%%%%		-*- 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:      02/14/2025
%%%%
%%%% Purpose:   Introduction to Laplace's Equation
%%%%

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\title{Introduction to Laplace's Equation}
\subtitle{\textit{Partial Differential Equations}} % (optional)


\author % (optional, use only with lots of authors)
{J~Robert~Buchanan}
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%   affiliation.

\institute[Millersville University of Pennsylvania]
% (optional, but mostly needed)
{
  Department of Mathematics
}
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% - Keep it simple, no one is interested in your street address.

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

\subject{Talks}
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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
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% of between 15min and 45min and the theme suggest that you stick to
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% - 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 about
\begin{itemize}
\item the partial differential equations and boundary value problems
  known as Laplace's and Poisson's equations,
\item techniques for solving Laplace's and Poisson's equations, and
\item applications of Laplace's and Poisson's equations.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{The Laplacian Operator}

Let $\mathbf{x} = (x_{1}, x_{2}, \ldots, x_{n}) \in \mathbb{R}^{n}$
and let function $u \, : \, \Omega \subset \mathbb{R}^{n} \to
\mathbb{R}$, then the \textbf{Laplacian} of $u$ is
\[
\Delta u
	= \nabla^{2} u
	= \frac{\partial^{2} u}{\partial x_{1}^{2}} +
\frac{\partial^{2} u}{\partial x_{2}^{2}} +
\cdots + \frac{\partial^{2} u}{\partial x_{n}^{2}}	.
\]
If $n = 2$ we will often write $\mathbf{x} = (x, y)$ and
\[
\Delta u
	= \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}	.
\]
If $n = 3$ we will often write $\mathbf{x} = (x, y, z)$ and
\[
\Delta u
	= \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2}
  u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}}	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Open Sets}

A set $\Omega \subset \mathbb{R}^{n}$ is \textbf{open} if for every
$\mathbf{x} \in \Omega$ there exists $\delta > 0$ such that
\[
\{ \mathbf{y} \, | \, \| \mathbf{x}-\mathbf{y} \| < \delta \} \subset \Omega	.
\]
\begin{center}
\resizebox{0.60\textwidth}{!}{\includegraphics{openset}}
\end{center}

\end{frame}
%%

\begin{frame}\frametitle{Connected Sets}

A set $\Omega \subset \mathbb{R}^{n}$ is \textbf{connected} if for every
$\mathbf{x}, \mathbf{y} \in \Omega$ there exists a path connecting
$\mathbf{x}$ and $\mathbf{y}$ which lies completely within $\Omega$.
\begin{center}
\resizebox{0.60\textwidth}{!}{\includegraphics{connectedset}}
\end{center}

\end{frame}
%%

\begin{frame}\frametitle{Laplace's Equation}

If $u \, : \, \Omega \subset \mathbb{R}^{n} \to \mathbb{R}$ where
$\Omega$ is an open, connected set then $u$ satisfies
\textbf{Laplace's equation} if
\[
\Delta u
	= 0	.
\]
Function $u$ is said to be a \textbf{harmonic function}.

\vspace*{0.10in}

If function $f$ is defined on $\Omega$ then $u$ satisfies
\textbf{Poisson's equation} if
\[
\Delta u
	= f	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Example}

Let $u(x, y) = e^{x} \cos y$ and show $u$ is harmonic on $\Omega =
\mathbb{R}^{2}$.

\vspace*{0.10in}

\uncover<2->{%
\[
\Delta u
	= e^{x} \cos y + (e^{x} (-\cos y)) = 0
\]
for all $(x, y) \in \Omega$.
}

\end{frame}
%%

\begin{frame}\frametitle{Boundary Conditions}

\begin{itemize}
\item
If $\Omega \subset \mathbb{R}^{n}$ is an open set then the set of
boundary points of $\Omega$ will be denoted $\partial\Omega$.
\item
Laplace's and Poisson's equations must be supplemented with boundary
conditions.
\end{itemize}
\uncover<2->{%
\begin{description}
\item[Dirichlet BCs:] $\displaystyle u(\mathbf{x}) = \phi(\mathbf{x})$
  for $\mathbf{x} \in \partial\Omega$.
\item[Neumann BCs:] $\displaystyle \frac{\partial u}{\partial
    \mathbf{n}} (\mathbf{x}) = \phi(\mathbf{x})$
  for $\mathbf{x} \in \partial\Omega$ where $\mathbf{n}$ is the unit
  outward normal vector to $\partial\Omega$.
\item[Mixed BCs:] $\displaystyle u(\mathbf{x}) + \alpha \frac{\partial u}{\partial
    \mathbf{n}} (\mathbf{x}) = \phi(\mathbf{x})$
  for $\mathbf{x} \in \partial\Omega$ where $\alpha$ is constant and
  $\mathbf{n}$ is the unit outward normal vector to $\partial\Omega$.
\end{description}
}

