EML 5060 Analysis in Mechanical Engineering 12/12/96
Closed book Van Dommelen 10:00-12:00am

Show all reasoning and intermediate results leading to your answer. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used.

1.

(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 $\lambda_n$. Solution

2.

(30 points) Solve the following wave propagation problem:

$$\begin{array} {c} u_{tt} + 2 u_t + u = u_{xx} \mbox{ for } 0... ... x; \ u(x,0) = u_t(x,0) = 0; \quad u_x(0,t) = f(t).\end{array}$$

Solution

3.

(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 $G(\cdot,\cdot)$. Solution