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\centerline{Mathematics 373 -- Applied Mathematics (PDE)}
\centerline{Final Problem Set}
\centerline{April 30, 2001}
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{\it Groundrules and Directions}
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This is an open-book, open-notes, open-library, open Swords 335,
open Internet, unlimited 
time assignment.  If you use something you find in a book other than our
text or on the web in one of your solutions, please footnote it (give
the full URL or ``web address'' for any sites you use).
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 a series solution of a BVP is asked for, 
give your answer {\it together with formulas
for the coefficients}.

Don't kill trees unnecessarily for the Maple output you submit:  Please
delete irrelevant work and {\it resize your plots} before printing.
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\noindent
{\bf Due:} No later than 5:00 pm, Friday, May 11.
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\noindent
I. (35) In this problem, we will reconsider solutions of the 1D wave 
equation on finite intervals and compare the information we get
from Fourier series solutions with the information we get from
D'Alembert's approach to the wave equation on ${\bf R}$.
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\item{A)}  Consider the problem
$$\cases{u_{tt} = u_{xx}& \cr
         u(0,t) = u(5,t) = 0& \cr
         u(x,0) = \cases{5(x-2)(3-x) & if $2\le x \le 3$\cr
                            0 & otherwise\cr} & \cr
         u_t(x,0) = 0& \cr}\leqno(1)$$
on the interval $[0,5]$.  Using Maple, compute and plot a Fourier
(sine) series solution to this problem.  (Be sure to take enough terms
in your partial sum so that there is good agreement with the initial
conditions.)  Describe the behavior of the solution as $t$ ranges from 
$0$ to $10$.  What happens when a wave reaches one of the endpoints of 
the string?
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\noindent
Now we want to study the relation between solutions of a general problem
of the same type as (1):
$$\cases{u_{tt} = u_{xx}& \cr
         u(0,t) = u(L,t) = 0& all $t$\cr
         u(x,0) = f(x) & \cr
         u_t(x,0) = g(x) &\cr}\leqno(2)$$
and the D'Alembert solution of the general initial
value problem problem for the 1D wave equation on ${\bf R}$
$$\cases{u_{tt} = u_{xx}& \cr
         u(x,0) = F(x) & \cr
         u_t(x,0) = G(x).&\cr}\leqno(3)$$
Recall, D'Alembert's solution is given by
$$u(x,t) = {1\over 2}(F(x-t) + F(x+t)) + {1\over 2}\int_{x-t}^{x+t} G(s)\ ds$$
(we have taken the speed of wave propagation, $c$, to be $ = 1$
in the general formula).  D'Alembert's formula describes a solution
on the whole real line; it {\it does not} take into account the fixed
ends of the string, say at $x=0, x = L$.  To bring those additional
conditions into play, we will proceed as follows.
First, we extend $f,g$ from (2) to {\it odd functions} on the interval $[-L,L]$
as in the construction of the Fourier sine series.  Then we extend the odd
extensions to {\it odd periodic} functions on ${\bf R}$ with 
period $2L$.  
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\item{B)}  Sketch the odd periodic extension of the initial data
in the given problem (1) above.
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\noindent
Apply the D'Alembert formula with $F,G$ equal to the odd periodic 
extensions of the initial data from (2) and get the solution of (3)
on ${\bf R}$ for the periodic initial conditions.  
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\item{C)}  Show that the {\it restriction of the D'Alembert solution\/}
of (3) to the interval $[0,L]$, with initial data 
given by the odd periodic extensions $F,G$ is the corresponding solution 
of (2).
\item{D)}  Show how part C leads to the form of the solution you 
saw in part A for the specific problem (1).
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\noindent
II. (30) Consider a Sturm-Liouville form differential equation:
$${d\over dx}\left(p(x){dy\over dx}\right) + q(x) y = 0,
\leqno(4)$$
and assume $p(x) > 0$ for all $x \in [a,b]$ (as in a regular Sturm-Liouville
eigenvalue problem).  
In this problem, we will study the {\it zeroes} of solutions of (4), and
prove a result called the {\it Sturm Separation Theorem} -- part C below.
Let $y_1,y_2$ be any two solutions of (4). As we saw
in the ODE class, the {\it Wronskian} of $y_1, y_2$ is defined to be:
$$W[y_1,y_2](x) = \det\pmatrix{y_1(x)&y_2(x)\cr
                               {dy_1\over dx}(x)&{dy_2\over dx}(x)\cr}$$
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\item{A)} Show that $W[y_1,y_2](x) = C/p(x)$ for some constant $C$. 
({\bf Hint:} What is the derivative ${d\over dx}W[y_1,y_2](x)$?)  
\item{B)}  Use part A to show that if the Wronskian of $y_1$ and $y_2$ 
is zero at any $c \in [a,b]$, then $y_1$ and $y_2$ are {\it linearly dependent}.
You will need to use the fact that for all $c \in [a,b]$, and all 
$A,B\in {\bf R}$, there is a {\it unique} solution of (4), together 
with the initial conditions $y(c) = A$, ${dy\over dx}(c) = B$.
\item{C)}  Let $y_1,y_2$ be two linearly independent solutions of (4).  
The Sturm Separation Theorem is the statement that 
between any two consecutive zeroes of $y_1$ in $[a,b]$, there
is precisely one zero of $y_2$.  Argue as follows:  Assume $x_1 < x_2$ are the
consecutive zeroes of $y_1$, but that $y_2$ is not zero on the 
interval $[x_1,x_2]$.  Consider the function ${y_1\over y_2}$ and 
apply the ``most valuable theorem'' (MVT) of calculus.
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III. (35) 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
$$\cases{u_t = k \nabla^2 u &\cr
         u_r(a,\theta,t) = 0 &\cr
         u(r,\theta,0) = \alpha(r,\theta) &\cr}$$
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\item{A)}  If we look for product
solutions $u(r,\theta,t) = R(r)\Theta(\theta)T(t)$, what ODE's must
$R$, $\Theta$, and $T$ satisfy, and what
boundary value problem for $R(r)$ results?
\item{B)} Show that for each Bessel function
of the first kind $J_m(z)$ there is an infinite sequence of 
numbers $\mu_{mn} \ge 0$ where $J'_m(\mu_{mn}) = 0$.  (Note: I'm looking
for an actual proof here, not just a plausibility argument from the 
shape of the graph of $J_m(z)$.) 
\item{C)}  What are the eigenvalues and eigenfunctions of the 
1D boundary value problem for $R(r)$?  
\item{D)}  Give a Fourier-Bessel expansion for the solution of 
the problem:
$$\cases{u_t = k \nabla^2 u &\cr
         u_r(a,\theta,t) = 0 &\cr
         u(r,\theta,0) = e^{-(r-1/2)^2}. &\cr}$$
Taking $a = k = 1$, plot a partial sum approximation to the solution
using Maple.  Show the temperature function at the times 
$t = 0, 0.5, 1.0, 1.5$, and $2.0$.  (Or do a Maple animation!)
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{\bf Extra Credit.} (10) Who first posed the question, ``can you hear the
shape of a drum?''  When, and by whom, was the question answered?  
What is the answer?  What areas of mathematics were involved in
finding the answer?
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\centerline{\it Have an enjoyable and productive summer!}
\bye