-- 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.