\documentclass[a4paper,12pt]{amsart}
\oddsidemargin 0cm
\evensidemargin 0cm
\topmargin -20mm
\textheight 260mm
\textwidth 16cm
\pagestyle{empty}
\begin{document}
\title{\bf Initial--boundary value problems approximation\\
with the {\emph{Mathematica\/}} system}
\author{\emph{Ryszard Walenty{\'n}ski PhD}}
\address {Silesia University of Technology\\
Faculty of Civil Engineering\\
Department of Building Structures' Theory\\
ul.~Akademicka~5, PL44-101~Gliwice, Poland}
\email {\tt rwal@kateko.bud.polsl.gliwice.pl}
\date{Printed: \today}
\maketitle
\section*{Abstract}
The paper deals with approximation of initial--boundary value problems.
The refined method of least squares is applied. The system
\emph{Mathematica\/} is used to build the algorithm and to solve tasks.
Let us consider an arbitrary boundary, initial or initial--boundary value problem. It
can be, to focus our attention, described with the equation
\begin{equation}
\mathcal{L}(U)=b\label{equation},
\end{equation}
and boundary conditions
\begin{equation}
x \in \Gamma_I:\ U \mid_{\Gamma_I} = T_d,
\quad x \in \Gamma_{I\!I}:\ q \mid_{\Gamma_{I\!I}} = q_d,\label{conditions}
\end{equation}
where $x$ is a point on the boundary of the domain $D$ and $q=n
\cdot \mathrm{grad} U$. Vector $n$ is normal to the boundary.
Let us introduce, for the above example, functions of residuum
(error or defect of the solution)
\begin{eqnarray}
R=\mathcal{L}(U_o)-b,\label{residuum}\\
R_I=U_o-T_d,\label{residuum1}\\
R_{I\!I}=q_o-q_d.\label{residuum2}
\end{eqnarray}
If the $U_o$ is an approximate solution of the boundary value
problem then residuum functions are not equal to zero. We can
build for the functions (\ref{residuum}), (\ref{residuum1})
and (\ref{residuum2}) the following functional
\begin{equation}
\mathcal{F}=\int_{(D)} (R\,s)^2 \,\mathrm{d}D
+\int_{(\Gamma_I)} (R_I\,s_I)^2\,\mathrm{d}\Gamma_I+
\int_{(\Gamma_{I\!I})} (R_{I\!I}\,s_{I\!I})^2\,\mathrm{d}\Gamma_{I\!I}
\geq 0\label{functional}
\end{equation}
where $s$, $s_I$ and $s_{I\!I}$ are scale coefficients which can
shape the minimization path.
The functional can be minimized with the Ritz method. The obtained
approximation usually satisfy neither the equation nor the
initial--boundary condition.
In fact, it is not very easy to find a set of function that would
satisfy all initial--boundary conditions especially for
multidimensional tasks. Presented above approach neglects that
classical requirement. It results in the very short algorithm in
\emph{Mathematica\/}.
The presented method can be easily applied to any initial--boundary
value problem. The method is stable. It is not sensitive on high gradient
of functions or type of equations. Problems with singularities
can be approximated, too.
Analytical integration with \emph{Mathematica\/} was applied if
possible. It is specially useful for ill conditioned tasks. The
solution can be made with an arbitrary precision.
The result is given in form of function. It can be easily presented
with \emph{Mathematica\/} and used in another tasks solved with it.
The error estimation is very easy. One should only put result into
residue (\ref{residuum}), (\ref{residuum1}) or (\ref{residuum2}) to
see ''local satisfaction'' of the problem or into functional
(\ref{functional}) to estimate overall error.
\end{document}