math 232 Spring 2002

Approximate Syllabus

  • Introduction and examples of common linear and nonlinear PDEs
  • Well posedness for ODEs vs PDEs
  • The Fourier Transform
  • The heat equation in R^n
  • The fundamental solution of the heat equation and mollifiers
  • The heat equation on bounded domains
  • simple time stepping schemes for solving the heat equation
  • Sobolev spaces
  • similarity solutions for linear and nonlinear diffusion equations
  • Hopf-Cole transformation for parabolic equations with quadratic nonlinearity
  • Transport equations, material derivative, and method of characteristics
  • Conservation laws in 1D, weak solutions, and the Rankine-Hugoniot jump condition
  • Viscous regularization, traveling waves, and admissible shocks
  • The Lax entropy condition and the Riemann problem
  • Finite difference schemes for conservation laws
  • Von Neumann stability analysis, consistency, and modified equation analysis
  • Second order equations: parabolic, elliptic and hyperbolic
  • Derivation of the wave equation
  • 1D wave equation on the line (D'Alembert's formula)
  • 1D wave equation on an interval and harmonics of musical instruments
  • Wave equation on a half space
  • method of spherical means for the wave equation in higher D
  • Laplace's and Poisson's equations
  • The Newtonian potential, distribution derivatives, and introduction to singular integral operators
  • properties of harmonic functions