math 266C Spring 2007

Approximate Syllabus

  • Week 1: Introduction and examples of common linear and nonlinear PDEs, Well posedness for ODEs vs PDEs, The Fourier Transform
  • 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: Simple time stepping schemes for solving the heat equation, Hopf-Cole transformation for parabolic equations with quadratic nonlinearity
  • Week 5: Transport equations, material derivative, and method of characteristics
  • Week 6: Conservation laws in 1D, weak solutions, and the Rankine-Hugoniot jump cond ition
  • Week 7: Viscous regularization, traveling waves, and admissible shocks
  • The Lax entropy condition and the Riemann problem
  • Week 8: Finite difference schemes for conservation laws
  • Von Neumann stability analysis, consistency, and modified equation analysis
  • Week 9: Laplace's and Poisson's equations
  • The Newtonian potential, distribution derivatives, and introduction to singular integral operators
  • Week 10: properties of harmonic functions

    Texts: Evans Partial Differential Equations, first four chapters, Strickwerda Finite difference schemes, some handouts in class.