Lecture Schedule

Here is a lecture by lecture list of topics covered in class, with references for further reading.

Introduction

• Introduction, and model PDE’s.
• Derivation of model PDE’s from physical principles. (Strauss 1.3, 372-wiki.)
  • Transport equation
  • Heat equation
  • Wave equation
  • Laplace and Poisson equation
• Boundary conditions (Strauss 1.4, 372-wiki.)
• Uniqueness via Energy methods
  • Energy decay for the heat equation (372-wiki)
  • Conservation of energy for the wave equation (Strauss 2.2)

A Glimpse at Numerical Methods

Reference: Hall and Porsching, Chapters 1, 4.

• Approximating derivatives
• The explicit Euler scheme for the heat equation
  • Stability analysis

Separation of Variables and Fourier Series

References. Strauss: Chapters 4, 5. Pinchover and Rubinstein: Chapter 5. 372-wiki: Separation of Variables, Fourier Series

• Series form solutions for the heat and wave equations.
• Eigenfunctions of the Laplacian
• Computing Fourier Coefficients
• Bessel’s and Parseval’s inequality
• Basic convergence results

Harmonic Functions

References. Strauss: Chapter 6. 372-wiki

• Separation of variables in a disk.
  • Poisson Kernel
  • Poisson Formula in two dimensions
  • Mean value property, Strong Maximum Principle
  • Smoothness of solutions
  • Approximate identities, and convergence at the boundary
• Separation of variables in a square, annulus and a wedge.
• Eigenfunctions of the Laplacian
• Maximum principle with convection terms.

Greens Functions

References. Strauss: Chapter 7, 372-wiki.

• Newton Potentials.
  • The Poisson equation in $\R^d$.
  • A representation formula for Harmonic functions.
  • Mean value property.
• Greens functions.
  • Existence and Uniqueness
  • Symmetry
  • Computation in half space and spheres.

The Heat Equation

References. Strauss: Chapters 2,3. 372-wiki

• The heat kernel on $\R^d$.
• Duhamel’s principle
• Parabolic maximum principle.
• Strong maximum principle.

The Wave Equation

• D’Alembert’s formula
• Duhamel’s principle
• Kirchoff’s formula
• Huygens principle

Optional topics

(These were covered in the last week and a half of class for interest only. No homework will be assigned on these topics, and they will not appear on the final.)

An introduction to asymptotic expansions

• Periodic divergence form equations
• Computing the effective diffusivity

The Fourier Transform

• Elementary Properties
• Inversion
• Applications to linear PDE’s
• Uncertainty Principle

An introduction to Fluid Dynamics

• The Euler and Navier-Stokes equations
• D’Alembert’s paradox: why planes shouldn’t fly.