\end{frame}
%%

\begin{frame}\frametitle{Dirichlet Problem on a Rectangle}

Define
\[
\Omega
	= \{ (x, y) \, | \, 0 < x < a, \, 0 < y < b \} = (0,a) \times (0,b)
\]
where $a$ and $b$ are constants.
In this case
\[
\partial\Omega
	= \{ (x, y) \, | \, x=0, \,a \mbox{ and } 0
\le y \le b, \, \mbox{ or } y=0, \, b \mbox{ and } 0 \le x
\le a \}	.
\]
Consider the following example of Laplace's equation.
\begin{align*}
\Delta u
	&= 0
\mbox{ for $(x, y) \in \Omega$}	\\
u(0, y)
	&= 0	\\
u(a, y)
	&= f(y)	\\
u(x, 0)
	&= 0	\\
u(x, b)
	&= 0
\end{align*}
This problem can be solved using separation of variables and Fourier series.

\end{frame}
%%

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

Assume $u(x, y) = X(x) Y(y)$, then
\begin{align*}
X''(x) Y(y) + X(x) Y''(y)
	&= 0	\\
\frac{X''(x) Y(y)}{X(x) Y(y)} + \frac{X(x) Y''(y)}{X(x) Y(y)}
	&= 0	\\
-\frac{X''(x)}{X(x)}
	&= \frac{Y''(y)}{Y(y)}
	= c
\end{align*}
where $c$ is a constant.

\vspace*{0.10in}

\uncover<2->{%
The boundary conditions become
\begin{align*}
u(0, y)
	&= X(0) Y(y) = 0
\implies X(0) = 0	\\
u(x, 0)
	&= X(x) Y(0) = 0
\implies Y(0) = 0	\\
u(x, b)
	&= X(x) Y(b) = 0
\implies Y(b) = 0	.
\end{align*}
}

\end{frame}
%%

\begin{frame}\frametitle{Boundary Value Problem for $Y(y)$}

Separation of variables and the Dirichlet boundary conditions imply
the following BVP for $Y(y)$:
\begin{align*}
Y(y) - c \, Y(y)
	&= 0
\mbox{ for $0 < y < b$}	\\
Y(0)
	&= 0	\\
Y(b)
	&= 0	.
\end{align*}
Previously we have seen the only nontrivial solutions occur when
\[
c
	= -\lambda_{n}^{2}
	= -\frac{n^{2} \pi^{2}}{b^{2}}
\]
and $\displaystyle Y_{n}(y) = \sin \frac{n \pi y}{b}$ for $n \in \mathbb{N}$.

\end{frame}
%%

\begin{frame}\frametitle{Boundary Value Problem for $X(x)$}

Function $X(x)$ must satisfy the following boundary value problem.
\begin{align*}
X''(x) + c \, X(x)
	&= X''(x) - \frac{n^{2} \pi^{2}}{b^{2}} X(x) = 0  \\
X(0)
	&= 0
\end{align*}
The general solution takes the form
\[
X(x)
	= A \cosh \frac{n \pi x}{b} + B \sinh \frac{n \pi x}{b}
\]
and $A = 0$ in order to satisfy the boundary condition.
Thus
\[
X_{n}(x)
	= \sinh \frac{n \pi x}{b}	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Product Solution}

\begin{itemize}
\item
Define the function
\[
u_{n}(x, y)
	= \sinh \frac{n \pi x}{b} \sin \frac{n \pi y}{b}
\]
for $n \in \mathbb{N}$.
\item
Each $u_{n}(x, y)$ satisfies Laplace's equation on the rectangle and
the boundary conditions for $(x, y) = (x, 0)$ and $(x, y) =  (x, b)$ for
$0 \le x \le a$ and for $(x, y) = (0, y)$ for $0 \le y \le b$.
\item
By the Principle of Superposition
\[
u(x, y)
	= \sum_{n=1}^{N} c_{n} \sinh \frac{n \pi x}{b} \sin \frac{n \pi y}{b}
\]
will also satisfy these equations for any $N \ge 1$ and constants $c_{n}$.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Boundary $(a, y)$ for $0 \le y \le b$}

