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 Rn
- 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
