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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 %{Class Time~}\underline{~~~~~~~~}
 \newline


\begin{center}
\bf Math 3150 Problems \\
{\small\bf Chapter 3}
\end{center}

{\small\sf
 {\bf Due date}: See the internet due date. Problems are
collected once a week. Records are locked when the stack is
returned. Records are only corrected, never appended.

{\bf Submitted work}. Please submit one package per problem.
 Label each problem with its corresponding problem number,
e.g., \fbox{Prob3.1-4} or \fbox{Xc1.2-4}. Kindly label extra credit
problems with label \fbox{\bf Extra Credit}. You may attach this
printed sheet to simplify your work.

{\bf Labeling}. The label \fbox{Probx.y-z} means the problem is for
chapter \fbox{x}, section \fbox{y}, problem \fbox{z}. When $y=0$,
then the problem does not have a textbook analog, it is a
\BF{background problem}. Otherwise, the problem number should match
a corresponding problem in the textbook. The same labeling applies
to extra credit problems, e.g., \fbox{Xc1.0-4}, \fbox{Xc1.1-2}. }

\section*{Chapter 3: 3.1 --     Examples in Physics and Engineering}

\Prob{Prob3.1-3}{(Classification)} Classify $u_{xx}-u_t =2u$ as
linear, nonlinear, homogeneous, non-homogeneous, and report the
order of the equation.

\Prob{Prob3.1-7}{(Laplace Equation)} Verify that $u(x,y)=e^y\cos x +
x + y$ is a solution of Laplace's partial differential equation.

\section*{Chapter 3: 3.2-3.3 -- One Dimensional Wave Equation}

\Prob{Prob3.2-1}{(Wave Equation)} Derive the equation
$u_{tt}=10^5u_{xx}$ for the vibrations of a stretched homogeneous
string with linear density $\rho=0.001$ kg/m and tension $\tau=100$
N, with no forces other than the tension. State all assumptions used
to obtain the model. Make the presentation brief, by referencing a
textbook for derivation details and results.

\Prob{Prob3.3-9a}{(Separation of Variables)} Solve $u_{tt}=u_{xx}$,
$u(0,t)=u(1,t)=0$, $u(x,0)=x(1-x)$, $u_t(x,0)=\sin \pi x$, $t\ge 0$,
$0 \le x \le 1$. The model is for a guitar string of unit length.

\Prob{Prob3.3-9b}{(Filmstrip Plots)} Plot partial sums of the answer
to the previous problem,
$$u(x,t) = \frac{1}{\pi}\sin (\pi x) \sin (\pi t) + \sum_{m=0}^\infty \frac{8}{\pi^3(2m+1)^3}\sin (2m\pi x+\pi
x)\cos(2m\pi t+\pi t),$$ at
$t=0,1,2,3$. Choose the number of series terms for the four graphics
by making the first graphic match $x(1-x)$ on $0\le x \le 1$. This filmstrip
has 4 frames, each frame corresponding to a time $t$. A frame has
graph window $0 \le x \le 1$, $a \le u \le b$ (you must choose $a,b$).

\Prob{Prob3.3-9c}{(Surface Plot)} Plot a specific partial sum of the
answer
$$u(x,t)=\frac{1}{\pi}\sin (\pi x) \sin (\pi t)+\sum_{m=0}^\infty \frac{8}{\pi^3(2m+1)^3}\sin (2m\pi x+\pi
x)\cos(2m\pi t+\pi t)$$ on
the domain $0\le x \le 1$, $0\le t \le 4$. Use all features possible
of the 3D graphics program in order to produce the best plot with
fine accuracy, view and colors.

