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\title{Final Study Guide}
\date{}


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\begin{document}
\maketitle

\section{Concepts}
\subsection{First order linear}

Solution by characteristic ODE/ change of variable

\subsection{Heat Equation}
\begin{itemize}
\item Heat kernel solution for infinite and half-life
\item With initial conditions
\item Homogeneous and nonhomogeneous
\item Separable solutions for bounded domain. This will have boundary conditions as well as initial conditions
\item When is maximum principle applicable?
\end{itemize}
\subsection{Wave Equation}

\begin{itemize}
\item d'Alembert solution for the infinite line, even/odd extension solution for half-line
\item homogeneous/nonhomogeneous.
\item two initial conditions
\item Separable solution for bounded interval.  This will have boundary conditions.
\end{itemize}

\subsection{Laplace Equation}

\begin{itemize}
\item Maximum principle.
\item Separable Solutions.
\end{itemize}

\section{Some Exercises}
\begin{enumerate}
\item Solve the eigenvalue problem
\begin{equation*}
y'' + \lambda y = 0 \ \ \text{ on } [0,L]
\end{equation*}
with 
\begin{enumerate}
\item Dirichlet boundary $y(0) = y(L) = 0$
\item Neumann boundary $y'(0) = y'(L) = 0$.
\end{enumerate}
\item Show that the eigenvalue is nonnegative for Neumann boundary and strictly positive for Dirichlet boundary.  Note that $y=0$ is the trivial case, which we disregard.
\item Write the Laplacian $\Delta = \frac{\dd^2}{\dd x^2} + \frac{\dd^2}{\dd y^2}$ in polar coordinates.
\item Solve Laplace's equation
\begin{equation*}
\Delta u = 0
\end{equation*}
for the unit ball $ x^2+y^2 =1$ with Dirichlet boundary condition. (Hint: maximum principle).
\item Assuming the maximum principle for harmonic functions, show that the solution to the Dirichlet problem of the Poisson equation
\begin{equation*}
\begin{cases}
\Delta u = f & \text{ in }D\\
u = h & \text{ on }\dd D.
\end{cases}
\end{equation*}
where $f$ and $h$ are appropriate given functions.
\item Section 4.1, 1--6
\item Section 4.2, 1--4
\end{enumerate}
\end{document}


