and exterior sphere conditions | #** Interior and exterior sphere conditions | ||

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

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

## Revision as of 09:29, 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
- Cauchy problem
- Problems with boundary data
- Huygens' principle
- Applications

- Elliptic equations
- 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
- Poisson integral formula
- Existence for the Dirichlet Problem on a ball
- Fundamental solutions
- Green's functions
- Harnack inequality
- Liouville's Theorem
- Harnack's Convergence Theorem
- Existence for the Dirichlet Problem on domains with regular boundaries and for continuous boundary data
- Interior and exterior sphere conditions

- 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):