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\newcommand{\erf}{\mathop{\rm erf}\nolimits}                                  
                                           
                                       
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 \centerline{\large Math.\ 401, Sec.\ 501\hfill    Spring, 2006}
 \bigskip
\centerline{\bfseries\large Homework 10, due April 5}
\bigskip
                                           
                                       
\begin{enumerate}


\item {\it [Cf.\ remark on p.\ 84 of notes.]\/} \
Find the Parseval equation for the Fourier cosine series.  
\begin{enumerate}
\item Use the definition of the Fourier cosine series on pp.\ 
78--79 of class notes.
\item How would your answer change if you used the other 
convention (p.\ 81 of notes)?
\end{enumerate}


\item {\it [Schaum's, p.\ 46, Ex.\ 2.56 and 2.57]\/} \ 
\begin{enumerate}
\item A square plate of side $L$ has one side maintained at 
temperature $f(x)$ and the others at zero.
Find the steady-state temperature at any point of the plate
(as a Fourier series of appropriate type).
\item Explain how to solve the problem if the four sides 
are maintained at temperatures $f_1(x)$, $g_1(y)$, $f_2(x)$, and 
$g_2(y)$.
(Write out the answer in full for the case $f_2 =0=g_2\,$.)
\end{enumerate}

\item {\it [Schaum's, p.\ 46, Ex.\ 2.58(a)]\/} \
 An infinitely long plate of width $L$ has its two parallel sides 
maintained at temperature $0$ and its other side at constant 
temperature $T$.
Find the steady-state temperature.

\item {\it [Schaum's, p.\ 46, Ex.\ 2.63]\/} \ 
Solve the boundary-value problem
$$\frac{\partial u}{\partial t} = \frac{\partial^2u}{\partial 
x^2} - \alpha^2u, \qquad
u(0,t)=u_1\,, \quad u(L,t)=u_2\,, \quad
u(x,0)=0,$$
where $0<x<L$, $0<t$, and $\alpha$, $L$, $u_1\,$, and $u_2\,$  
are constants.

                                          
                                      
                                      


\item  {\it [Schaum's, p.\ 94, Ex.\ 5.42]\/} \
An infinite thin bar ($-\infty <x<\infty$) whose surface is 
insulated 
has an initial temperature
$$f(x) = \left\{\begin{array}{ll} u_0 &\mbox{if $|x|<a$}, \\
\noalign{\smallskip}                  0 & \mbox{if $|x|\ge a$}. 
\end{array}\right.
$$
\begin{enumerate}
\item Solve the heat equation
$\displaystyle\frac{\partial u}{\partial t} = 
\kappa\,\frac{\partial^2u}{\partial x^2}$
 by separation of variables, 
obtaining a double integral.
\item Exchange the order of integration and evaluate the 
inner integral as the heat equation Green function
(called ``Gauss--Weierstrass kernel'' by Constanda).
Work the answer into the final form
$$u(x,t) = \frac{u_0}2 \left[ \erf\left( {x+a\over 2 \sqrt{\kappa 
t}}\right)
- \erf\left( {x-a\over 2 \sqrt{\kappa t}}\right) \right].$$
The {\sl error function\/} \,erf\, is defined by 
$$\erf(x) = \frac2{\sqrt{\pi}} \int_0^x e^{-u^2}\,du.$$
\end{enumerate}

\goodbreak
\item  {\it [Schaum's, p.\ 94, Ex.\ 5.43]\/} \
A semiinfinite solid ($x>0$) has initial temperature 
$f(x) = u_0 e^{-bx^2}$. 
The plane face, $x=0$, is insulated.
Show that the temperature is
$$u(x,t) = \frac{u_0}{\sqrt{1+4\kappa bt}}\,
e^{-bx^2/ (1+4\kappa bt)}.$$
(Follow the same two steps as in the previous problem.)

\goodbreak
\item  {\it [Schaum's, p.\ 94, Ex.\ 5.46]\/} \
Solve the potential problem
$$\frac{\partial^2u}{\partial x^2} + 
\frac{\partial^2u}{\partial y^2} =0$$
for $y>0$ with the boundary data
$$u(x,0) = \left\{\begin{array}{ll}
 0 &\mbox{if $x<-1$ or $x>1$}, 
\\ \noalign{\smallskip}
                  1 & \mbox{if $-1<x<1$}. \end{array}\right.
$$

\item  {\it [Schaum's, p.\ 94, Ex.\ 5.49]\/} \
The lines $y=0$ and $y=a$ in the $xy$ plane are kept at 
potentials $0$ and $f(x)$ respectively.
In the strip between, the potential $u(x,y)$ obeys Laplace's 
equation (as in the previous problem).
Show that the solution is
$$u(x,y) = \frac1{\pi} \int_{\lambda=0}^\infty 
\int_{\zeta=-\infty}^\infty
f(\zeta) \, {\sinh \lambda y\over \sinh \lambda a}\, \cos 
\lambda(\zeta-x)\,
d\zeta\, d\lambda.$$

\end{enumerate}
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