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

-- Solving ODEs: separable, linear, by substitution. Second order ODEs with constant coefficients
-- Conservation laws, the characteristics method. Interpretation of PDEs, BCs, and ICs.
-- Derivation of conservation laws, heat, wave and Laplace equations. The divergence theorem.
-- The Laplace equation in polar coordinates. The Laplace equation in the radial case (n - dimensions).
-- The maximum principle.
-- Classification of PDEs - bringing a PDE to canonical form when the transformation is given.
-- The solution of Cauchy problem for the heat equation in terms of the heat kernel G(x,t).
-- D'Alembert's formula for the wave equation. Domain of dependence.
-- 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).
-- Sources on bounded domains
-- Finite difference methods - approximating derivatives by difference qoutients; computational
atoms.

Formulas that will be given if necessary:
- variation of parameters formula for second order differential equations with constant coefficients
- the heat kernel G(x,t) and the erf function
- definition of the Fourier transform and transformation of derivartives under FT
- tables of Laplace or Fourier transforms
- Poisson's formula
- stability conditions for finite difference methods for the heat and wave equations