%%%%		-*- Mode: LaTeX -*-
%%%% Author:    J. Robert Buchanan
%%%% Address:   Department of Mathematics
%%%%		Millersville University
%%%%            P.O. Box 1002
%%%%            Millersville, PA 17551-0302
%%%% Phone:     717-871-7305
%%%% FAX:	717-871-7948
%%%%
%%%% Email:     Robert.Buchanan@millersville.edu
%%%%
%%%% Date:      02/18/2025
%%%%
%%%% Purpose:   Zeroes of Eigenfunctions
%%%%

\documentclass[10pt]{beamer}

\mode<presentation>
{
  %\usetheme{Warsaw}
  \usetheme{default}
  %\usetheme{CambridgeUS}
  % or ...

  %\setbeamercovered{transparent}
  % or whatever (possibly just delete it)
}


\usepackage[english]{babel}
% or whatever

\usepackage[latin1]{inputenc}
% or whatever

\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.


\title{Zeros of Eigenfunctions}
\subtitle{\textit{Partial Differential Equations}} % (optional)


\author % (optional, use only with lots of authors)
{J~Robert~Buchanan}
% - Use the \inst{?} command only if the authors have different
%   affiliation.

\institute[Millersville University of Pennsylvania]
% (optional, but mostly needed)
{
  Department of Mathematics
}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date{ % (optional)
Fall
%Spring
%Summer
\the\year}

\subject{Talks}
% This is only inserted into the PDF information catalog. Can be left
% out. 



% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:

% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
% \logo{\pgfuseimage{university-logo}}



% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
\AtBeginSubsection[]
{
  \begin{frame}<beamer>
    \frametitle{Outline}
    \tableofcontents[currentsection,currentsubsection]
  \end{frame}
}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command: 

%\beamerdefaultoverlayspecification{<+->}


\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\begin{frame}
%  \frametitle{Outline}
%  \tableofcontents
  % You might wish to add the option [pausesections]
%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%\section{Introduction}

\begin{frame}\frametitle{Objectives}

In this lesson we will learn:
\begin{itemize}
\item about the placement of zeros of eigenfunctions,
\item the relationship between zeros of different eigenfunctions,
\item the relationship between zeros of eigenfunction to different
  regular Sturm-Liouville boundary value problems.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Isolated Zeros}

Consider the Sturm-Liouville BVP:
\begin{align*}
[ p(x) y' ]' + (q(x) + \lambda r(x)) y
	&= 0
\mbox{ for $a < x < b$}	\\
\alpha_{1} y(a) + \beta_{1} y'(a)
	&= 0	\\
\alpha_{2} y(b) + \beta_{2} y'(b)
	&= 0
\end{align*}
with $\alpha_{1}^{2} + \beta_{1}^{2} > 0$ and $\alpha_{2}^{2} +
\beta_{2}^{2} > 0$.

\begin{theorem}
%\label{thm:SL:isolation}
Let $\phi(x) \in \mathcal{C}^{1}[a, b]$ be a nontrivial solution to
the regular Sturm-Liouville BVP.
If $\phi(x_{0}) = 0$ for some $x_{0} \in (a, b)$ then there is an open
interval $(x_{0} - \delta, x_{0} + \delta)$ with $\delta > 0$ which
contains no other zero of $\phi(x)$.
\end{theorem}

\end{frame}
%%

\begin{frame}\frametitle{Example}

While not a proof of the previous theorem, the following example
exhibits the behavior of the eigenfunctions described.

