Math 547: Partial Differential Equations 1

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Catalog Information

Title

Partial Differential Equations 1. [Recommended change: Applied Partial Differential Equations]

Credit Hours

3

Prerequisite

Math 334, 342; 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

It is proposed that the focus of this course be applied partial differential equations. Thus, the students ought to be less concerned with knowing existence/uniqueness proofs than with understanding the properties and representation of solutions and with using PDEs to model important phenomena.

Prerequisites

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
    • The wave equation
    • Cauchy problem
    • Problems with boundary data
    • Huygens' principle
    • Applications
  7. Elliptic equations
    • Laplace's equation
    • Poisson's equation
    • Green's functions
    • Maximum principles
    • Applications
  8. Parabolic equations
    • The heat equation
    • Green's functions
    • The heat kernel
    • Maximum principles
    • Applications

Textbooks

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

  • John Ockendon, Sam Howison, Andrew Lacey, and Alexander Movchan, Applied Partial Differential Equations (Revised Edition), Oxford University Press, 1999.

Additional topics

Courses for which this course is prerequisite

Currently this course is listed as a prerequisite for Math 647. It is recommended that this connection be dropped.