A graduate course in Mathematics at the Department of Mathematics, National Tsing Hua University.
Instructor: Professor Wei-Cheng Wang,
Meeting: General III building, room 707, T5R5R6.
First Class: September 16, 1999
Grading: The grade will be based on weekly homework assignments
Textbook and References: Ref: L. C. Evans: Partial Differential Equations ; F. John: Partial Differential Equations. (Evans will be the prime reference, our course outline)
Course description:
Part I : Difinition of PDE; Notation; Linear PDE's; Four impotatant linear PDE's: linear transport equation, Wave equation, Laplace equation and heat equation.
Part II : General techniques for solving linear PDE's: exact formula, reduction to ODE (method of characteristics), Fourier transform and Laplace transform. Separation of variables (Eigenfunction expansions), ...
Part III : Theory of Linear PDE: Generalized functions (distributions); Weak dereivatives, Weak solutions (distribution-valued solutions), Well posedness, Initial value problem, boundary value problem, Initial boundary value problem, Hilbert space method, Maximum princeple, Energy method, Smooth approximations, Fourier analysis revisited, ...
Part V : Nonlinear PDE's: (Details later).
Lectures:
Week 1: Lecture 1.
Week 4: Lecture 6, Yet another variation on the theme of linearity: representing solutions of the Dirichlet problems using Green's function. Lecture 7. Properties of harmonic functions: mean value property, maximum principle, regularity and local estimates.
Homework assignments
Assignment 1, in postscript format, in LaTeX format, or in HTML (a last resort).
Assignment 2, in postscript format, in LaTeX format, or in HTML.
Assignment 3, in postscript format, in LaTeX format, and (soon) HTML format, along with the data file
Assignment 4, in postscript format, in LaTeX format.
Make sure to check the homework comments and hints page often while doing assignments.