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
-- Fourier Method and Orthogonal Expansions; L^2 inner-product;
(pointwise, uniform, mean-square); generalized Fourier series
Bessel's inequality; Parseval's identity;
-- Classical Fourier series (on [-L,L]); convergence of Fourier series
pointwise, uniform (Dirichlet's theorem)
-- Sturm-Liouville Problems; eigenvalues and eigenfunctions; theorem
of infinitely many eigenvalues and eigenfunctions for a
SLP associated with the
-- Separation of variables: the general method for the heat and wave
Neumann, Robin boundary conditions and their
-- energy arguments
-- Solving Laplace's equation with separation of variables; Green's
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
Formulas that will be given if
- 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