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 \centerline{\large Math.\ 401, Sec.\ 501\hfill    Spring, 2006}
 \bigskip
\centerline{\bfseries\large Homework 8, due March 22}
\bigskip
                                                                               
\begin{enumerate}
                                                                               
%\item  {\it [Logan, p.\ 195,  Ex.\ 3.1(d)]}\quad
%Find the eigenvalues and eigenfunctions for the problem
%$$y'' + \lambda y = 0 \quad\mbox{on $0<x<L$}, $$
%$$y'(0) = 0, \qquad y(L) =0.$$
                                                                               


 \item Suppose that the boundary conditions (``BC:'')
on p.~68 of the notes are replaced by
$$   \frac{\partial w}{\partial x}(t,0) = 0, \qquad 
\frac{\partial w}{\partial x}(t,1) = 0.$$
What changes would be necessary in pp.\ 68--73
(and corresponding lecture on separation of variables in the
heat equation)?
{\sl Warning:\/}  Pay close attention to the possibility
that $\lambda$ may not always be positive.

 
\item {\it [Logan, p.\ 195,  Ex.\ 3.4]\/} \quad
Find the Fourier sine series of $f(x) =1$ on $0\le x\le\pi$.
 To what value does the series converge at $x=0\,$?
at $x=\frac{\pi}2\,$?

\item {\it [Logan, p.\ 195,  Ex.\ 3.9]\/} \quad
Solve by  separation of variables (getting a Fourier sine series):
$$u_{xx} + u_{yy} =0  \qquad\mbox{on \quad$0<x<a$, \quad 
$0<y<b$,}$$
$$u(x,0) =0\quad \mbox{and}\quad u(x,b) = f(x) \qquad\mbox{for 
$0<x<a$,}$$
$$u(0,y) = u(a,y) =0 \qquad\mbox{for $0<y<b$.}$$



\item {\it [Schaum's, p.\ 46, Ex.\ 2.43]\/}

\begin{enumerate}
\item Show that for $-\pi <x<\pi$,
$$x= 2\left({\sin x\over1} -{\sin 2x\over 2} +{\sin 3x\over 3} - 
\cdots \right).$$
\item By integrating the result of (a), show that for 
$ -\pi \le x\le\pi$,
$$x^2 = {\pi^2\over 3} -4\left({\cos x\over 1^2} - {\cos 2x\over 
2^2 
} + 
{\cos 3x \over 3^2} - \cdots\right).$$
(Determine the constant of integration by requiring that 
the series formula give the right answer for $\int_0^\pi 
x^2\,dx$.)

\item By integrating once more and being careful about 
constants of integration, show that
$$x(\pi-x)(\pi+x) =
12\left({\sin x\over 1^3} - {\sin 2x\over 2^3} +{\sin 3x \over 
3^3} 
- \cdots\right).$$
\end{enumerate}

\goodbreak
\item {\it [Schaum's, p.\ 46, Ex.\ 2.37]\/}

\begin{enumerate}
\item Expand $f(x)=\cos x$, $0<x<\pi$, in a Fourier sine 
series.
\item
How should $f(x)$ be defined at $x=0$ and $x=\pi$ so that the 
series will converge to $f(x)$ for $0\le x\le \pi$?
\end{enumerate}


\item {\it [Schaum's, p.\ 46, Ex.\ 2.50(a)]\/} \quad
Solve the boundary-value problem
$$\frac{\partial u}{\partial t} = 
2\,\frac{\partial^2u}{\partial x^2}\,, \qquad
u(0,t)=u(4,t)=0, \quad u(x,0)=25x.$$
It is understood that $0<x<4$ and $t>0$.


 
\bigskip
\noindent{\bf Instructions for Exercises 7 and 8:}
J. B. Fourier was ridiculed by some of the mathematicians of his 
day
when he first announced his discovery that an arbitrary function
on the interval $0<x<\pi$, such as $f(x) = x^2$, can be expanded
in a series of sine functions.
Some of the criticisms were like the two statements which follow.
In each case explain in a short essay how the mathematicians were
confused (and Fourier was right).
 
 
\item ``$x^2$ is an even function;
but any fool can see that a sum of sines will always be an
odd function.''
 
\item
``$x^2$ is not zero at the right endpoint ($\pi$);
but any fool can see that a sum of the functions
$\,\sin nx$ will always vanish at $x=\pi$.
The same criticism applies if we consider the limits of functions
as $x\to \pi$
instead of the values of the functions when $x=\pi$.''


\end{enumerate}
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