\Prob{Prob3.3-13}{(Damped Vibrations of a String)} Solve the problem
$$ \begin{array}{rcl}
\dd u_{tt}(x,t) + u_t(x,t) &=&\dd u_{xx}(x,t), \\
 u(0,t) &=& 0,\\
 u(\pi,t)&=& 0,\\
 u(x,0) &=& \sin x,\\
\dd u_t(x,0) &=& 0.
 \end{array}
$$


\section*{Chapter 3: 3.4 --     d'Alembert's Method}

\Prob{Prob3.4-15}{(d'Alembert's Solution)} Consider the problem
$$
 \begin{array}{rcl}
\dd u_{tt} &=& \dd u_{xx},\,\, 0\le x \le 1, \, t\ge 0, \\
 u(0,t) &=& 0,\\
 u(1,t)&=& 0,\\
 u(x,0) &=& f(x),\\
 u_t(x,0) &=& 0.
 \end{array}
$$
Assume $f(x)=4x$ on $0\le x \le 0.25$, $f(x)=2-4x$ on $0.25 < x \le
0.5$, $f(x)=0$ on $0.5<x \le 1$.

 (a) Find a solution formula for $u(x,t)$ using d'Alembert's method.

 (b) Plot a 3-frame filmstrip of the string shape at times $t=0,0.25,0.5$.

\begin{verbatim}
# EXAMPLE. Let f(x)=4x on [0,.25], f(x)=2-4x on [.25,.5], f(x)=0 otherwise
# Asmar 3.4-15, D'Alembert's solution of the wave equation, f=pulses,g=0
pulse:=(x,a,b)->piecewise(x<a,0,x<b,1,0);
f:=x->4*x*pulse(x,0,1/4)+(2-4*x)*pulse(x,1/4,1/2);
#plot(f(x),x=0..1);
F:=x->piecewise(x<0,-f(-x),f(x)); # Odd extension of f(x)
plot(F(x),x=-1..1);
u:=(x,t)->(1/2)*(F(x+t)+F(x-t));
#plot(u(x,0.7),x=-2..2);
plots[animate]( plot, [u(x,t),x=-3..3], t=0..1.5, trace=0, frames=50 );
\end{verbatim}
See the web site links for updates to this sample maple code.

\Prob{Xc3.4-18}{(Energy Conservation and d'Alembert's Solution)}
Define
$$ E(t) = \frac12 \int_0^L \left(u_t^2(x,t)+c^2u_x^2(x,t)\right)dx.$$
Prove the energy conservation law, which says that the energy during
free vibrations of a string is constant for all time.

{\small\sf \BF{Hint}: Show $dE/dt=0$.}

\section*{Chapter 3: 3.5-3.6 -- One Dimensional Heat Equation}

\Prob{Prob3.5-13}{(Nonhomogeneous Heat Equation)} Consider the
one-dimensional heat conduction problem
$$
 \begin{array}{rcl}
\dd u_{t} &=&\dd u_{xx},\,\, 0\le x \le \pi, \, t> 0, \\
 u(0,t) &=& 100,\\
 u(\pi,t)&=& 50,\\
 u(x,0) &=& f(x).
 \end{array}
$$
Assume $f(x)=33x$ on $0< x \le \pi/2$, $f(x)=33\pi-33x$ on $\pi/2 <
x < \pi$. Find a solution formula for the temperature $u(x,t)$. 


 \Prob{Prob3.6-3}{(Heat Conduction in an Insulated Bar)} Consider the
one-dimensional heat conduction problem
$$
 \begin{array}{rcl}
\dd u_{t} &=&\dd u_{xx},\,\, 0\le x \le 1, \, t> 0, \\
\dd u_x(0,t) &=& 0,\\
\dd u_x(1,t)&=& 0,\\
 u(x,0) &=& \cos \pi x
 \end{array}
$$
Find a solution formula for the temperature $u(x,t)$ at location $x$
along the bar at time $t$. Hint: Don't integrate!

