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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 12/11/02}\\
{\it Closed book \hfill Van Dommelen \hfill 12:30-2:30 pm}
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Show all reasoning and intermediate results leading to your answer, or
credit will be lost. I must be able to see clearly {\em how} you
derived everything, and you must state {\em what} the result you
derived is {\em in terms of what is asked.}  Answer exactly what is
asked; you do {\em not} get credit for making up your own questions
and answering those.  You {\em must} use the systematic procedures
{\em followed in class,} not mess around randomly until you get some
answer.  Clean up as much as possible.  One book of mathematical
tables, such as Schaum's Mathematical Handbook, may be used, as well
as a calculator and a handwritten letter-size formula sheet.

\begin{enumerate}
  
\item Find the temperature in a bar of length 1 if the bar is at zero
  temperature at time $t=0$, but for $t>0$ the temperature of the ends
  is equal to t.  Take the heat conduction coefficient to be 1.  From
  the results, give a simple approximation for the temperature at
  large times that has a vanishingly small error.

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

\item Find the decay of an asymptotic suction boundary layer when the
  suction is terminated and the wall is allowed to move with the flow.
  This problem is governed by the following equations for the unknown
  $u(x,t)$:
  \begin{displaymath}
    u_t = u_{xx} \mbox{ for } 0 \le x \le \infty; 0 \le t \le \infty
  \end{displaymath}
  \begin{displaymath}
    u(x,0)= e^{-x} \qquad u(0,t) = 0.
  \end{displaymath}
  Note that, unlike usually, the initial condition is here nonzero.

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

\item The initial displacement $u(x,0)$ of a string is 1, (really only
  by approximation), except at the ends at $x=0$ and $x=1$, where the
  displacement is zero at all times; $u(0,t)=u(1,t)=0$.  The initial
  velocity is zero.  The wave propagation speed $a=1$.  Find the
  displacement $u$ at $x=0.5$ and $t=3$.  What sort of values can
  $u(x,t)$ have in general?

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

\item Answer the preassigned PDE problem.

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

\end{enumerate}

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