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

Revision as of 09:36, 9 May 2008

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

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+== Catalog Information ==
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+=== (Credit Hours:Lecture Hours:Lab Hours) ===
+=== Offered ===
+=== Prerequisite ===
+=== Description ===
+=== Note ===
 == Chris Grant's Proposed Core Topics for Math 547/548 ==
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