Topics covered on the second exam in PDE-Math 324 (Fall 06):

-- topics that were included on the first exam (e.g. solving ODEs, conservation laws,
interpretation of PDEs and BCs)
-- Ill-Posed Problems - definition; show that a problem is ill-posed
-- Solving PDEs on semi-infinite domains; odd and even reflections
-- Duhamel's Principle for ODEs and PDEs; source terms
-- Laplace transform; definition; applicability; transformation of derivatives under LT;
convolution product; the convolution theorem;
-- Fourier transform; applicability; convolution product; the convolution theorem;
-- Fourier Method and Orthogonal Expansions; L^2 inner-product; convergences
(pointwise, uniform, mean-square); generalized Fourier series and coefficients;
Bessel's inequality; Parseval's identity;
-- Classical Fourier series (on [-L,L]); convergence of Fourier series - mean-square,
pointwise, uniform (Dirichlet's theorem)
-- Sturm-Liouville Problems; eigenvalues and eigenfunctions; theorem about existence
of infinitely many eigenvalues and eigenfunctions for a SLP associated with the
diffusion equation
-- Separation of variables: the general method for the heat and wave equations; Dirichlet,
Neumann, Robin boundary conditions and their interpretation;
-- energy arguments
-- Solving Laplace's equation with separation of variables; Green's identities; uniqueness
of solutions; Dirichlet's principle
-- Cooling of a sphere (solving the radial heat equation in 3D).

Formulas that will be given if necessary:
- the fundamental solution of the heat equation (the erf function)
- D'Alembert formula (solution of the wave equation)
- definition of the Fourier transform and transformation of derivartives under FT
- tables of Laplace or Fourier transforms
- Poisson's formula

Caution!!! This list applies only for the second exam! Some of the above formulas
may not be given on the final exam.