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\begin{document}

\begin{center}
\noindent Topics for final exam
\end{center}

\bigskip

\begin{itemize}
\item First order systems 

\begin{itemize}
\item linear and nonlinear

%\item conservation laws

\item characteristic curves , compatibility condition
\end{itemize}

\item Fourier Expansion

\item Wave Equation - 1D

\begin{itemize}
\item infinite domain - D'Alembart

\item semi-infinite, even and odd functions

\item bounded domain - Fourier

\item Dirichlet, Neumann and Robin boundary conditions

\item forcing term - Duhamel's principle

\item properties

\begin{itemize}

\item characteristics

\item well-posedness

\item domain of influence

\item domain of dependence

\item conservation of energy

\item propagation of singularities
\end{itemize}
\end{itemize}

\item Parabolic Equation - 1D

\begin{itemize}
\item explicit solution via convolution with Gaussian

\item maximum principle

\item energy

\item uniqueness

\item infinite and semi-infinite domains

\item bounded domain - Fourier

\item forcing terms, Duhamel
\end{itemize}

\item Sturm Liouville

\begin{itemize}
\item properties of eigenvalues and eigenvectors

\item convergence of series of eigenfunctions

\item integration \& differentiation of series

\item Gibb's phenomena

\newpage
\end{itemize}

\item Multi-dimensions

\begin{itemize}
\item separation of \ variables - convergence

\item forcing terms, eigenfunction expansion, Duhamel
\end{itemize}

\item Elliptic Equations

\begin{itemize}

\item Separation of Variables

\item Maximum Principle

\item Poisson formula

\item Rayleigh Quotient
\end{itemize}

\item Bessel \&  spherical harmonic functions

%\item Fourier \& Laplace transforms

\item Helmholtz equation

\item Green's functions

\item Distributions

\begin{itemize}

\item definitions

\item PDEs with delta functions

\end{itemize}

\end{itemize}

\end{document}
