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\centerline{{\sc Problem set 3}}
\centerline{Math 207B, Winter 2016}
\centerline{Due: Fri, Feb. 5}


\bigskip\noindent
{\bf 1.} Suppose that $u(x)$ is a non-zero solution of the eigenvalue problem
\begin{align*}
-u'' = \lambda u\qquad 0<x<1,
\\
u(0) = 0,\qquad u(1) = 0.
\end{align*}
Show that
\[
\lambda = \frac{\int_0^1 (u')^2\, dx}{\int_0^1 u^2\, dx}.
\]
Deduce that every eigenvalue $\lambda$ is strictly positive.


\bigskip\noindent
{\bf 2.} Heat flows in a rod of length $L$ with a heat source ($a>0$) or sink ($a<0$) whose density $au$ is proportional to the temperature $u$. Suppose that
$u(x,t)$ satisfies the IBVP
\begin{align*}
&u_t = Du_{xx} + a u\qquad 0<x<L, \quad t>0,
\\
&u(0,t) = 0,\qquad u(L,t) = 0,
\\
&u(x,0) = f(x).
\end{align*}
(a) Nondimensionalize the problem, and show that the IBVP can be written in nondimensional form as
\begin{align*}
&u_t = u_{xx} + \alpha u\qquad 0<x<1, \quad t>0,
\\
&u(0,t) = 0,\qquad u(1,t) = 0\qquad t>0,
\\
&u(x,0) = f(x)\qquad 0<x<1,
\end{align*}
where $\alpha$ is a suitable nondimensional parameter. Give a physical interpretation of $\alpha$.

\smallskip\noindent
(b) Solve the IBVP in (a) by the method of separation of variables.

\smallskip\noindent
(c) How does your solution behave as $t\to\infty$? For what values of $\alpha$ does $u(x,t) \to 0$ as $t\to\infty$? What happens for larger values of $\alpha$?
Give a physical explanation of this behavior in terms of the thermal energy.



\newpage\noindent
{\bf 3.} Solve the following eigenvalue problem for the linear operator $- d^2/dx^2$ with Neumann BCs:
\begin{align*}
-u'' = \lambda u\qquad 0<x<1,
\\
u'(0) = 0,\qquad u'(1) = 0.
\end{align*}
(a) Find the eigenvalues $\lambda=\lambda_n$, where $n=0,1,2,3,\dots$, and the corresponding eigenfunctions $u_n(x)$.

\smallskip\noindent
(b) Show that the eigenfunctions can be normalized so that
\[
\int_0^1 u_m(x) u_n(x)\, dx = \begin{cases} 1 &\text{if $m=n$}
\\
0 &\text{if $m\ne n$}
\end{cases}
\]
(c) Does your argument in Problem 1 that $\lambda\ne 0$ work in this case?

\bigskip\noindent
{\bf 4.} (a) Solve the following IBVP by the method of separation of variables
\begin{align*}
&u_t = u_{xx}\qquad\qquad 0<x<1, \quad t>0,
\\
&u_x(0,t) = 0,\qquad u_x(1,t) = 0\qquad  t>0,
\\
&u(x,0) = f(x)\qquad 0<x<1.
\end{align*}
(b) How does your solution behave as $t\to \infty$?

\smallskip\noindent
(c) Show directly from the IBVP in (a) that
\[
\int_0^1 u(x,t)\, dx = \int_0^1 f(x)\, dx\qquad \text{for all $t \ge 0$}.
\]
Is this result consistent with your answer in (b)? Give a physical explanation of the long-time behavior of $u(x,t)$.

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