Math 547: Partial Differential Equations 1

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

Minimal learning outcomes

1. General Cauchy problem
   • Cauchy-Kowalevski Theorem
   • Lewy Example
2. Method of characteristics for first-order equations
   • Semilinear case
   • Quasilinear case
   • General case
3. Quasilinear systems of conservation laws on a line
   • Riemann problem
   • Rankine-Hugoniot jump condition
   • Entropy condition
   • Shocks
   • Rarefaction waves
4. Classification of general second-order equations
5. Canonical forms for semilinear second-order equations
6. Hyperbolic equations
   • Cauchy problem
   • Problems with boundary data
   • Huygens' principle
   • Applications
7. Elliptic equations
8. Parabolic equations
9. 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
   • Heat equation
     • Fourier transforms
     • The heat kernel
     • Existence for the IVP
     • Weak and strong maximum principles
     • Uniqueness for the IBVP

Textbooks

Possible textbooks for this course include (but are not limited to):