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\centerline{Mathematics 174 -- Applied Mathematics 2}
\centerline{Midterm Problem Set}
\centerline{February 24, 2000}
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\noindent
{\it Groundrules and Directions}
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This is an open-book, open-notes, open-library, open Swords 335, unlimited 
time assignment.  If you use something you find in a book other than Haberman
in one of your solutions, please footnote it.
You must work independently, and you may not consult other faculty
members in working on these problems.  {\it When you have completed
the problem set, please write a brief statement saying that you have
abided by these ground rules, sign it, and turn it in with your solutions.}

In all parts of problems where solution of a BVP is asked for, 
give your answer in series form, {\it together with formulas
for the coefficients}.

Don't kill trees unnecessarily for the Maple output you submit:  
delete irrelevant work and {\it resize your plots} before printing!!  
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\noindent
{\bf Due:} No later than 5:00 pm, Thursday, March 2.
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\noindent
{\it Standing Assumptions:}  In all parts of both problems, $m$ is 
an integer and $m \ge 0$.
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\noindent
I.  For the 2D heat equation on a circular disk of radius $a$, if the outer
edge is {\it insulated}, we are led to consider the problem
$$\cases{u_t = k \nabla^2 u &\cr
         u_r(a,\theta,t) = 0 &\cr
         u(r,\theta,0) = \alpha(r,\theta) &\cr}$$
\item{A)}  If we look for product
solutions $u(r,\theta,t) = f(r)g(\theta)h(t)$, what ODE's must
$h$ and $g$ satisfy, and what
boundary value problem for $f(r)$ results?
\item{B)} Show that for each Bessel function
of the first kind $J_m(z)$ there is an infinite sequence of 
numbers $w_{mn} \ge 0$ where $J'_m(w_{mn}) = 0$.  (Note: I'm looking
for an actual proof here, given the facts we know about $J_m(z)$,
its zeroes, etc.  What theorem from 1-variable calculus guarantees the
existence of places where the derivative of a function is zero?) 
\item{C)}  What are the eigenvalues and eigenfunctions of the 
boundary value problem for $f(r)$?  
\item{D)}  Solve the problem:
$$\cases{u_t = k \nabla^2 u &\cr
         u_r(a,\theta,t) = 0 &\cr
         u(r,\theta,0) = r\sin(3\theta) &\cr}$$
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\noindent
II.  In this problem, you will study the eigenvalues and eigenfunctions
of the Sturm-Liouville problem:
$$\cases{
{d\over dr}\left(r{df\over dr}\right) - {m^2\over r}f = -\lambda r f &\cr
f(a) = 0 & where $a > 0$\cr
f(b) = 0 &\cr}\leqno(1)$$
arising from problems on annular (ring-shaped) regions,
and use them to solve several boundary value problems.
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\noindent
In class on 2/22, we saw that the change of variables $z^2=\lambda r^2$
takes this to a new type of boundary value problem for the Bessel equation
of order $m$:
$$\cases{z^2 {d^2 f\over dz^2} + z {df \over dz} + (z^2 - m^2)f = 0&\cr
         f(\sqrt{\lambda}a) = 0 &\cr
         f(\sqrt{\lambda}b) = 0 &\cr}\leqno(2)$$
(Note: the square roots are justified here since all eigenvalues satisfy
$\lambda \ge 0$, as we will see shortly.)

