How can I position \twocolumn bottom of the page

Question

On the following example, long equation shows up on the top of the second page.

The equation example is taken from https://tex.stackexchange.com/a/321805/127048.

\documentclass[5p,times]{elsarticle}
\usepackage{lipsum}
\usepackage{amsmath}
\begin{document}
\section{Start}
Hello World
\twocolumn[
{
    \begin{equation}\begin{split}
f_{r:n}(x)&=\frac{n!}{(r-1)!\,(n-r+1)!}
   \Biggl[ \biggl( \frac{x-1}{k}  \biggr)^{\!r-1}
           \biggl( \frac{k-x+1}{k}\biggr)^{\!n-r+1} \\
  &\qquad -\biggl( \frac{x}{k}    \biggr)^{\!r-1}
           \biggl( \frac{k-x}{k}  \biggr)^{\!n-r+1} \,
   \Biggr]+ f_{r-1:n}(x)
\end{split}\end{equation}
       }\bigskip]
\end{document}

output:

Is it possible to force the long equation to show up at the bottom of the first page or right after the Hello World text? Please note that I want to show the long equation in both of the two columns rather than fitting it into single column of the double-column.

Answer 1

Adding the package cuted to Micos`s answer, the wide equation can be put after "Hello world".

\documentclass[5p,%times % don't specify the 'times' option
]{elsarticle}
\usepackage{amsmath} % for 'aligned' environment
\usepackage{newtxtext,newtxmath} % for a more up to date Times Roman math font
\usepackage{cuted}% strip environment
\usepackage{lipsum}
\begin{document}
\section{Start}

\noindent
Hello world.

\begin{strip}
\begin{equation} % see https://tex.stackexchange.com/a/321805/5001
    \begin{aligned}[b]
        f_{r:n}(x)
        &=\binom{n}{r-1} %\frac{n!}{(r-1)!\,(n-r+1)!}
        \biggl[ \biggl( \frac{x-1}{k}  \biggr)^{\!r-1}
        \biggl( \frac{k-x+1}{k}\biggr)^{\!n-r+1} \\
        &\quad  -\biggl( \frac{x}{k}    \biggr)^{\!r-1}
        \biggl( \frac{k-x}{k}  \biggr)^{\!n-r+1} \,
        \biggr ] + f_{r-1:n}(x)
    \end{aligned}
\end{equation}
\bigskip

\begin{equation}
    f_{r:n}(x)  =\binom{n}{r-1} %\frac{n!}{(r-1)!\,(n-r+1)!}
    \biggl[ \biggl( \frac{x-1}{k}  \biggr)^{\!r-1}
    \biggl( \frac{k-x+1}{k}\biggr)^{\!n-r+1} -\biggl( \frac{x}{k}    \biggr)^{\!r-1}
    \biggl( \frac{k-x}{k}  \biggr)^{\!n-r+1} \,
    \biggr ] + f_{r-1:n}(x)
\end{equation}

\end{strip}

\noindent
Hello other world.
\begin{multline} % see https://tex.stackexchange.com/a/321863/5001
    f_{r:n}(x)
    =\binom{n}{r-1} %\frac{n!}{(r-1)!\,(n-r+1)!}
    \biggl[ \biggl( \frac{x-1}{k}  \biggr)^{\!r-1}
    \biggl( \frac{k-x+1}{k}\biggr)^{\!n-r+1} \\
    -\biggl( \frac{x}{k}    \biggr)^{\!r-1}
    \biggl( \frac{k-x}{k}  \biggr)^{\!n-r+1} \,
    \biggr ] + f_{r-1:n}(x)
\end{multline}

\lipsum[1-7]


\end{document}

Answer 2

You asked,

Is it possible to force the long equation to show up ... right after the Hello World text?

Short answer: Yes.

Longer answer: Simply omit the \twocolumn[{...}\bigskip] "wrapper". Since you employ the elsarticle document class with the option 5p, the document is already in two-column mode. It's not clear to me what you're trying to achieve by issuing the \twocolumn directive to begin with -- except, probably involuntarily, force an unnecessary and unwanted page break while also, likely quite unexpectedly, switching to single-column mode.

While on the subject of the elsarticle document class: If at all possible, don't use the times document class option, as it loads the borderline obsolete txfonts package. Instead, do yourself a favor and load the newtxtext and newtxmath packages explicitly, as is done in the code shown below.

The following screenshot shows two approaches -- one with the help of an equation/aligned combination, the other with a multline environment -- to making the long equation fit inside the fairly narrow column. The results of both approaches are fine; which one you may prefer is mainly a matter of tast (about which there is no arguing, right?).

\documentclass[5p,%times % don't specify the 'times' option
              ]{elsarticle}
\usepackage{amsmath} % for 'aligned' environment
\usepackage{newtxtext,newtxmath} % for a more up to date Times Roman math font
\begin{document}
\section{Start}
\noindent
Hello world.
\begin{equation} % see https://tex.stackexchange.com/a/321805/5001
\begin{aligned}[b]
f_{r:n}(x)
  &=\binom{n}{r-1} %\frac{n!}{(r-1)!,(n-r+1)!}
   \biggl[ \biggl( \frac{x-1}{k}  \biggr)^{!r-1}
           \biggl( \frac{k-x+1}{k}\biggr)^{!n-r+1} \
  &\quad  -\biggl( \frac{x}{k}    \biggr)^{!r-1}
           \biggl( \frac{k-x}{k}  \biggr)^{!n-r+1} ,
   \biggr ] + f_{r-1:n}(x)
\end{aligned}
\end{equation}
\noindent
Hello world.
\begin{multline} % see https://tex.stackexchange.com/a/321863/5001
f_{r:n}(x)
  =\binom{n}{r-1} %\frac{n!}{(r-1)!,(n-r+1)!}
   \biggl[ \biggl( \frac{x-1}{k}  \biggr)^{!r-1}
           \biggl( \frac{k-x+1}{k}\biggr)^{!n-r+1} \
          -\biggl( \frac{x}{k}    \biggr)^{!r-1}
           \biggl( \frac{k-x}{k}  \biggr)^{!n-r+1} ,
   \biggr ] + f_{r-1:n}(x)
\end{multline}
\end{document}