Some LaTeX comments
Preamble
Use the standard article documentclass
Almost all of your definitions in the preamble were not used---omit them.
use full conventional names for standard environments theorem, lemma, corollary, proposition, example, conjectures, etc---it makes your latex much more portable
Do not explicitly override the equation numbering convention of the article style as I have to undo your changes so the journal style can work its magic unimpeded by you.
put definitions of newtheorems and newcommands in the preamble
Title
use \and to separate authors within an \author command
the international standard is mailto:e-mail-address, not e-mail: e-mail-address
\usepackage{url} then \protect\url{...} for URLs
Do not use gratuitous capitals in the titles and captions.
I prefer you to use the \thanks command to typeset each authors address
Fonts
Use \textsc{tla} instead of TLA
Use \mathbb not the outdated \Bbb
Do not use math mode to get italics, the interletter spacing in mathematics is different to that for words, and ligatures are not recognised in maths mode.
despite being Latin, 'et~al.' is usually typeset in normal font, not italic.
Style
Use the enumerate environment to give a list of sequentially labelled alternatives or cases
Avoid writing something 'clearly' or 'it is easy' as such a phrase insults anyone who cannot see it.
Never use \\ outside of a tabular or maths array environment
Never underline, instead use logical commands such as \paragraph or the description environment or \emph for emphasis
Shun wishy-washy indefinite phrases such as 'can be', 'wish to' or 'may be'; if you are, then say so definitely; if not, say so; be explicit
A list of consecutive citations is '\cite{KRA0, KRA1}' not '\cite{KRA0},~\cite{KRA1}'
avoid forcing \flushleft, let the journal style take care of such issues
Avoid gratuitous verbiage such as 'We should mention that' and 'It is worth noting'
never use \noindent
Use logical \paragraph{Case 1}, not the physical \textbf{Case 1}
$i$th not $i^{th}$
Computer algorithms and their parameters are much better typeset using verbation environment or \verb.
Do not explicitly put {$\hfill \rule{2.5mm}{2.5mm}$} in at the end of theorems etc, let the style of the journal do what they want at the end of a theorem, etc.
Spacing
generally separate maths from punctuation by a thin space \, (including inline mathematics)
Have punctuation outside inline maths environment
Enter and leave maths mode appropriately, for example, '$t$, $k$, $\gamma$ and~$h$' not '$ t,\: k,\: \gamma $ and $ h$'; and '$0 \leq \beta \leq 2+h$\,, $i=1,2$\,.' not '$0 \leq \beta \leq 2+h , i=1,2. $'
Use a non-breaking space in Region~II, Section~3, Figure~1, in~(22), Conjecture~3, reactant~$A$ etc.
Generally separate differentials from an integrand and other differentials by a thin space \,
Do not have gratuitous blank lines forcing paragraphs about displayed equations.
separate maths statements in displayed equations by one \quad space
Do not have gratuitous nonbreaking spaces in (~\ref{eqn 1.6}--~\ref{eqn 1.8}) nor 'in ~\cite{ref1-Gawad1}' for example
almost never use forced spacing commands such as \enskip, \bigskip, \hspace, \vspace, etc
If you do need to force a bit of extra vertical space in the array environment, use units of ex, not mm, as an ex stretches naturally with the font size.
Mathematics
use eqnarray to align equations that appear immediately after one another.
Have ellipses at the same height as the operator they are to continue. For example a comma separated list is continued with \ldots, not \cdots; whereas an equality list is \cdots. And always remember to put in the operators on both sides of the ellipsis for a finite list.
'the following equation' or 'given by:' or 'expressed by' followed by an equation is usually a waste of words
Use \mbox{some text} to put some text into a maths environment, not \rm
Multi character variables are usually typeset in roman font such as \mbox{Re} or \mbox{Sat}
Do not put a hat over a variable and its subscript as in '\hat{u_0}', instead just put it over the variable as in '\hat u_0'; same for tildes.
Use \left(...\right) to surround maths stuff to get parentheses about the correct size
Generally avoid using \frac in sub- and super-scripts
Also avoid \frac in inline mathematics except for the simplest fractions
prefer 'for all' to \forall
{\cal O} is much nicer than \bigcirc
do not use \mid for the vertical bar in absolute value
$\mbox{P}=\mbox{NP}$ not P=NP as then the spacing is correct
Do not wreck TeX's spacing about relations by '~\leq~' (nor about arrows)
Do not use \vert as a separator, use \mid
\pmod 4 usefully typesets ' (mod 4)'
Use eqnarray* to layout maths, not forcing linebreaks in inline maths.
Use \times when you want to emphasise the multiplication of two things.
Use \max for abbreviation of maximise within maths environment.
Figures and tables
Use the figure environment in conjunction with the graphicx package for figures. Also use labels and ref.
\label needs to come after a \caption in figure or table environments
It is pointless using '\begin{center}\begin{figure}...\end{figure}\end{center}'. Instead just use \begin{figure}\centering...\end{figure}
Minimize the use of ruled lines in a table, let the tabular structure speak for itself at the lowest level of structure.
Backmatter
Use the logical \paragraph{Acknowledgements:} not the physical "\bf Acknowledgements".
Do not say "would like to thank"; either you do, or you do not, say which.
Journal names are generally typeset in italic---I recommend you use use BiBTeX.