math 266BC Winter/Spring 2005

Proposed Syllabus Winter 2005 Math 266B, Parabolic and Elliptic PDE

  • Week 1: Introduction and examples of common linear and nonlinear PDEs, Well posedness for ODEs vs PDEs, The Fourier Transform, distribution derivatives
  • Week 2: The heat equation in R^n, Sobolev spaces, The fundamental solution of the heat equation and mollifiers,
  • Week 3: Similarity solutions for linear and nonlinear diffusion equations, The heat equation on bounded domains.
  • Week 4: Energy methods
  • Week 5: Second order parabolic equations and the maximum principle
  • Week 6: Simple time stepping schemes for solving the heat equation, stability and monotonicity
  • Week 7 and 8: Linear stability analysis and Calculus of variations
  • Week 9: Laplace's and Poisson's equations
  • Week 10: The Newtonian potential and introduction to singular integral operators
  • Note we will include examples from current research topics including Porous Media equation, TV minimization in image processing (ROF, etc), etc.

    Proposed Syllabus Spring 2005, Hyperbolic PDE

  • Week 1: Transport equations, material derivative, and method of characteristics
  • Week 2: Conservation laws in 1D, weak solutions, and the Rankine-Hugoniot jump condition
  • Week 3: The Lax entropy condition and the Riemann problem, Hopf-Cole transformation for parabolic equations with quadratic nonlinearity
  • Week 4: Viscous regularization, traveling waves, and admissible shocks, non-convex flux functions and the Oleinik Chord condition.
  • Week 5: The wave equation in 1D, the line, semi-infinite line and bounded domain (musical instruments)
  • Week 6: The wave equation in multiple dimensions (spherical means etc)
  • Week 7: Elementary finite differences for conservation laws: Von Neumann stability analysis, consistency
  • Week 8: Nonlinear numerical techniques conservation laws: modified equation analysis and Godunov's method
  • Weeks 9,10: Systems of scalar conservation laws
  • Note we will include examples related to current research topics including fluid equations, traffic flow, Hamilton-Jacobi equations, etc.

    Texts: Evans Partial Differential Equations, first four chapters, Strickwerda Finite difference schemes, some handouts in class, Leveque's book for godunov, Haberman elementary PDE, McOwen, Majda/Bertozzi (excerpts).