%%%%		-*- 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:      03/28/2025
%%%%
%%%% Purpose:   Green’s Functions in Higher Dimensions
%%%%

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\title{Green's Functions in Higher Dimensions}
\subtitle{\textit{Partial Differential Equations}} % (optional)


\author % (optional, use only with lots of authors)
{J~Robert~Buchanan}
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\institute[Millersville University of Pennsylvania]
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{
  Department of Mathematics
}
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% - Keep it simple, no one is interested in your street address.

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

\subject{Talks}
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\begin{frame}
  \titlepage
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%\begin{frame}
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%  \tableofcontents
  % You might wish to add the option [pausesections]
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% Since this a solution template for a generic talk, very little can
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%   15 and 30 frames, all told.

%\section{Introduction}

\begin{frame}\frametitle{Objectives}

In this lesson we will:
\begin{itemize}
\item introduce the second-order linear differential operator in two
  spatial dimensions,
\item introduce a second-order linear differential operator in three
  spatial dimensions,
\item present Green's identities,
\item introduce the notion of the principal solution.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Second-Order Linear BVP}

\begin{align*}
%\label{eq:green:higher:2ndOrder}
L[u]
	&= A u_{xx} + B u_{xy} + C u_{yy} + D u_{x} + E u_{y} + F u
	= \Gamma	\\
%\label{eq:green:higher:generalBCs}
B[u]
	&= a \, u + b (\nabla u \cdot \mathbf{N})
	= c
\end{align*}
\begin{itemize}
\item $A$, $B$, $C$, \ldots, $F$, and $\Gamma$ are sufficiently smooth
functions of $x$ and $y$.
\item The PDE is assumed to hold over some region
$R$ in the $xy$-plane.
\item The boundary conditions are defined on $\partial R$ where
  $\mathbf{N}$ is the unit outward normal 
vector to $\partial R$ at coordinates $(x, y)$.
\item Coefficients $a$, $b$, and $c$ are permitted to be functions of
$(x, y)$ along $\partial R$.
If $c = 0$ the boundary conditions are homogeneous.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Flux Form of Green's Theorem}

\begin{theorem}[Flux Form]
%\label{thm:green:higher:fluxform}
Let $\mathbf{F}(x, y) = M(x, y) \mathbf{i} + N(x, y) \mathbf{j}$ be a
vector field where $M$ and $N$ are continuous and have continuous
first partial derivatives on some open region $D \subset \mathbb{R}^{2}$.
Let $R \subset D$ with a piecewise-smooth, simple closed boundary
curve $\partial R$, then
\[
%\label{eq:green:higher:fluxform}
\oint_{\partial R} \mathbf{F} \cdot \mathbf{N} \, ds
	= \iint_{R} \left( \frac{\partial M}{\partial x} +
          \frac{\partial N}{\partial y} \right) \, dA	.
\]
\end{theorem}
The expression $ds$ is the differential arc length along the boundary
curve of region $R$ and $dA$ is the differential area element of
region $R$.

\end{frame}
%%

\begin{frame}\frametitle{Adjoint Operator}

Define the quantities,
\begin{align*}
M
	&= v A u_{x} - (v A)_{x} u + v B u_{y} + v D u	\\
N
	&= -(v B)_{x} u + v C u_{y} - (v C)_{y} u + v E u	,
\end{align*}
then
\[
%\label{eq:green:higher:generalformaladjoint}
\iint_{R} v L[u] \, dA
	= \oint_{\partial R} [ M \mathbf{i} + N \mathbf{j} ] \cdot
        \mathbf{N} \, ds +
\iint_{R} L^{\ast}[v] u \, dA
\]end{equation}
where
\[
L^{\ast}[v]
	= (A v)_{xx} + (B v)_{xy} + (C v)_{yy} - (D v)_{x} - (E v)_{y}
        + F v	.
\]
The Green's function approach will be to solve the partial
differential equation
\[
L^{\ast}[G](x; y)
	= \delta(x-\xi, y-\eta)
\]
where $\delta$ is now the
multivariable version of the Dirac delta function and to force the
Green's function $G$ to satisfy boundary conditions such that the
surface terms vanish.

\end{frame}
%%

\begin{frame}\frametitle{Laplacian Operator}

\[
L[u]
	= u_{xx} + u_{yy}
\]
In this setting $A = C = 1$ and $B = D = E = F = 0$.
Thus $L^{\ast}[v] = L[v]$.

\vspace*{0.10in}

The adjoint equation simplifies to the following.
\[
\iint_{R} v L[u] \, dA
%\label{eq:green:higher:laplacianadj}
	= \oint_{\partial R} (u_{\mathbf{N}} v - u v_{\mathbf{N}}) \, ds +
\iint_{R} L^{\ast}[v] u \, dA
\]
The notations $u_{\mathbf{N}}$ and $v_{\mathbf{N}}$ denote the
directional derivative of $u$ and $v$ in the direction of the outward
normal vector to the boundary of region $R$.

\end{frame}
%%

\begin{frame}\frametitle{Helmholtz Operator}

\[
L[u]
	= u_{xx} + u_{yy} + k^{2} u
\]
In this setting $A = C = 1$ and $B = D = E = 0$ and $F = k^{2}$.
Thus $L^{\ast}[v] = L[v]$.

