I suggest you use a split
environment, which in contrast to multline
may be used a subenvironment of equation
. You need to specify an alignment point on each line with &
and separate lines with \\
. In this case the first line should be move left relative to the others and the package mathtools
provides a convenient command for this:

\documentclass{article}
\usepackage{mathtools}
\begin{document}
\begin{equation}
\label{wave kinematic}
\begin{split}
\MoveEqLeft
\frac{\partial N(\sigma;\lambda;\theta;t)}{\partial t}
+ \frac{\partial C_{g,\lambda}N(\sigma;\lambda;\theta;t)}
{\partial \lambda} \\
&+ \cos\phi^{-1} \cdot
\frac{\partial C_{f,\phi}N(\sigma;\lambda;\theta;t)}{\partial
\phi} \\
&+ \frac{\partial C_{f,\theta}N(\sigma;\lambda;\theta;t)}{\partial
\theta}
+ \frac{\partial C_{f,\sigma}N(\sigma;\lambda;\theta;t)}{\partial
\sigma}
= \frac{S(\sigma;\theta;\lambda;\varphi;t)}{\sigma}
\end{split}
\end{equation}
\end{document}
I have split across three lines for clarity. If you wanted to just split in two parts, then multlined
(notice the extra d
) from the mathtools
package would be a simpler solution:

\documentclass{article}
\usepackage{mathtools}
\begin{document}
\begin{equation}
\label{wave kinematic}
\begin{multlined}
\frac{\partial N(\sigma;\lambda;\theta;t)}{\partial t}
+ \frac{\partial C_{g,\lambda}N(\sigma;\lambda;\theta;t)}
{\partial \lambda}
+ \cos\phi^{-1} \cdot
\frac{\partial C_{f,\phi}N(\sigma;\lambda;\theta;t)}{\partial
\phi} \\
+ \frac{\partial C_{f,\theta}N(\sigma;\lambda;\theta;t)}{\partial
\theta}
+ \frac{\partial C_{f,\sigma}N(\sigma;\lambda;\theta;t)}{\partial
\sigma}
= \frac{S(\sigma;\theta;\lambda;\varphi;t)}{\sigma}
\end{multlined}
\end{equation}
\end{document}
All the above works with elsarticle
class in your updated question. E.g. the first version becomes:
\documentclass[authoryear,preprint,review,12pt]{elsarticle}
\usepackage{mathtools}
\begin{document}
\begin{equation}
\label{wave kinematic}
\begin{split}
\MoveEqLeft
\frac{\partial N(\sigma;\lambda;\theta;t)}{\partial t}
+ \frac{\partial C_{g,\lambda}N(\sigma;\lambda;\theta;t)}
{\partial \lambda} \\
&+ \cos\phi^{-1} \cdot
\frac{\partial C_{f,\phi}N(\sigma;\lambda;\theta;t)}{\partial
\phi} \\
&+ \frac{\partial C_{f,\theta}N(\sigma;\lambda;\theta;t)}{\partial
\theta}
+ \frac{\partial C_{f,\sigma}N(\sigma;\lambda;\theta;t)}{\partial
\sigma}
= \frac{S(\sigma;\theta;\lambda;\varphi;t)}{\sigma}
\end{split}
\end{equation}
\end{document}
You can make use of mathtools
' multlined
environment:
\documentclass{article}
\usepackage{amsmath,mathtools}
\begin{document}
\[
\begin{split}
x &= a + a \\
&= b + b + b + b \\
&= \!\begin{multlined}[t]
c + c + c + c + c + c + c + c + c + c + c + c \\
+ c + c + c + c + c + c + c + c + c + c
\end{multlined}
\end{split}
\]
\end{document}

Note: the \!
before the environment is to get correct spacing between =
and c
.
It's also possible to specify the total width of the two lines (from left margin at first line to right margin at last line) as an optional argument to multlined
:
\documentclass{article}
\usepackage{amsmath,mathtools}
\begin{document}
\[
\begin{split}
x &= a + a \\
&= b + b + b + b \\
&= \!\begin{multlined}[t][10cm]
c + c + c + c + c + c + c + c + c + c + c + c \\
+ c + c + c + c + c + c + c + c + c + c
\end{multlined}
\end{split}
\]
\end{document}

If you want the two lines right aligned, you can set the mathtools
key firstline-afterskip
to 0pt
, either globally or locally:
\documentclass[border=5pt,preview]{standalone}
\usepackage{amsmath,mathtools}
\begin{document}
\mathtoolsset{firstline-afterskip=0pt}
\[
\begin{split}
x &= a + a \\
&= b + b + b + b \\
&= \!\begin{multlined}[t]
c + c + c + c + c + c + c + c + c + c + c + c \\
+ c + c + c + c + c + c + c + c + c + c
\end{multlined}
\end{split}
\]
\end{document}

Best Answer
Use either breqn to break lines automatically or use amsmath and its many environments exactly for this purpose. For example, with breqn:
Note, the expression around
\mid
required braces to prevent it from breaking at this point; I'm sure there is a better way to do that; anyway, here's the output:With amsmath, you need to specify the break points manually: (as others have also mentioned)
The users guide to amsmath is called amsldoc.pdf, but you can access it by typing
texdoc amsmath
on the command line. The main environments you'll use there would bealign
,split
, andmultline
.