# math 266BC Winter/Spring 2005

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. **
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).