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\begin{document}

\begin{center}
{\bf EML 4930/5930 \hfill Analysis in M.E. II \hfill 04/22/08}\\
{\it Closed book \hfill Van Dommelen \hfill 3-5 pm}
\end{center}

Show all reasoning and intermediate results leading to your answer, or
credit will be lost. One book of mathematical tables, such as Schaum's
Mathematical Handbook, may be used, as well as a calculator and one
handwritten letter-size single formula sheet.

\begin{enumerate}

\item (20\%) Solve the PDE and initial condition
  \begin{displaymath}
    y^3 u_x - x^3 u_y = y^3 u \quad \mbox{ for } y \ge 0;
    \qquad u(x,0) = 1 \quad \mbox{ for } x \ge 0.
  \end{displaymath}
  Why is the initial condition only given for $x\ge0$?

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}

\item (20\%) Use D{\rq}Alembert to find the pressure $u$ to the
  problem of acoustics in a pipe of length $\frac12\pi$ with one end
  open and the other end closed,
  \begin{displaymath}
    u_{tt} = u_{xx} \qquad u(0,t)=0 \quad u_x({\textstyle\frac12}\pi,t)=0
  \end{displaymath}
  if the initial conditions are
  \begin{displaymath}
    u(x,0)=x\quad u_t(x,0) = \sin x.
  \end{displaymath}
  In particular evaluate $u$ at $x=1.5$ and $t=2.5$.  Note that the
  extension of $u_t(x,0)$ to all $x$ will take a simple analytical
  form.

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}

\item (40\%) Solve the following problem of heat conduction in a bar
  of length 1 using separation of variables:
  \begin{displaymath}
    u_t = u_{xx} \qquad u(x,0)=x \quad u(0,t)=0 \quad u(1,t)=e^{-t}.
  \end{displaymath}
  Write out the fully worked-out solution precisely and completely.

\begin{rawhtml}
<A HREF="figures/3.jpg">Solution.</A>
\end{rawhtml}

\item (20\%) Solve the following problem of vibrations in a
  semi-infinite string using the Laplace transform method:
  \begin{displaymath}
    u_{tt}=a^2 u_{xx} \qquad u(x,0)=u_t(x,0)=0\quad u(0,t)=f(t)
  \end{displaymath}
  where $f(t)$ is to be considered to be some given function and
  $x\ge0,t\ge0$.  Show all steps and reasoning. Your answer should not
  have weird mathematics, but be in simple terms that anyone with a
  basic understanding of functions can understand.  Does your solution
  agree with the general solution to the wave equation?

\begin{rawhtml}
<A HREF="figures/4.jpg">Solution.</A>
\end{rawhtml}

\end{enumerate}

\newpage
\begin{center}
  \renewcommand{\arraystretch}{2}
  {\bf Table 6.3: Properties of the Laplace Transform}\\
  \begin{tabular}{|l|l|}
    \hline\hline
    \multicolumn{1}{|c|}{$\displaystyle f(t)_{\strut}^{\strut}$} &
    \multicolumn{1}{c|}{$\displaystyle \hat f(s)
      =\int_{\strut0}^{\strut\infty}f(t)e^{-st}\D t$} \\
    \hline\hline
    {\bf 1.} $\displaystyle C_1 f_1(t) + C_2 f_2(t) $ &
    $\displaystyle C_1 \hat f_1(s) + C_2 \hat f_2(s)$ \\
    \hline
    {\bf 2.} $\displaystyle f(at)$ &
    $\displaystyle a^{-1} \hat f(s/a)\qquad (a>0)$ \\
    \hline
    {\bf 3.} $\displaystyle f^{(n)}(t)$ &
    $\displaystyle s^n \hat f(s) - s^{n-1} f(0) - \ldots -
     f^{(n-1)}(0) \qquad (n=1,2,\ldots)$ \\
    \hline
    {\bf 4.} $\displaystyle t^n f(t)$ &
    $\displaystyle (-1)^n
    \hat f^{(n)}(s)\qquad (n=1,2,\ldots)$ \\
    \hline
    {\bf 5.} $\displaystyle e^{ct} f(t)$ &
    $\displaystyle \hat f(s-c)\qquad (s = \mbox{const.})$ \\
    \hline
    {\bf 6.}
    \begin{tabular}[t]{l}
      $\displaystyle H(t-b) f(t-b)$, where \\
      {\renewcommand{\arraystretch}{1.5}
        $\displaystyle \quad H(t) =
        \bigg\{_{\strut}^{\strut}
        \begin{array}{lr} 0\ & t<0 \\ 1\ & t>0 \end{array}$}
    \end{tabular} &
    $\displaystyle e^{-bs} \hat f(s)\qquad (b>0)$ \\
    \hline
    {\bf 7.} $\displaystyle f * g(t)\equiv
    \int_{{\strut}0}^{{\strut}t} f(t-\tau)g(\tau)\D\tau$ &
    $\displaystyle \hat f(s)\hat g(s)$ \\
    \hline\hline
  \end{tabular}
  \\
  \vspace{5pt}
  {\bf Table 6.4: Laplace Transform Pairs}\\
  \begin{tabular}{|l|l|}
    \hline\hline
    \multicolumn{1}{|c|}{$\displaystyle f(t)$} &
    \multicolumn{1}{c|}{$\displaystyle \hat f(s)
      =\int_{\strut0}^{\strut\infty}f(t)e^{-st}\D t$} \\
    \hline\hline
    {\bf 1.} $\displaystyle 1 $ &
    $\displaystyle \frac{1}{s}_{\strut}$ \\
    \hline
    {\bf 2.} $\displaystyle t^n$ &
    $\displaystyle \frac{n!}{s^{n+1}}_{\strut}\qquad(n=1,2,\ldots)$ \\
    \hline
    {\bf 3.} $\displaystyle e^{kt}$ &
    $\displaystyle \frac1{s-k}_{\strut}$ \\
    \hline
    {\bf 4.} $\displaystyle \sin(at)$ &
    $\displaystyle \frac{a}{s^2+a^2}_{\strut}$ \\
    \hline
    {\bf 5.} $\displaystyle \cos(at)$ &
    $\displaystyle \frac{s}{s^2+a^2}_{\strut}$ \\
    \hline
    {\bf 6.} $\displaystyle \frac{1}{\sqrt{\pi t}}_{\strut}$ &
    $\displaystyle \frac{1}{\sqrt{s}}$ \\
    \hline
    {\bf 7.} $\displaystyle \frac{1}{\sqrt{\pi t}}_{\strut} e^{-k^2/4t}$ &
    $\displaystyle \frac{1}{\sqrt{s}} e^{-k\sqrt{s}}\qquad(k>0)$ \\
    \hline
    {\bf 8.} $\displaystyle \frac{k}{\sqrt{4\pi t^3}}_{\strut} e^{-k^2/4t}$ &
    $\displaystyle e^{-k\sqrt{s}}\qquad(k>0)$ \\
    \hline
    {\bf 9.} 
    \begin{tabular}[t]{l}
      $\displaystyle \hbox{erfc}\left(k/2\sqrt{t}\right)_{\strut}$,
      where\\
      $\displaystyle \hbox{erfc}(z) \equiv
      \frac2{\sqrt{\pi}}\int_{\strut z}^\infty e^{-u^2} \D u$
    \end{tabular} &
    $\displaystyle \frac1s e^{-k\sqrt{s}}\qquad(k>0)$ \\
    \hline\hline
  \end{tabular}
\end{center}

\end{document}
