Difference between revisions of "Math 547: Partial Differential Equations 1"
From MathWiki
Line 2: | Line 2: | ||
=== Title === | === Title === | ||
+ | Partial Differential Equations 1. | ||
− | === | + | === Credit Hours === |
− | + | 3 | |
− | + | ||
=== Prerequisite === | === Prerequisite === | ||
+ | [[Math 214]], [[Math 334|334]]; or equivalents. | ||
=== Description === | === Description === | ||
− | + | Topics include the method of characteristics, elliptic equations, potential theory, parabolic equations and systems, maximum principles, linear and nonlinear waves, Hamilton-Jacobi equations, Fourier transforms, Green's functions, distributions, and energy methods. | |
− | + | ||
== Chris Grant's Proposed Core Topics for Math 547/548 == | == Chris Grant's Proposed Core Topics for Math 547/548 == |
Revision as of 12:00, 9 May 2008
Contents
Catalog Information
Title
Partial Differential Equations 1.
Credit Hours
3
Prerequisite
Math 214, 334; or equivalents.
Description
Topics include the method of characteristics, elliptic equations, potential theory, parabolic equations and systems, maximum principles, linear and nonlinear waves, Hamilton-Jacobi equations, Fourier transforms, Green's functions, distributions, and energy methods.
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