Remark. {\small\sf The book's problem 3.6-3 has a piecewise example,
using $u(x,0)=f(x)$. See the maple advice for problem 3.5-13, to handle
that case.}

\section*{Chapter 3: 3.7 --     Two Dimensional Equations}

\Prob{Prob3.7-5a}{(Vibrations of a Membrane)} Consider the
rectangular drumhead problem, in which we assume $0 < x < 1$, $0 <
y< 1$, $t> 0$:
$$
 \begin{array}{rcl}
\dd u_{tt}(x,y,t) &=&\dd \frac{1}{\pi^2}\left(u_{xx}(x,y,t)+u_{yy}(x,y,t)\right),\\
 u(0,y,t) &=& 0,\\
 u(1,y,t)&=& 0,\\
 u(x,0,t) &=& 0,\\
 u(x,1,t) &=& 0,\\
 u(x,y,0) &=& 0,\\
 u_t(x,y,0)&=&1.
 \end{array}
$$
Solve for the drumhead defection $u(x,y,t)$.

\Prob{Prob3.7-5b}{(Membrane Snapshots)} Consider the solution of the
rectangular drumhead problem given by the series
$$
 u(x,y,t) = \frac{16}{\pi^2}\sum_{n\mbox{~odd}} \sum_{m\mbox{~odd}}
 \frac{1}{nm\sqrt{n^2+m^2}}\,\sin (m\pi x)\sin (n\pi y)
 \sin\left(t\sqrt{m^2+n^2}\right).$$
 Illustrate the various shapes of the drumhead during vibration, by
 plotting suitable surface snapshots at times $t=1,2,3$. The
 snapshot at $t=0$ should be the initial flat membrane shape $u=0$.
 Choose suitable partial sums to reveal adequate detail in the
 plots.

\Prob{Prob3.7-12}{(Heat Conduction in a Plate)} Consider the
rectangular plate heat conduction problem in which we assume $ 0\le
x \le 1$, $0 < y < 1$, $t> 0$:
$$
 \begin{array}{rcl}
\dd u_{t}(x,y,t) &=&\dd u_{xx}(x,y,t)+u_{yy}(x,y,t), \\
 u(0,y,t) &=& 0,\\
 u(1,y,t)&=& 0,\\
 u(x,0,t) &=& 0,\\
 u(x,1,t) &=& 0,\\
 u(x,y,0) &=& x(1-x)y(1-y).
 \end{array}
$$
Solve for the plate temperature $u(x,y,t)$.

\Prob{Xc3.7-12}{(Heat Conduction in a Plate)} Find a series estimate
for the solution $u(x,y,t)$ of the rectangular plate heat conduction
problem which shows that $|u(x,y,t)|\le M e^{-\alpha t}$ for some
number $M>0$ and some constant $\alpha>0$. Then conclude that
$$\lim_{t\to\infty} u(x,y,t)=0,$$
which implies the plate temperature $u$ stabilizes to the edge
temperature $u=0$ as $t$ approaches infinity.

\begin{quote}
{\small\sf {\bf Problem notes}.

The Cauchy-Schwartz inequality is used to find a upper estimate of
$|u|^2$ as a product of two positive series. One series is numeric,
and Bessel's inequality can be used to determine an upper bound
$M_1$ for it. The other series in the product is a series of
functions, each function an exponential function bounded above by
$e^{-\beta t}$, where $\beta>0$ is a fixed constant. A clever
analysis of the exponential factors, using the geometric series
formula $\sum_{n=0}^\infty r^n = \frac{1}{1-r}$, shows that the
second series is bounded by some constant $M_2>0$ times $e^{-\rho
t}$, where $\rho=\beta/2$. Taking square roots across $|u|^2\le
M_1M_2e^{-\rho t}$ implies that constants $M=\sqrt{M_1M_2}>0$ and
$\alpha=\rho/2>0$ satisfy $|u|\le Me^{-\alpha t}$.}
\end{quote}

\section*{Chapter 3: 3.8-3.9 -- Laplace's and Poisson's Equations}

\Prob{Prob3.8-2}{(Steady-State Temperature in a Plate)} Consider the
rectangular plate steady-state heat conduction problem in which we
assume the plate is given by $0\le x \le 1$, $0 < y < 1$:
$$
 \begin{array}{rcl}
\dd u_{xx}(x,y)+u_{yy}(x,y) &=& 0, \\
 u(x,0) &=& 0,\\
 u(x,1) &=& 100,\\
 u(0,y) &=& 0,\\
 u(1,y)&=& 100.
 \end{array}
$$

 (a) Draw a figure for the Dirichlet problem, showing the edge
 temperatures on the plate. Break the problem into two subproblems,
 decomposing $u=u_1+u_2$. Draw figures for each subproblem.

