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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 12/11/01}\\
{\it Closed book \hfill Van Dommelen \hfill 5:30-7:30 pm}
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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 standard size formula sheet.

\begin{enumerate}
  
\item Classify the PDE
  \begin{displaymath}
    y^2 u_{xx} - 2 x y u_{xy} + x^2 u_{yy} +
    y u_x + x u_y = 0
  \end{displaymath}
  Find new coordinates $\xi$ and $\eta$ that simplify this PDE.  Draw
  the coordinate lines (lines of constant $\xi$ and of constant
  $\eta$).  Indicate the form of the PDE in terms of the new
  coordinates.

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

\item Find the temperature $u(x,t)$ in a bar initially at zero
  temperature if one end is kept at zero temperature but the
  temperature at the other end is linearly increased in time:
  \begin{displaymath}
    u_t = \kappa u_{xx}
  \end{displaymath}
  \begin{displaymath}
    u(x,0) = 0 \qquad u(0,t) = 0 \quad u(1,t) = t
  \end{displaymath}
  What is the form of the temperature distribution after long times?

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

\item Solve the convection equation
  \begin{displaymath}
    u_t = - u_x  \qquad x \ge 0 \quad t \ge 0
  \end{displaymath}
  with initial and boundary conditions
  \begin{displaymath}
    u(x,0) = 0 \qquad u_x(0,t) = f(x)
  \end{displaymath}
  (where $f(x)$ is a known function) using the Laplace transform
  method.  Clean up the answer.

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

\end{enumerate}

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