%%%%		-*- 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:      02/10/2025
%%%%
%%%% Purpose:   Derivations of Common PDEs
%%%%

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\title{Derivations of Laplace's Equation}
\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}
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% out. 



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% the beginning of each subsection:
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{
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\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 examine a time-independent partial differential
equation known as Laplace's equation.
It often arises as
\begin{itemize}
\item a steady-state solution to the heat or wave equation, and
\item in as a potential for a vector field.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Steady-State Solutions}
\fontsize{8pt}{9.6}\selectfont

Consider the initial, boundary value problems for the heat and wave equations.
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{center} Heat Equation \end{center}
\begin{align*}
u_{t}
	&= \kappa u_{xx} + Q(x, t)
\mbox{ for $0 < x < L$, $t > 0$}	\\
K_{0} u_{x}(0, t)
	&= \alpha (u(0, t) - g_{1}(t))
\mbox{ for $t > 0$}	\\
K_{0} u_{x}(L, t)
	&= -\beta (u(L, t) - g_{2}(t))
\mbox{ for $t > 0$}	\\
u(x, 0)
	&= f(x)
\mbox{ for $0 < x < L$}
\end{align*}
\end{column}
%
\begin{column}{0.5\textwidth}
\begin{center} Wave Equation \end{center}
\begin{align*}
u_{tt}
	&= c^{2} u_{xx} + F(x, t)
\mbox{ for $0 < x < L$, $t > 0$}	\\
T_{0} u_{x}(0, t)
	&= k_{1} (u(0, t) - d_{1}(t))
\mbox{ for $t > 0$}	\\
T_{0} u_{x}(L, t)
	&= -k_{2} (u(L, t) - d_{2}(t))
\mbox{ for $t > 0$}	\\
u(x, 0)
	&= f(x)
\mbox{ for $0 < x < L$}	\\
u_{t}(x, 0)
	&= g(x)
\mbox{ for $0 < x < L$}
\end{align*}
\end{column}
\end{columns}

A $t$-independent solution to the PDE and boundary conditions is known
as a steady-state solution.

\end{frame}
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\begin{frame}\frametitle{Steady-State Solutions}
\fontsize{9pt}{10.8}\selectfont

For either the heat or wave equation if $u(x, t) = U(x)$, then we have:

\begin{columns}
\begin{column}{0.5\textwidth}
\begin{center} Heat Equation \end{center}
\begin{align*}
0
	&= \kappa U_{xx} + Q(x, t)
\mbox{ for $0 < x < L$}	\\
K_{0} U_{x}(0)
	&= \alpha U(0)	\\
K_{0} U_{x}(L)
	&= -\beta U(L)
\end{align*}
\end{column}
%
\begin{column}{0.5\textwidth}
\begin{center} Wave Equation \end{center}
\begin{align*}
0
	&= c^{2} U_{xx} + F(x)
\mbox{ for $0 < x < L$}	\\
T_{0} U_{x}(0)
	&= k_{1} U(0)	\\
T_{0} U_{x}(L)
	&= -k_{2} U(L)
\end{align*}
\end{column}
\end{columns}

In either case the differential equation can be written in the general
form as
\[
U''(x) + f(x)
	= 0
\mbox{ for $0 < x < L$.}
\]
\begin{itemize}
\item If $f(x) \equiv 0$ this is the one-dimensional Laplace's equation.
\item If $f(x) \not\equiv 0$ this is the one-dimensional Poisson's equation.
\end{itemize}

\end{frame}
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\begin{frame}\frametitle{Higher Dimensional Versions}

In two dimensions,
\[
\Delta u
	= u_{xx} + u_{yy}
	= f(x, y)
\mbox{ for $(x, y) \in \Omega \subset \mathbb{R}^{2}$.}
\]
In three dimensions,
\[
\Delta u
	= u_{xx} + u_{yy} + u_{zz}
	= f(x, y, z)
\mbox{ for $(x, y, z) \in \Omega \subset \mathbb{R}^{3}$.}
\]

\uncover<2->{%
If the right-hand side of the equation is $0$ on $\Omega$ this is
Laplace's equation, otherwise it is Poisson's equation.
}

\end{frame}
%%

\begin{frame}\frametitle{Boundary Conditions}

Suppose
\[
\Delta u
	= u_{xx} + u_{yy}
	= f(x, y)
\mbox{ for $(x, y) \in \Omega \subset \mathbb{R}^{2}$,}
\]
where $\Omega$ is an open, connected set.

\vspace*{0.10in}

Dirichlet boundary conditions can be expressed as
\[
u(x, y)
	= \phi(x, y)
\mbox{ for $(x, y) \in \partial \Omega$}
\]
where $\partial \Omega$ denotes the boundary of curve of $\Omega$.

\vspace*{0.10in}

Neumann boundary conditions can be expressed as
\[
\nabla u(x, y) \cdot \mathbf{n}
	= \psi(x, y)
\mbox{ for $(x, y) \in \partial \Omega$}
\]
where $\mathbf{n}$ is the unit outward normal vector to $\partial \Omega$.

\end{frame}
%%

\begin{frame}\frametitle{Boundary Conditions}

Suppose
\[
\Delta u
	= u_{xx} + u_{yy} + u_{zz}
	= f(x, y, z)
\mbox{ for $(x, y, z) \in \Omega \subset \mathbb{R}^{3}$,}
\]
where $\Omega$ is an open, connected set.

\vspace*{0.10in}

Dirichlet boundary conditions can be expressed as
\[
u(x, y, z)
	= \phi(x, y, z)
\mbox{ for $(x, y, z) \in \partial \Omega$}
\]
where $\partial \Omega$ denotes the boundary of surface of $\Omega$.

\vspace*{0.10in}

Neumann boundary conditions can be expressed as
\[
\nabla u(x, y, z) \cdot \mathbf{n}
	= \psi(x, y, z)
\mbox{ for $(x, y, z) \in \partial \Omega$}
\]
where $\mathbf{n}$ is the unit outward normal vector to $\partial \Omega$.

\end{frame}
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%\begin{frame}\frametitle{Homework}

%\begin{itemize}
%\item Read Section $1.5$
%\item Exercises: 7--11
%\end{itemize}

%\end{frame}
%%

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%%%%
%%%% EOF
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