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

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

+ | <div style="-moz-column-count:2; column-count:2;"> | ||

+ | # General Cauchy problem | ||

+ | #* Cauchy-Kovalevskaya 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 | ||

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

+ | #* 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 | ||

+ | </div> | ||

+ | |||

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

## Revision as of 14:11, 8 May 2008

## Contents

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

- General Cauchy problem
- Cauchy-Kovalevskaya 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
- 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