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\title[\sc Dirichlet problem for the wave equation]{\large Dirichlet problem for the wave equation}
\author[Yu.I. Sukharev, V.P. Tanana, B.A. Markov]{{\bf \large \textsc{{Yu.I. Sukharev, V.P. Tanana, B.A. Markov}}}\protect\orcidicon{0000-0001-8827-2630}\\ \\
\textit{Представлено} \textsc{П.П. Петровым}
%\textit{(заполняется редактором)}
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\address{Yurii Ivanovich Sukharev  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Chelyabinsk state University,
\newline\hphantom{iii} Br. Kashirinikh str., 129,
\newline\hphantom{iii} 454001, Chelyabinsk, Russia}%
\email{\textcolor{blue}{yuri\_sucharev@mail.ru}}%
\address{Vitalii Pavlovich Tanana   % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} South Urals state Univercity,
\newline\hphantom{iii} Lenin pr., 78,
\newline\hphantom{iii} 454080, Chelyabinsk, Russia}%
\email{\textcolor{blue}{smpx1969@mail.ru}}%
\address{Boris Anatolievich Markov   % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} South Urals state Univercity,
\newline\hphantom{iii} Lenin pr., 78,
\newline\hphantom{iii} 454080, Chelyabinsk, Russia}%
\email{\textcolor{blue}{smpx1969@mail.ru}}%

\thanks{\sc Sukharev, Yu.I., Tanana V.P., Markov B.A.
On the solution of the Dirichlet problem for the wave equation}
\thanks{\copyright \ 2024 Yu.I. Sukharev, V.P. Tanana, B.A. Markov}
\thanks{}
\thanks{\it Поступила 22 октября 2024 г., опубликована 31 декабря 2024 г.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\bf Abstract:} The paper considers the Dirichlet problem for the wave equation on a finite time interval. The difficulty in solving the problem lies in the fact that under natural conditions the solution to this problem may not exist, which complicates the problem formulation.

The paper solves this problem by comparing the Dirichlet problem with the Cauchy problem. Under certain boundary conditions, it is possible to construct a solution to the Dirichlet problem by reducing it to the Cauchy problem for the wave equation, thus proving the existence of a solution. Due to the uniqueness and stability of the solution to the Cauchy problem, it is possible to prove the uniqueness of the solution to the Dirichlet problem under rather strong constraints on the functions in the boundary conditions.\medskip

\noindent{\bf Keywords:} Dirichlet problem, finite time interval, wave equation, conditionally ill-posed problem, Cauchy problem, non-equivalence of the Cauchy problem and the Dirichlet problem.
 \end{quote}
}

\bigskip

\section*{Introduction}
\input{Introduction.tex}
\section{Mathematical statement of the problem} 
\input{MathematicalProblemStatement.tex}
\subsection{Sturm-Liouville problem}
\input{SturmLiuvilleProblem.tex}
\subsection{Construction of the solution to the problem (\ref{eqn1-1b}) and some of its properties}
\input{SeriesOfAuxiliaryProblem.tex}
\subsection{Solution to the problem (\ref{eqn1-1})}
\input{FormalSolution.tex}
%\section{Построение регуляризации. Оценка погрешности}
%\input{Regularization.tex}
\section*{Conclusion}
\input{Conclusions.tex}
\bigskip
\input{bbs.tex}
\end{document}