Math 547: Partial Differential Equations 1

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Catalog Information


Partial Differential Equations 1.

Credit Hours



Math 334, 342; or equivalents.


Math 314, 341; or equivalents.


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
    • 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
  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
    • Heat equation
      • Fourier transforms
      • The heat kernel
      • Existence for the IVP
      • Weak and strong maximum principles
      • Uniqueness for the IBVP


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

Additional topics

Courses for which this course is prerequisite