\documentclass[10pt]{article}

\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}





\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
            linkcolor = blue,
            urlcolor  = blue,
            citecolor = blue,
            anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
  \errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother


\usepackage{natbib}




\renewenvironment{abstract}
  {{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
  {\bigskip}

\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}


\usepackage{authblk}


\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%

\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}



\usepackage{float}








\begin{document}

\title{Inverse coefficient problem for a time - fractional wave equation with
initial - boundary and integral type overdetermination conditions}



\author[1]{H.H. Turdiev}%
\author[2]{D. K. Durdiev}%
\affil[1]{Bukhara State University}%
\affil[2]{Bukhara branch of the institute of Mathematics named after VI Romanovskiy at the Academy of sciences of the Republic of Uzbekistan}%


\vspace{-1em}



  
  \date{May 26, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
This paper considers the inverse problem of determining the
time-dependent coefficient in the time-fractional diffusion-wave
equation. In this case, an initial boundary value problem was set for
the fractional diffusion-wave equation, and an additional condition was
given for the inverse problem of determining the coefficient from this
equation. First of all, it was considered the initial boundary value
problem. By the Fourier method, this problem is reduced to equivalent
integral equations. Then, using the Mittag-Leffler function and the
generalized singular Gronwall inequality, we get apriori estimate for
solution via unknown coefficient which we will need to study of the
inverse problem. The inverse problem is reduced to the equivalent
integral of equation of Volterra type. The principle of contracted
mapping is used to solve this equation. Local existence and global
uniqueness results are proved. The stability estimate is also obtained.%
\end{abstract}%



\sloppy




\selectlanguage{english}
\FloatBarrier
\end{document}

