Math 547: Partial Differential Equations 1
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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 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
Textbooks
Possible textbooks for this course include (but are not limited to):