-- 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