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



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
    v)  gradient flow
 
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
    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 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