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\documentclass[a4paper,12pt]{report}
\usepackage[american]{babel}
\usepackage{amsfonts,amssymb,amsmath,amsthm}
\usepackage[table]{xcolor}
\usepackage{pifont}
\usepackage{marvosym}
\usepackage{fancybox,fancyhdr}
\usepackage{graphicx}
\usepackage{eso-pic}
\usepackage{yfonts}
\usepackage{pdfsync}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[dvips]{epsfig}
\usepackage{caption,subcaption}
\usepackage{algorithmicx}
\usepackage[ruled]{algorithm}
\usepackage{algpseudocode}

\usepackage[backend=biber,style=apa]{biblatex}

\usepackage{booktabs, collcell, makecell, tabularx, threeparttable}
    \newcommand{\tclr}[1]{\textcolor{black!70!black}{#1}}
    \newcolumntype{L}{>{\collectcell\tclr\raggedright}X<{\endcollectcell}}
    \renewcommand\TPTtagStyle{\bfseries} % optional
\usepackage{siunitx}
\usepackage{xparse}
\NewExpandableDocumentCommand\mcc{O{1}m}
    {\multicolumn{#1}{c}{#2}}
\newcommand\tot{\mathrm{tot}}
begin{document}
\begin{table}%[!htp]
    \small
    \sisetup{table-format=6.2e-1,per-mode=symbol}
    \setlength\tabcolsep{3pt}
\caption{The QPSO update equations for different potential energy types}\label{3wavfun}
\centering
\begin{threeparttable}
\begin{tabularx}{\textwidth}{@{} L l *{3}{>{$}l<{$}S} @{}}
\toprule
 {\bf Delta potential}  & {\bf QPSO update equation}\\
 \midrule
 Delta potential well & $x_i(t+1)=p(t)\pm\frac{\ln(1/u)}{2q\ln\sqrt{2}}\parallel x_i(t)-p(t)\parallel$\\
 \midrule
 Harmonic oscillator & $x_i(t+1)=p(t)\pm\frac{\sqrt{\ln1/u}}{0.47694q}\parallel x_i(t)-p(t)\parallel$\\
 \midrule
 Square well & $x_i(t+1)=p(t)+\frac{0.6574}{\xi q}\cos^{-1}(\pm\sqrt u)\parallel x_i(t)-p(t)\parallel$\\
\bottomrule
\end{tabularx}
\end{threeparttable}
\end{table}
 \end{document}

enter image description here

1

You wrote in a comment,

i need the "table 1:" to be in bold in order to respect APA format

All you need to do, then, is to add the instruction \captionsetup{labelfont=bf} in the preamble -- after loading the caption package.

However, you should do yourself (and your readers) a lot more than just that. For the sake of making the code more maintainable, you should remove a lot of cruft. You should also make it more readable by rendering the equations in the second column in displaymath mode. And please don't use \parallel -- at least not in this context; use \lVert and \rVert instead. Better yet, define a \norm macro. Finally, don't overuse bold-facing.

You should clean up and simplify the document's preamble. I strongly doubt that you need many of packages you're loading at present. For example, the yfonts package provide text-mode fraktur-type fonts -- do you really need them?

enter image description here

\documentclass[a4paper,12pt]{report}
\usepackage[american]{babel}
\usepackage[T1]{fontenc}
\usepackage{booktabs,tabularx}

\usepackage{mathtools}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}

\usepackage{caption}
\captionsetup{labelfont=bf,skip=0.5\baselineskip,
              justification=raggedright,
              singlelinecheck=false}

\begin{document}
\begin{table}[!ht]
\caption{The QPSO update equations for different potential energy types}
\label{3wavfun}
\begin{tabularx}{\textwidth}{@{} >{\raggedright\arraybackslash}X >{$\displaystyle}l<{$} @{}}
\toprule
Delta potential  
& $QPSO update equation$\\
\midrule
Delta potential well 
& x_i(t+1)=p(t)\pm \frac{\ln(1/u)}{2q\ln\sqrt{2}} \norm{x_i(t)-p(t)}\\[3ex]

Harmonic oscillator 
& x_i(t+1)=p(t)\pm \frac{\sqrt{\ln(1/u)}}{0.47694q} \norm{x_i(t)-p(t)}\\[3ex]

Square well 
& x_i(t+1)=p(t)+ \frac{0.6574}{\xi q}\cos^{-1}(\pm\sqrt{u}\,) \norm{x_i(t)-p(t)}\\
\bottomrule
\end{tabularx}
\end{table}
\end{document}
| improve this answer | |
  • all the package i load are to my 100pages of dissertation. and thx for your concern – Chamanga Mar 26 at 6:55

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