cp
math4ft2015
Mathematics 4FT: Topics in Differential Equations
20152016: 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 (Takehome) 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 R^{1}
 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
v) gradient flow
4) Laplace's equation on
R^{n}_{+}
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
viii) Hadamard variational formula
5) Analysis of the Fourier transform
Course notes Chapter 5:
i) Hilbert space
ii) Schwartz class and distributions
6) Wave equations in R^{n}
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