\vspace*{0.10in}

Find the eigenfunctions and their zeros for the following
Sturm-Liouville BVP.
Show that the zeros of the eigenfunctions are isolated,
\begin{align*}
x^{2} y'' + x y' + \lambda y
	&= 0
\mbox{ for $1 < x < 2$}	\\
y(1)
	&= 0	\\
y(2)
	&= 0
\end{align*}

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (1 of 5)}

This is an example of Euler's differential equation.
Make the change of variable $x = e^{z} \iff z = \ln x$.
This produces the boundary value problem:
\begin{align*}
\frac{d^{2} y}{d z^{2}} + \lambda y
	&= 0	\\
y(0)
	&= 0	\\
y(\ln 2)
	&= 0	.
\end{align*}

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (2 of 5)}

\textbf{Case}: $\lambda = -k^{2} < 0$

\vspace*{0.10in}

The general solution is $y(z) = c_{1} e^{k z} + c_{2} e^{-k z}$ where
$c_{1}$ and $c_{2}$ are constants.
The boundary conditions imply,
\begin{align*}
y(0)
	&= c_{1} + c_{2}
	= 0	\\
y(\ln 2)
	&= 2^{k} c_{1} + \frac{1}{2^{k}} c_{2}
	= 0	.
\end{align*}
The only solution is $c_{1} = c_{2} = 0$ and thus there are no
nontrivial solutions (eigenfunctions) when $\lambda < 0$.

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (3 of 5)}

\textbf{Case}: $\lambda = 0$

\vspace*{0.10in}

The general solution is $y(z) = c_{1} z + c_{2}$ where
$c_{1}$ and $c_{2}$ are constants.
The boundary conditions imply,
\begin{align*}
y(0)
	&= c_{2}
	= 0	\\
y(\ln 2)
	&= (\ln 2) c_{1} + c_{2}
	= 0	.
\end{align*}
The only solution is $c_{1} = c_{2} = 0$ and thus there are no
nontrivial solutions (eigenfunctions) when $\lambda = 0$.

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (4 of 5)}

\textbf{Case}: $\lambda = k^{2} > 0$

\vspace*{0.10in}

The general solution is $y(z) = c_{1} \cos (k z) + c_{2} \sin (k z)$ where
$c_{1}$ and $c_{2}$ are constants.
The boundary conditions imply,
\begin{align*}
y(0)
	&= c_{1}
	= 0	\\
y(\ln 2)
	&= c_{1} \cos (k \ln 2) + c_{2} \sin (k \ln 2)
	= 0	.
\end{align*}
Certainly $c_{1} = 0$.
If $c_{2} = 0$ then once again there are no nontrivial solutions.
However if
\[
k \ln 2
	= n \pi
\iff
k
	= \frac{n \pi}{\ln 2}
\]
for $n \in \mathbb{N}$ then
\[
y(z) = c_{2} \sin \left( \frac{n \pi z}{\ln 2} \right)
\implies
y(x) = c_{2} \sin \left( \frac{n \pi \ln x}{\ln 2} \right)
\]
is a nontrivial solution (eigenfunction) for any $c_{2} \ne 0$.
The corresponding eigenvalues are
\[
\lambda
	= \lambda_{n}
	= \left( \frac{n \pi}{\ln 2} \right)^{2}	.
\]

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (5 of 5)}

The zeros of the eigenvalues are isolated from one another in the
interval $[1, 2]$.
For instance if $n = 10$ the graph of $y_{10}(x)$ resembles the following.
\begin{center}
\resizebox{0.90\textwidth}{!}{\includegraphics{isolated}}
\end{center}

\end{frame}
%%

\begin{frame}\frametitle{Separation of Zeros}

\begin{theorem}[Sturm Separation Theorem]
%\label{thm:SL:separation}
Suppose $\phi(x)$ and $\psi(x)$ are linearly independent solutions to
the regular Sturm-Liouville BVP,
then $\psi(x)$ has a zero strictly between any two
zeros of $\phi(x)$.
\end{theorem}

\end{frame}
%%

\begin{frame}\frametitle{Example}

Find two linearly independent solutions $y_{1}$ and $y_{2}$ to the following
Sturm-Liouville BVP and show that between any two zeros of $y_{1}$
there is a zero of $y_{2}$.
\begin{align*}
x^{2} y'' + x y' + \lambda y
	&= 0
\mbox{ for $1 < x < 2$}	\\
y(1)
	&= 0	\\
y(2)
	&= 0
\end{align*}

