<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass {slides}
\pagestyle {plain}
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% psnup -4 talk1.ps &gt; talk1.ps4

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\textwidth 6.6in
\textheight 8.5in

%\renewcommand{\baselinestretch}{0.8}

%rest is the same as for regular latex files

\usepackage{graphics}
%\usepackage{subfigure, epsfig}
\usepackage{rotate}

\newtheorem{theorem}{\scshape Theorem}
\newtheorem{lemma}{\scshape Lemma}
\newtheorem{corollary}{\scshape Corollary}
\newtheorem{remark}{\scshape Remark}

\newcommand{\Lee}{\mathbf{L}}
\newcommand{\Bee}{\mathbf{B}}
\newcommand{\Cee}{\mathbf{C}}
\newcommand{\Pee}{\mathrm{P}}
\newcommand{\Iee}{\mathbf{I}}
\newcommand{\hs}{\hspace}
\newcommand{\mc}{\multicolumn}

\begin{document}

\begin{slide}
\begin{center}
\LARGE\bfseries
My Wonderful Talk\\
using Slides in Latex\\
and Portrait Format\\
\mdseries by Student Name\\
%Supervised by Prof. C. Christara
\end{center}
\end{slide}

\begin{slide}
{\Large \bf Outline}
\begin{itemize}
\item Background
\item Spline collocation on non-uniform partitions
\item Numerical results
\item Summary
\end{itemize}
\end{slide}

\begin{slide}
{\Large\bfseries
Background}

{\large \bf Cubic Spline Collocation (CSC)}
\begin{itemize}
\item {\bf BVP }\\
Determine $u(x)$ that satisfies
\begin{equation}
\Lee u \equiv ru_{xx} + pu_x + qu = g~\mbox{ in }~\Omega \equiv
(0,1),\label{eq:pde}
\end{equation}
subject to
\begin{equation}
\Bee u \equiv \alpha u + \beta u_x = \gamma \mbox{ on  $\partial\Omega$},
\label{eq:bc}
\end{equation}
where $r,p,q,g, \alpha, \beta, \gamma$ are given functions of $x$.
\end{itemize}
\end{slide}

\begin{slide}
\begin{itemize}
\item
{\bf Domain Discretization}\\
Let $\Delta \equiv \{x_0=0&lt;x_1\ldots&lt;x_N=1\}$ be a partition for $\Omega$.
\item
{\bf Approximation Space}\\
One-dimensional cubic splines ($\mathbf{ C^2}$) wrt $\Delta$.\\
Basis functions $\phi_i(x), i = -1,\ldots,N+1$.
\item
{\bf Cubic Spline Approximation}\\ $U(x)=\sum_i \mathbf{c}_{i}\phi_i(x)$
\item
{\bf Standard Collocation Method}\\
Let $T_\partial \equiv \{x_0,x_N\}$.\\
Determine the approximation $U$ to $u$ such that
\begin{eqnarray*}
\Lee U =&amp; g &amp;\mbox{ on } \Delta,\\
\Bee U =&amp; \gamma &amp;\mbox{ on } T_\partial.
\end{eqnarray*}
\end{itemize}
\end{slide}

\begin{slide}
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%\vspace{-0.5cm}
\begin{center}
My bleeding-edge caption.
\end{center}
\end{slide}

\begin{slide}
{\Large \bf Summary}
\begin{itemize}
\item
We developed optimal spline collocation methods for
non-uniform partitions by
\begin{enumerate}
\item using a mapping function between uniform and non-uniform
partitions and
\item developing expansions of the error at the non-uniform data points
of appropriately defined spline interpolants.
\end{enumerate}
\item
We have shown mathematically and experimentally that our mapping method
for non-uniform partitions produces optimal convergence.
\item
Our method can be incorporated into adaptive techniques.
\end{itemize}
\end{slide}

\end{document}
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