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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 12/12/96 }\\
{\it Closed book \hfill Van Dommelen \hfill 10:00-12:00am }
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Show all reasoning and intermediate results leading to your answer.
One book of mathematical tables, such as Schaum's Mathematical Handbook,
may be used.

\begin{enumerate}

\item
(30 points)
In the separation-of-variables solution of a heat conduction problem
with a mixed and a Dirichlet boundary condition, you encounter the
following Sturm-Liouville problem:
\[ \begin{array}{c}
X_n'' + \lambda_n X_n = 0; \\
X_n'(0) - p X_n(0) = 0, \quad X_n(\ell) = 0,
\end{array}\]
with $p$ a positive constant.
Show that there are no nontrivial solutions to this Sturm-Liouville
problem for negative or zero $\lambda_n$. Find an approximation for
the values of the large eigenvalues $\lambda_n$.
\begin{rawhtml}
  <a href="figures/1.gif">Solution</a>
\end{rawhtml}

\item
(30 points)
Solve the following wave propagation problem:
\[ \begin{array}{c}
u_{tt} + 2 u_t + u = u_{xx} \mbox{ for } 0 \le t, 0 \le x; \\
u(x,0) = u_t(x,0) = 0; \quad u_x(0,t) = f(t).
\end{array}\]
\begin{rawhtml}
  <a href="figures/2.gif">Solution</a>
\end{rawhtml}

\item
(40 points)
Find the steady temperature $u(r,\theta)$ in a circular plate of
radius one if the normalized heat flow out of the edge of the
plate $u_r(1,\theta)$ is a given function $f(\theta)$.
Use separation of variables.
List the Sturm-Liouville problem that gives rise to the
eigenfunctions, and list the complete set of resulting eigenfunctions.
Show that function $f(\theta)$ must satisfy a constraint
in order for a solution to exist.  What is the physical reason
for the constraint?
Also show that the solution can be written in the form
\[ u(r,\theta) =
\int_0^{2\pi} f(\bar\theta) G(r,\theta-\bar\theta) \/ {\rm d}\bar\theta,\]
for some function $G(\cdot,\cdot)$.
\begin{rawhtml}
  <a href="figures/3.gif">Solution</a>
\end{rawhtml}

\end{enumerate}

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