\vspace*{0.10in}

The adjoint equation simplifies to the following.
\[
\iint_{R} v L[u] \, dA
%\label{eq:green:higher:laplacianadj}
	= \oint_{\partial R} (u_{\mathbf{N}} v - u v_{\mathbf{N}}) \, ds +
\iint_{R} L^{\ast}[v] u \, dA
\]
The notations $u_{\mathbf{N}}$ and $v_{\mathbf{N}}$ denote the
directional derivative of $u$ and $v$ in the direction of the outward
normal vector to the boundary of region $R$.

\end{frame}
%%

\begin{frame}\frametitle{Second-Order Linear BVP}

%\label{eq:green:higher:3DBVP}
\begin{align*}
L[u]
	&= A u_{xx} + B u_{yy} + C u_{zz} + D u_{x} + E u_{y} + F
        u_{z} + G u
	= H	\\
B[u]
	&= a u + b (\nabla u \cdot \mathbf{N})
        = c
\end{align*}
is defined on a region $Q \subset \mathbb{R}^{3}$.
\begin{itemize}
\item The functions $A$ through $H$ depending on $(x, y, z)$ are
sufficiently smooth.
\item Functions $a$, $b$, and $c$ present in the boundary conditions also
depend on $(x, y, z)$.
\item Vector $\mathbf{N}$ is the unit outward normal vector to the
boundary surface of region $Q$.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Divergence Theorem}

\begin{theorem}[Divergence]
%\label{thm:green:higher:divergencethm}
Let $Q$ be a closed, bounded subset of $\mathbb{R}^{3}$ with a
piecewise smooth boundary $\partial Q$ and let $\mathbf{F}(x, y, z)$
be a vector field with continuous first partial derivatives.
Then,
\[
%\label{eq:green:higher:divergencethm}
\iint_{\partial Q} \mathbf{F} \cdot \mathbf{N} \, dS
	= \iiint_{Q} (\nabla \cdot \mathbf{F}) \, dV	.
\]
\end{theorem}
\begin{itemize}
\item The Divergence Theorem is also known as Gauss's
Theorem.
\item The symbol $dV$ is the differential volume element.
\item The symbol $\mathbf{N}$ is the unit outward normal vector to the
surface of $Q$.
\item The symbol $dS$ is the differential surface area element.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Adjoint Operator}

Define
\begin{align*}
M
	&= v A u_{x} - (v A)_{x} u + v D u	\\
N
	&= v B u_{y} - (v B)_{y} u + v E u	\\
P
	&= v C u_{z} - (v C)_{z} u + v F u	.
\end{align*}
The adjoint equation is then
\[
%\label{eq:green:higher:adjoint3D}
\iiint_{Q} v L[u] \, dV
	= \iint_{\partial Q} [ M \mathbf{i} + N \mathbf{j} + P
        \mathbf{k} ] \cdot \mathbf{N} \, dS +
\iiint_{Q} L^{\ast}[v] u \, dV
\]
where
\[
L^{\ast}[v]
	= (A v)_{xx} + (B v)_{yy} + (C v)_{zz} - (D v)_{x} - (E v)_{y}
        - (F v)_{z} + G v	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Green's Identities}

Green's First Identity:
\[
%\label{eq:green:higher:1stID}
\iint_{\partial Q} (g \nabla f) \cdot \mathbf{N} \, dS
	= \iiint_{Q} (\nabla f) \cdot (\nabla g) \, dV + \iiint_{Q} g
        (\Delta f) \, dV	.
\]

\vspace*{0.10in}

Green's Second Identity:
\[
%\label{eq:green:higher:2ndID}
\iint_{\partial Q} \left[ (g \nabla f) - (f \nabla g) \right] \cdot \mathbf{N} \, dS
	= \iiint_{Q} \left[ g (\Delta f) - f (\Delta g) \right] \, dV	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Inner Product}

Define the inner product for functions $f(x, y)$ and $g(x, y)$ defined
on region $R \subset \mathbb{R}^{2}$ as
\[
%\label{eq:green:higher:2Dinnerproduct}
\langle f, g \rangle
	= \iint_{R} f(x, y) g(x, y) \, dA	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Strategy}

Denote the Green's function as $G(x, y; \xi, \eta)$ where $x$
and $y$ will be thought of as the coordinates of a particular point
$(x, y) \in R$ and $\xi$ and $\eta$ will be variables of integration.
Suppose a Green's function can be found so that
\begin{itemize}
\item the integrand of the surface terms vanishes, and
\item $L^{\ast}[G] = \delta(\xi - x, \eta - y)$.
\end{itemize}
If function $v$ in the adjoint equation is
replaced by the Green's function $G$, then
\[
\iint_{R} L[u] G \, dA
	= \iint_{R} u L^{\ast}[G] \, dA
\]
and thus
\[
\iint_{R} G(x, y; \xi, \eta) \Gamma(\xi, \eta) \, dA
	= \iint_{R} u(\xi, \eta) \delta(\xi - x, \eta - y) \, dA
	= u(x, y)	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Principal Solution}

Suppose the Green's function is the sum of
two functions,
\[
%\label{eq:green:higher:greensum}
G(x, y; \xi, \eta)
	= U(x, y; \xi, \eta) + g(x, y; \xi, \eta)	.
\]
\begin{itemize}
\item Function $U$, known as the \textbf{principal
  solution} satisfies the partial
differential equation $L^{\ast}[U] = \delta(\xi-x, \eta-y)$ but need
not satisfy the boundary conditions imposed on the Green's function.
\item Function $g$ will satisfy the differential equation $L^{\ast}[g] = 0$
and the boundary conditions required of the Green's function.
\end{itemize}

\end{frame}
%%

%\begin{frame}\frametitle{Homework}

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

%\end{frame}
%%

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