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\begin{document}

\centerline{\textbf{Sample Midterm Questions}}
\centerline{\textbf{Math 121}, \textbf{Fall 2004}}

\bigskip\bigskip
\noindent
\textbf{1.} Use Fourier series to find the solution $u(x,y)$ of the following boundary value problem
for Laplace's equation in the semi-infinite strip $0<x<1$, $y>0$:
\beann
&&\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0,\\
&&u(0,y) = u(1,y) = 0,\\
&&u(x,0) = 1,\\
&&u(x,y) \to 0\quad \mbox{as $y\to\infty$}.
\eeann

\bigskip\bigskip
\noindent
\textbf{2.} Use Fourier series to find the solution $u(x,t)$ of the following initial-boundary value problem
for the wave equation in $0<x<1$ and $t >0$:
\beann
&&\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0,\\
&&\frac{\partial u}{\partial x}(0,t) = \frac{\partial u}{\partial x}(1,t) = 0,\\
&&u(x,0) = 0,\\
&&\frac{\partial u}{\partial t}(x,0) = x.
\eeann



\bigskip
\noindent
\textbf{3.} Use Fourier transforms to solve the following initial value problem
for $u(x,t)$ in $-\infty < x < \infty$, $t >0$:
\beann
&&\frac{\partial u}{\partial t} = -\frac{\partial^4 u}{\partial x^4},\\
&&u(x,0) = f(x).
\eeann
Write the solution for $u(x,t)$ as a convolution, but do compute any inverse transforms explicitly.
How smooth is the solution for $t>0$?



\newpage
\noindent
\textbf{4.} (a) Give the formulas for the Fourier transform $\widehat{f}(k)$ of a function $f(x)$ and the
inverse Fourier transform.

\smallskip\noindent
(b) Compute the Fourier transform of $e^{-|x|}$.


\smallskip\noindent
(c) State Parseval's theorem, and use it to evaluate
\[
\int_0^\infty\frac{1}{(1+k^2)^2}\, dk.
\]


\bigskip
\noindent
\textbf{5.} Use Laplace transforms to solve the following initial value problem:
\beann
&&y^{\prime\prime} +2 y^\prime + 2 y = 1,\\
&&y(t) = 0,\quad y^\prime(0) = 1.
\eeann

\bigskip
\noindent
\textbf{6.}
(a) Say what jump conditions the solution of
$y(t)$ of the following initial value problem satisfies
at $t=0$, and find the solution directly (do not use Laplace transforms):
\beann
&&y^{\prime\prime} - 4 y = \delta(t),\\
&&y(t) = 0\quad\mbox{for $t<0$}.
\eeann

\smallskip\noindent
(b) Write the solution of the following initial value problem, where $f(t)$ is an arbitrary function,
as a convolution (you don't need to derive your answer):
\beann
&&y^{\prime\prime} - 4 y = f(t),\\
&&y(0) = y^\prime(0) = 0.
\eeann

\end{document}