\begin{itemize}
\item Extend $f(y)$ as an odd, $2b$-periodic function.
\item For $0 \le y \le b$
\begin{align*}
f(y)
	&= u(a, y)	\\
	&= \sum_{n=1}^{\infty} c_{n} \sinh \frac{n \pi a}{b} \sin \frac{n \pi y}{b}	.
\end{align*}
\item<2-> Multiply both sides of the equation by $\displaystyle \sin
  \frac{m \pi y}{b}$ and integrate over $[0,b]$.
\begin{align*}
\int_{0}^{b} f(y) \sin \frac{m \pi y}{b} \, dy
	&= \sum_{n=1}^{\infty} c_{n} \sinh \frac{n \pi a}{b} \int_{0}^{b} \sin
              \frac{n \pi y}{b} \sin \frac{m \pi y}{b} \, dy	 \\
	&= \frac{b}{2} \, c_{m} \sinh \frac{m \pi a}{b}	\\
c_{m}
	&= \frac{2}{b \sinh \frac{m \pi a}{b}} \int_{0}^{b} f(y) \sin
            \frac{m \pi y}{b} \, dy
\end{align*}
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{General Solution}

The solution to
\begin{align*}
\Delta u
	&= 0
\mbox{ for $(x, y) \in \Omega$}	\\
u(0, y)
	&= 0	\\
u(a, y)
	&= f(y)	\\
u(x, 0)
	&= 0	\\
u(x, b)
	&= 0
\end{align*}
is therefore
\[
u(x, y)
	= \sum_{n=1}^{\infty} \left( \frac{2}{b \sinh \frac{n \pi a}{b}}
\int_{0}^{b} f(y) \sin \frac{n \pi y}{b} \, dy
\right) \sinh \frac{n \pi x}{b} \sin \frac{n \pi y}{b}	.
\]

\end{frame}
%%

\begin{frame}\frametitle{General Dirichlet Problem on a Rectangle}

The general boundary value problem:
\begin{align*}
\Delta u
	&= 0
\mbox{ for $(x, y) \in \Omega$}	\\
u(0, y)
	&= g_{1}(y)	\\
u(a, y)
	&= g_{2}(y)	\\
u(x, 0)
	&= f_{1}(x)	\\
u(x, b)
	&= f_{2}(x)
\end{align*}
can be decomposed into four simpler boundary value problems.

\end{frame}
%%

\begin{frame}\frametitle{Decomposition}

For $(x, y) \in \Omega$,
\begin{columns}
\begin{column}{0.25\textwidth}
\begin{align*}
\Delta u_{1}
	&= 0	\\
u_{1}(0, y)
	&= g_{1}(y)	\\
u_{1}(a, y)
	&= 0	\\
u_{1}(x, 0)
	&= 0	\\
u_{1}(x, b)
	&= 0
\end{align*}
\end{column}
%
\begin{column}{0.25\textwidth}
\begin{align*}
\Delta u_{2}
	&= 0	\\
u_{2}(0, y)
	&= 0	\\
u_{2}(a, y)
	&= g_{2}(y)	\\
u_{2}(x, 0)
	&= 0	\\
u_{2}(x, b)
	&= 0
\end{align*}
\end{column}
%
\begin{column}{0.25\textwidth}
\begin{align*}
\Delta u_{3}
	&= 0	\\
u_{3}(0, y)
	&= 0	\\
u_{3}(a, y)
	&= 0	\\
u_{3}(x, 0)
	&= f_{1}(x)	\\
u_{3}(x, b)
	&= 0
\end{align*}
\end{column}
%
\begin{column}{0.25\textwidth}
\begin{align*}
\Delta u_{4}
	&= 0	\\
u_{4}(0, y)
	&= 0	\\
u_{4}(a, y)
	&= 0	\\
u_{4}(x, 0)
	&= 0	\\
u_{4}(x, b)
	&= f_{2}(x)
\end{align*}
\end{column}
%
\end{columns}
If $u_{i}(x, y)$ solves the $i$th boundary value problem, then
\[
u(x, y)
	= u_{1}(x, y) + u_{2}(x, y) + u_{3}(x, y) + u_{4}(x, y)
\]
solves the general Dirichlet boundary value problem on $\Omega$.