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (1 of 2)}

In the previous example we found that $\displaystyle y(x) = \sin
\left( \frac{n \pi \ln x}{\ln 2} \right)$ is a nontrivial solution
to the BVP.
Let
\begin{align*}
y_{1}(x)
	&= \sin \left( \frac{\pi \ln x}{\ln 2} \right)	\\
y_{2}(x)
	&= \sin \left( \frac{2 \pi \ln x}{\ln 2} \right)	.
\end{align*}
Note that $y_{1}(x) = 0$ only for $x = 1$ and $x = 2$.

\vspace*{0.10in}

The eigenfunction $y_{2}(x) = 0$ for $1 < x = \sqrt{2} < 2$.

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (2 of 2)}

\begin{center}
\resizebox{1.00\textwidth}{!}{\includegraphics{between}}
\end{center}

\end{frame}
%%

\begin{frame}\frametitle{Interlacing of Zeros}

\begin{theorem}[Sturm Comparison Theorem]
%\label{thm:SL:comparison}
Suppose $\phi(x)$ and $\psi(x)$ are nontrivial solutions to the
ordinary differential equations:
\begin{align*}
\left[ p_{1}(x) \phi'(x) \right]'(x) + (q_{1}(x) + \lambda_{1} r_{1}(x)) \phi(x)
	&= 0	\\
\left[ p_{2}(x) \psi'(x) \right]'(x) + (q_{2}(x) + \lambda_{2} r_{2}(x)) \psi(x)
	&= 0
\end{align*}
on interval $(a, b)$.
Suppose further that $0 < p_{2}(x) \le p_{1}(x)$ and
$q_{1}(x) + \lambda_{1} r_{1}(x) \le q_{2}(x) + \lambda_{2} r_{2}(x)$
for all $x \in [a, b]$.
% with equality not holding on any subinterval of $[a, b]$.
If $\phi(x)$ has zeros at $x = \alpha$ and $x = \beta$
with $\alpha < \beta$,
then $\psi(x)$ must have a zero in $[\alpha, \beta]$.
\end{theorem}

\end{frame}
%%

\begin{frame}\frametitle{Example}

Consider the two boundary value problems:
\begin{columns}
\begin{column}{0.50\textwidth}
\[
y'' + (1 + \lambda_{1}) y
	= 0
\]
\end{column}
%
\begin{column}{0.50\textwidth}
\[
y'' + (2 + \lambda_{2}) y
	= 0
\]
\end{column}
\end{columns}
for $0 < x < 1$ subject to the boundary conditions:
\begin{align*}
y(0)
	&= 0	\\
y(1)
	&= 0	.
\end{align*}
Find the eigenfunctions for each boundary value problem and show that
their zeros satisfy the Sturm Comparison Theorem.

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (1 of 2)}

We can check that $\phi(x) = \sin (m \pi x)$ solves the first BVP when
$1 + \lambda_{1} = m^{2} \pi^{2}$ while
$\psi(x) = \sin (n \pi x)$ when $1 + \lambda_{2} = n^{2} \pi^{2}$
solves the second.

\vspace*{0.10in}

Note that the leading coefficient in each BPV is $p_{1}(x) = p_{2}(x)
= 1$.
If $1 + \lambda_{2} > 1 + \lambda_{1}$ or equivalently if $n > m$ then
between two consecutive zeros of $\phi(x)$ there must be a zero of
$\psi(x)$.

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution (2 of 2)}

\begin{center}
\resizebox{1.00\textwidth}{!}{\includegraphics{comparison}}
\end{center}

\end{frame}
%%

%\begin{frame}\frametitle{Homework}

%\begin{itemize}
%\item Read Section $x.y$
%\item Exercises: 
%\end{itemize}

%\end{frame}
%%

%%%
\end{document}

%%%%
%%%% EOF
%%%%
