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

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== Chris Grant's Proposed Core Topics for Math 547/548 == | == Chris Grant's Proposed Core Topics for Math 547/548 == | ||

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

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+ | === Prerequisites === | ||

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+ | === Minimal learning outcomes === | ||

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# General Cauchy problem | # General Cauchy problem | ||

− | #* Cauchy- | + | #* Cauchy-Kowalevski Theorem |

#* Lewy Example | #* Lewy Example | ||

# Method of characteristics for first-order equations | # Method of characteristics for first-order equations | ||

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# Classification of general second-order equations | # Classification of general second-order equations | ||

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

+ | # Hyperbolic equations | ||

+ | # Elliptic 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> | ||

#* Laplace's equation | #* Laplace's equation | ||

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

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

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

## Revision as of 10:23, 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.

## Chris Grant's Proposed Core Topics for Math 547/548

## 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
- 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

- Wave equation
- Method of spherical means
- Hadamard’s method of descent
- Huygen’s Principle
- Conservation of Energy
- Domain of Dependence

- 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):