\documentclass[12pt]{article}
\pagestyle{empty}
\usepackage{amsfonts,amsmath}

\def\Rl{\mathbb{R}}
\def\Cx{\mathbb{C}}

\begin{document}

\centerline{{\sc Problem set 4}}
\centerline{Math 207B, Winter 2016}
\centerline{Due: Fri, Feb. 12}


\bigskip\noindent
{\bf 1.} The following nonhomogeneous IBVP describes heat flow in a rod whose ends are held at temperatures $u_0$, $u_1$:
\begin{align*}
&u_t = u_{xx}\qquad 0<x<1, \quad t>0
\\
&u(0,t) = u_0,\qquad u(1,t) = u_1
\\
&u(x,0) = f(x)
\end{align*}
(a) Find the steady state temperature $U(x)$ that satisfies
\begin{align*}
&U_{xx} = 0\qquad 0<x<1
\\
&U(0) = u_0,\qquad U(1) = u_1
\end{align*}
(b) Write $u(x,t) = U(x) + v(x,t)$ and find the corresponding IBVP for $v$. Use separation of variables to solve for $v$ and hence $u$.

\smallskip\noindent
(c) How does $u(x,t)$ behave as $t\to\infty$?


\bigskip\noindent
{\bf 2.} Define a first-order differential operator with complex coefficients acting in $L^2(0,2\pi)$ by
\[
A = - i\frac{d}{dx}.
\]
(a) Show that $A$ is formally self-adjoint.

\smallskip\noindent
(b) Show that $A$ with periodic boundary conditions $u(0) = u(2\pi)$ is self-adjoint, and find the eigenvalues and eigenfunctions of the corresponding eigenvalue problem
\[
-iu'=\lambda u,\qquad u(0) = u(2\pi).
\]

\smallskip\noindent
(c) What are the adjoint boundary conditions to the Dirichlet condition $u(0)=0$ at $x=0$? Is $A$ with this Dirichlet boundary condition self-adjoint? Find all eigenvalues and eigenfunctions of the corresponding eigenvalue problem
\[
-iu'=\lambda u,\qquad u(0) = 0.
\]
How does your result compare with the properties of finite-dimensional eigenvalue problems for matrices?


\newpage\noindent
{\bf 3.} Let $A$ be a regular Sturm-Liouville operator, given by
\[
Au = -\left(pu'\right)' + q u,
\]
acting in $L^2(a,b)$. Verify that $A$ with the Robin boundary conditions
\[
\alpha u'(a) + u(a) = 0,\qquad u'(b) + \beta u(b) = 0
\]
is self-adjoint.

\bigskip\noindent
{\bf 4.} Show that the eigenvalues of the Sturm-Liouville problem
\begin{align*}
&-u'' = \lambda u\qquad 0<x<1
\\
&u(0) = 0, \qquad u'(1) + \beta u(1) = 0
\end{align*}
are given by $\lambda = k^2$ where $k > 0$ satisfies the equation
\[
\beta \tan k + k = 0.
\]
Show graphically that there is a infinite sequence of simple eigenvalues $\lambda_1 < \lambda_2 < \dots < \lambda_n < \dots$
with $\lambda_n\to\infty$ as $n\to \infty$. What is the asymptotic behavior of $\lambda_n$ as $n\to\infty$?


\bigskip\noindent
{\bf 5.}  The following IBVP describes heat flow in a rod whose left end is held at temperature $0$ and whose right end loses heat to the surroundings according to Newton's law of cooling (heat flux is proportional to the temperature difference):
\begin{align*}
&u_t = u_{xx}\qquad 0<x<1, \quad t>0
\\
&u(0,t) = 0,\qquad u_x(1,t) = -\beta u(1,t)
\\
&u(x,0) = f(x)
\end{align*}
Solve this IBVP by the method of separation of variables.

\end{document}             % End of document.
