You need to redefine \sectionmark
(refer to the package documentation for details); for example, you could use the following:
\documentclass[fleqn,12pt,a4paper]{article}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead[R]{\leftmark}
\renewcommand\sectionmark[1]{\markboth{\MakeUppercase{#1}}{}}
\begin{document}
\section{section-1}
\newpage
\section{section-2}
\end{document}
After processing the above exampe code you'll get the warning
Package Fancyhdr Warning: \headheight is too small (12.0pt):
Make it at least 14.49998pt.
We now make it that large for the rest of the document.
This may cause the page layout to be inconsistent, however.
which indicates that the defined header is too big; you can then add to the preamble something like
\renewcommand\headheight{14.5pt}
To remove the section numbering for the body of the document, you could use \titleformat
from the titlesec
package:
\documentclass[fleqn,12pt,a4paper]{article}
\usepackage{titlesec}
\usepackage{fancyhdr}
\titleformat{\section}
{\normalfont\Large\bfseries}{}{0em}{}
\pagestyle{fancy}
\fancyhead[R]{\leftmark}
\renewcommand\sectionmark[1]{\markboth{\MakeUppercase{#1}}{}}
\renewcommand\headheight{14.5pt}
\begin{document}
\section{section-1}
\newpage
\section{section-2}
\end{document}
Is is to be noticed that the above code only suppresses the section numbering in the actual titles, but the numbering will still appear in the ToC and in the lower level sectional units; if the section numbering must be removed from the ToC and from the lower level sectional units, the following lines need to be added to the preamble:
\renewcommand\thesection{}
\renewcommand\thesubsection{\arabic{subsection}}