Difference between revisions of "Math 547: Partial Differential Equations 1"
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# Canonical forms for semilinear second-order equations | # Canonical forms for semilinear second-order equations | ||
# Hyperbolic equations | # Hyperbolic equations | ||
| + | #* The wave equation | ||
#* Cauchy problem | #* Cauchy problem | ||
#* Problems with boundary data | #* Problems with boundary data | ||
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#* Applications | #* Applications | ||
# Elliptic equations | # Elliptic equations | ||
| + | #* Laplace's equation | ||
| + | #* Poisson's equation | ||
| + | #* Green's functions | ||
| + | #* Maximum principles | ||
| + | #* Applications | ||
# Parabolic equations | # Parabolic equations | ||
# Classical theory for the canonical second-order linear equations on '''R'''<sup>''n''</sup> | # Classical theory for the canonical second-order linear equations on '''R'''<sup>''n''</sup> | ||
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#** Weak and strong maximum principles | #** Weak and strong maximum principles | ||
#** Uniqueness for the Dirichlet problem | #** Uniqueness for the Dirichlet problem | ||
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#* Heat equation | #* Heat equation | ||
#** Fourier transforms | #** Fourier transforms | ||
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#** Uniqueness for the IBVP | #** Uniqueness for the IBVP | ||
</div> | </div> | ||
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=== Textbooks === | === Textbooks === | ||
Revision as of 10:35, 31 May 2011
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.
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
- The wave equation
- Cauchy problem
- Problems with boundary data
- Huygens' principle
- Applications
- Elliptic equations
- Laplace's equation
- Poisson's equation
- Green's functions
- Maximum principles
- Applications
- 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
- 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):
