\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{A Barycentric Rational Interpolation Collocation Method for Solving the
Helmholtz Equation}



\author[1]{Miaomiao Yang}%
\author[1]{Wentao Ma}%
\author[2]{Yongbin Ge}%
\affil[1]{Ningxia University}%
\affil[2]{Institute of Applied Mathematics and Mechanics}%


\vspace{-1em}



  
  \date{May 05, 2024}


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





\selectlanguage{english}
\begin{abstract}
In this paper, we developed a meshless collocation method by using
barycentric rational interpolation basis function based on the Chebyshev
to deduce the scheme for solving the Helmholtz equation~defined in
arbitrary domain with complex boundary shapes. Firstly, the spatial
variables and their partial derivatives are treated by interpolation
basis functions, and the collocation method for solving second order
differential equations is established. Then the differential matrix is
used to simplify the differential equations on a given test node.
Finally, numerical experiments based on three kinds of test nodes show
that the proposed method can be used to calculate not only the high wave
numbers problems, but also the variable wave numbers problems. Moreover,
the algorithm has the advantages of high calculation accuracy, good
numerical stability and the less CPU time consuming.%
\end{abstract}%



\sloppy


\textbf{Hosted file}

\verb`manuscript.doc`  available at \url{https://authorea.com/users/352543/articles/476743-a-barycentric-rational-interpolation-collocation-method-for-solving-the-helmholtz-equation}

\textbf{Hosted file}

\verb`Figure1.doc`  available at \url{https://authorea.com/users/352543/articles/476743-a-barycentric-rational-interpolation-collocation-method-for-solving-the-helmholtz-equation}

\textbf{Hosted file}

\verb`Figure2.doc`  available at \url{https://authorea.com/users/352543/articles/476743-a-barycentric-rational-interpolation-collocation-method-for-solving-the-helmholtz-equation}

\textbf{Hosted file}

\verb`Figure3.doc`  available at \url{https://authorea.com/users/352543/articles/476743-a-barycentric-rational-interpolation-collocation-method-for-solving-the-helmholtz-equation}

\textbf{Hosted file}

\verb`Figure4.doc`  available at \url{https://authorea.com/users/352543/articles/476743-a-barycentric-rational-interpolation-collocation-method-for-solving-the-helmholtz-equation}

\textbf{Hosted file}

\verb`Figure5.doc`  available at \url{https://authorea.com/users/352543/articles/476743-a-barycentric-rational-interpolation-collocation-method-for-solving-the-helmholtz-equation}

\textbf{Hosted file}

\verb`Figure6.doc`  available at \url{https://authorea.com/users/352543/articles/476743-a-barycentric-rational-interpolation-collocation-method-for-solving-the-helmholtz-equation}

\textbf{Hosted file}

\verb`Figure7.doc`  available at \url{https://authorea.com/users/352543/articles/476743-a-barycentric-rational-interpolation-collocation-method-for-solving-the-helmholtz-equation}

\textbf{Hosted file}

\verb`Figure8.doc`  available at \url{https://authorea.com/users/352543/articles/476743-a-barycentric-rational-interpolation-collocation-method-for-solving-the-helmholtz-equation}

\textbf{Hosted file}

\verb`Figure9.doc`  available at \url{https://authorea.com/users/352543/articles/476743-a-barycentric-rational-interpolation-collocation-method-for-solving-the-helmholtz-equation}

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