 (b) Solve for the temperatures $u_1(x,y)$ and $u_2(x,y)$.

 (c) Report the solution to the original problem, $u=u_1+u_2$.
 
 \medskip 
 
        \bf{Notes}. In the general problem of Nakhle's section 3.8,
$f_1(x)=g_1(y)=0$ and $f_2(x)=g_2(y)=100$.
 In the summary shaded display before the 3.8 exercises, $A_n=C_n=0$ and
$B_n$, $D_n$ are computed from equations (5), (6). Example 2 in Nakhle's
section 3.8 solves for $B_n$, therefore you have an easy answer check
for half the problem.

\Prob{Prob3.9-3}{(Poisson Problem)} Consider the rectangular plate
steady-state Poisson heat conduction problem in which we assume the
plate is given by $0\le x \le 1$, $0 < y < 1$:
$$
 \begin{array}{rcl}
 u_{xx}(x,y)+u_{yy}(x,y) &=& \sin \pi x, \\
 u(x,0) &=& 0,\\
 u(x,1) &=& x,\\
 u(0,y) &=& 0,\\
 u(1,y)&=& 0.
 \end{array}
$$

 (a) Draw a figure for the Poisson problem with zero boundary conditions [see (b)].
  Draw a second figure for the corresponding Dirichlet problem with
  identical boundary conditions [see (c)].

 (b) Solve for the temperature $u_1(x,y)$ satisfying the Poisson
 problem

$$
 \begin{array}{rcl}
\dd u_{xx}(x,y)+u_{yy}(x,y) &=& \sin \pi x, \\
 u(x,0) &=& 0,\\
 u(x,1) &=& 0,\\
 u(0,y) &=& 0,\\
 u(1,y)&=& 0.
 \end{array}
$$

 (c) Solve for the temperature $u_2(x,y)$ satisfying the Dirichlet
 problem

$$
 \begin{array}{rcl}
\dd u_{xx}(x,y)+u_{yy}(x,y) &=& 0, \\
 u(x,0)&=& 0,\\
 u(x,1) &=& x,\\
 u(0,y) &=& 0,\\
 u(1,y)&=& 0.
 \end{array}
$$

 (d) Report the solution to the original Poisson problem, which is
$u=u_1+u_2$.

\medskip

{\bf Notes}. Problem (c) is solved in section 3.8 of Nakhle's textbook,
with summary in the shaded display just before the exercises 3.8. In
this display, $A_n=C_n=D_n=0$ and $B_n$ must be computed from (5) using
$f_2(x)=x$. Problem (b) is solved from Nakhle's section 3.9 equations
(2) and (4). A similar problem is solved in Example 1. The challenge is
the double integration in (4) with $f(x,y)=\sin \pi x$. Luckily, this is
an iterated double integral, evaluated by two successive one-variable
integrations: $$E_{mn}=\frac{-4}{\lambda_{mn}}\int_0^1 \sin \pi x \sin
m\pi x dx\int_0^1 \sin n\pi y dy.$$
\end{document}
\endinput
 Zhu's problems
 • Section 2.1: 20;
 • Section 2.2: 8, 15;
 • Section 2.3: 5;
 • Section 2.4: 7(a), 15;
 • Section 3.3: 9, 12(a,b);
 • Section 3.4: 15;
 • Section 3.5: 13;
 • Section 3.6: 3;
 • Section 3.7: 5, 13;
 • Section 3.9: 3;
 • Section 4.4: 5, 15, 17;
 • Section 7.1: 8;
 • Section 7.2: 20, 31;
 • Section 7.3: 19.






