The {\it general solution} of Bessel's 
equation of order $m$ is a linear combination of the Bessel function 
of the first kind, $J_m(z)$,
and the Bessel function of the second kind, which is called $Y_m(z)$:
$$f(z) = c_1 J_m(z) + c_2 Y_m(z)$$  
For $m \ge 1$, $Y_m(z) \sim z^{-m}$ as $z \to 0$, so it is not defined for $z = 0$.
(Similarly, $Y_0(z) \sim c\ln(z)$ for a constant $c$, so it is not 
defined for $z = 0$ either.)
That is why we have not used (or needed) $Y_m(z)$ before -- in all
of our work on circular domains, we have wanted solutions $f$ of 
the Bessel equation with $|f(0)| < \infty$.  Now, however, our boundary
conditions are at $a,b >0$ and we are interested in functions on 
the interval $[a,b]$.  Their behavior at $z = 0$ is not relevant!
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\item{A)}  Show that for each integer
$m \ge 0$, we will get an infinite sequence of eigenvalues 
$\lambda_{mn}$ of (1), $n = 1, 2, 3, \ldots$, and corresponding eigenfunctions
$\phi_{mn}(r)$ determined uniquely up to a constant multiple. 
\item{B)}  Using the Rayleigh quotient for (1), show that 
all eigenvalues satisfy $\lambda_{mn} \ge 0$. 
\item{C)}  Is there a smallest eigenvalue overall?  If
so, for which $m$ is the minimum achieved?  If not, say why not.
\item{D)}  Show that if $f(z) = c_1 J_m(z) + c_2 Y_m(z)$ is
a nonzero function satisfying (2), then
$$J_m(\sqrt{\lambda}a)Y_m(\sqrt{\lambda}b) - 
J_m(\sqrt{\lambda}b)Y_m(\sqrt{\lambda}a) = 0\leqno(3)$$
\item{E)} The relation (3) in part D is actually {\it an equation for}
$\lambda$.  It cannot be solved exactly by any usual technique, but we
can get approximations to solutions by graphical and numerical methods 
in Maple.   Taking $a=1$ and $b=2$, by plotting, estimate the locations of the
first 4 solutions of (3) for $m = 0, 1, 2$.  Then use the
{\tt fsolve} command in Maple to generate accurate approximations
of each of your roots.  Notes: The Bessel function of the second kind
of order $m$ is called {\tt BesselY(m,z)} in Maple. 
The command {\tt fsolve(equation(var),var=c);}
will use a numerical method similar to Newton's Method to solve
the equation {\tt equation(var) = 0} for the variable $var$,
starting from the initial approximation
$var = c$.
\item{F)} Once you have an approximation for $\lambda_{mn}$, how
do you get an approximation to the corresponding eigenfunction 
$\phi_{mn}(r)$?
\item{G)} Studying the vibrations of an annular (ring-shaped) membrane
with inner ($r = a$) and outer ($r = b$) edges fixed at zero displacement leads
to consideration of the boundary value/initial value problems
$$\cases{ u_{tt} = c^2 \nabla^2 u &\cr 
          u(a,\theta,t) = 0 &\cr
          u(b,\theta,t) = 0 &\cr
          u(r,\theta,0) = \alpha(r,\theta) &\cr
          u_t(r,\theta,0) = \beta(r,\theta)&\cr}\leqno(4)$$
As always, the product solutions $u(r,\theta,t) =
f(r)g(\theta)h(t)$ are called the {\it normal modes of vibration}.
There are four types of normal modes (``cosine-cosine, cosine-sine, etc.'') 
of (4) for each pair of 
integers $(m,n)$ with $m\ge 0$, $n\ge 1$.  Incorporating 
your work from parts A through F, use Maple to plot (approximations to)
the ``cosine-cosine'' normal modes with $m = 0,1,2$, $n = 1,2,3,4$ for
a ring with $a = 1$ and $b = 2$.
(Hand in plots with $t = \pi/4$; you can try animations too if you like!)
Discuss the results, giving numbers and locations of nodal curves, etc.
\item{H)} Using your work from parts A through F again, 
solve the following BVP for the heat equation on an annular region
as simply as possible:
 $$\cases{ u_{t} = k \nabla^2 u &\cr 
          u(a,\theta,t) = 0 &\cr
          u(b,\theta,t) = 0 &\cr
          u(r,\theta,0) = (r-a)(b-r)\sin(2\theta) &\cr}$$

\bye
