cp math4ft-2015

## Mathematics 4FT: Topics in Differential Equations

### 2015-2016: Analysis of partial differential equations

Instructor: W. Craig
Meeting times: Tuesday, Thursday and Friday 14:30 - 15:20 in HH 102

Office hours: Thursday 11:00 - 12:30 or by appointment, HH 418

Synopsis: This course will focus on the principal partial differential equations that arise in physics. We will describe solution methods for the most commonly arising partial differential equations; this will involve the Fourier transform, the method of characteristics, the maximum principle, and Hilbert space techniques. Concurrently we will introduce aspects of analysis that are useful in their applications to the study of solutions; this includes the analytic theory of Fourier series and Fourier integrals, the geometry of Hilbert space and operator theory, and energy methods in Sobolev space. In the course of this analysis we will deduce many of the main features of these solutions, including finite propagation speed and conservation of energy for wave equations, maximum principles for Laplace's equation and for heat flow, shock formation for nonlinear conservation laws, and other mathematical aspects of familiar and less familiar laws of physics.

Course notes are planned for handout and library reserve.

Recommended text:

F. John, Partial Differential Equations, Fourth Edition

Course Outline:  here

Problem set assignments:

Due Friday September 18
Chapter 1, page 7: problems 1.2, 1.4, 1.6 (extra problems 1.3* and 1.5)
Chapter 2, page 15: problems 2.4, 2.5 2.6

Due Friday October 2
Chapter 3. Problem Set #2

Due Friday October 23
Chapter 4. Problem Set #3

Due Tuesday November 10
Chapter 5. Problem Set #4

Due Friday November 27
Chapter 6. Problem Set #5

Due Tuesday December 15
Final Exam (Take-home) final

Syllabus:

1) Introduction.      Course notes Chapter 1:

2) Wave equations      Course notes Chapter 2:

i)  Fourier transform - elementary properties
ii)  transport equations
iii)  wave equations in R1 - d'Alembert's formula
iv)  method of images

3)  heat equation      Course notes Chapter 3:

i)  heat kernel
ii)  convolution operators
iii)  maximum principle
iv)  conservation laws and moments

4)  Laplace's equation on Rn+      Course notes Chapter 4:

i)   Dirichlet, Neumann, and Poisson problems
ii)   Poisson kernel
iii)   maximum principle
iv)   oscillation and attenuation estimates
v)   fundamental solution
vi)   maximum principles again
vii)   Green's functions and the Dirichlet - Neumann operator

5)  Analysis of the Fourier transform           Course notes Chapter 5:

i)   Hilbert space
ii)  Schwartz class and distributions

6)  Wave equations in Rn        Course notes Chapter 6:

i)  wave propagator by Fourier synthesis
ii)  Hamiltonian PDEs
iii)  Paley - Wiener theory
iv)  method of spherical means
v)  Huygens' principles
vi)  conservation laws and shocks

7) Method of stationary phase
i)  Schrödinger's equation
ii)  Heisenberg uncertainty principle
iii)  Fourier multipliers and their symbols
iv)  phase and group velocities
v)  stationary phase estimates

last edits October 9, 2015