\end{frame}
%%

\begin{frame}\frametitle{Solutions}

\begin{align*}
u_{1}(x, y)
	&= \sum_{n=1}^{\infty} \left(
\frac{2}{b \sinh \frac{n \pi a}{b}}
\int_{0}^{b} g_{1}(y) \sin \frac{n \pi y}{b} \, dy
\right) \sinh \frac{n \pi (a-x)}{b} \sin \frac{n \pi y}{b}
	\\
u_{2}(x, y)
	&= \sum_{n=1}^{\infty} \left(
\frac{2}{b \sinh \frac{n \pi a}{b}}
\int_{0}^{b} g_{2}(y) \sin \frac{n \pi y}{b} \, dy
\right) \sinh \frac{n \pi x}{b} \sin \frac{n \pi y}{b}
	\\
u_{3}(x, y)
	&= \sum_{n=1}^{\infty} \left(
\frac{2}{a \sinh \frac{n \pi b}{a}}
\int_{0}^{a} f_{1}(x) \sin \frac{n \pi x}{a} \, dx
\right) \sinh \frac{n \pi (b-y)}{a} \sin \frac{n \pi x}{a}
	\\
u_{4}(x, y)
	&= \sum_{n=1}^{\infty} \left(
\frac{2}{a \sinh \frac{n \pi b}{a}}
\int_{0}^{a} f_{2}(x) \sin \frac{n \pi x}{a} \, dx
\right) \sinh \frac{n \pi y}{a} \sin \frac{n \pi x}{a}
\end{align*}

\end{frame}
%%

\begin{frame}\frametitle{Example}

Let $\Omega = (0,1) \times (0,1)$ and solve the Dirichlet boundary
value problem for Laplace's equation on $\Omega$.
\begin{align*}
\Delta u
	&= 0
\mbox{ for $(x, y) \in \Omega$}	\\
u(0, y)
	&= 0	\\
u(1,y)
	&= 1 - y	\\
u(x, 0)
	&= x	\\
u(x,1)
	&= 0	\\
\end{align*}

\end{frame}
%%

\begin{frame}\frametitle{Solution}

Let $a = b = 1$, then using the notation prescribed earlier the
solution can be expressed as
\[
u(x, y)
	= u_{2}(x, y) + u_{3}(x, y)
\]
where
\begin{align*}
u_{2}(x, y)
	&= \sum_{n=1}^{\infty} \left(
\frac{2}{\sinh (n \pi)}
\int_{0}^{1} (1-y) \sin (n \pi y) \, dy
\right) \sinh (n \pi x) \sin (n \pi y)
	\\
	&= \sum_{n=1}^{\infty} \frac{2}{n \pi \sinh (n \pi)}
\sinh (n \pi x) \sin (n \pi y)
	\\
u_{3}(x, y)
	&= \sum_{n=1}^{\infty} \left(
\frac{2}{\sinh (n \pi)}
\int_{0}^{1} x \sin (n \pi x) \, dx
\right) \sinh (n \pi (1-y)) \sin (n \pi x)	\\
	&= -\sum_{n=1}^{\infty}
\frac{2 (-1)^{n} }{n \pi \sinh (n \pi)}
\sinh (n \pi (1-y)) \sin (n \pi x)
\end{align*}

\end{frame}
%%

\begin{frame}\frametitle{Illustration}
\fontsize{8pt}{9.6}\selectfont

\[
u(x, y)
	= \sum_{n=1}^{\infty} \frac{2}{n \pi \sinh (n \pi)}
\sinh (n \pi x) \sin (n \pi y) -
\sum_{n=1}^{\infty} \frac{2 (-1)^{n} }{n \pi \sinh (n \pi)}
\sinh (n \pi (1-y)) \sin (n \pi x)
\]
\begin{center}
\resizebox{0.60\textwidth}{!}{\includegraphics{example01}}
\end{center}

\end{frame}
%%

%\begin{frame}\frametitle{Homework}

%\begin{itemize}
%\item Read Section $6.1$--$6.2$
%\item Exercises: 1--7
%\end{itemize}

%\end{frame}
%%

%%%
\end{document}

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