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			% Type the title of the article here
\centerline{\bf \small{THE COMPLETE THEORY OF MATHEMATICS}} 
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     %Type the name(s) of the author(s)
\centerline{\scriptsize{S. O. BOSS\footnote{Corresponding 
author} AND T. K. POSTDOC\footnote{Supported by ICTP, grant number T4538}}}
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\scriptsize{
\noindent % Insert the abstract of your paper here
ABSTRACT. This study focuses on the functional framework  $x(z)=\lambda z^{\mu }$ 
for a non-linear PDE arising from a fluid in a sphere. 
}
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\noindent % Insert a maximum of 5 Keywords and phrases 
{\bf Keywords and phrases:} Third grade fluid, PDE, Sphere
\par
\noindent % Insert the AMS classification
2010 Mathematical Subject Classification: A80
\par
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{\small{
\centerline{1. INTRODUCTION}}
}
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\noindent %Insert the content of your introduction
During the past few decades, research in flows involving heat 
transfer of non-Newtonian fluids has been intensified. 
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{\small{
\centerline{2. PRELIMINARY}}
}
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\noindent
We define some basic results from Kim \cite{journ2.1} for the benefit 
of all.
\par
\noindent
{\bf Definition 1:} A solution $u\in \Omega$ is called a $\alpha $-weak 
solution of the boundary value problem \ldots
\par
\noindent
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{\small{
\centerline{3. THE HEART OF THE MATTER} }}
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\noindent
\par			%paragraph in the article
The heart of the matter is to state without proof the following: 
\par
\noindent				% Example of equation with numbering
{\bf Lemma 1:} (A Sobolev's embedding) Let $u\in \Omega$. Then we have the estimate
$$\int{r^mu'v'dr} + (a-m) \int{r^{m-1}u'vdr} = \int{r^m[h-g(u)]vdr},\eqno (1)$$
Next we proof the main theorem:
\par
\noindent
{\bf Theorem 1:} (A priori estimates). Let $u$ be a solution of \ldots
\par
\noindent
{\bf Proof:} We split the proof into two steps.
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The solution to equation (1) is investigated numerically and \ldots 
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\small{{\bf Table 1.} Comparison of perturbation and numerical result with \\ $\Lambda = \Gamma = 1$ and $\delta = 0.2$.}	
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\noindent
					% How to generate a table
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\begin{tabular}{|l|rrr|}
\hline  
$\gamma$ & $\theta_{max}$(PS) & $\theta_{max}$(NS) &  Rel. error \\
\hline
0  & 0.6439 & 0.6635 & $1.94 \%$ \\
1  & 0.6458 & 0.6684 & $2.39 \%$ \\
2  & 0.6476 & 0.6737 & $3.87 \%$ \\
4  & 0.6513 & 0.6846 & $4.87 \%$ \\
5  & 0.6531 & 0.6904 & $5.40 \%$ \\
\hline 
\end{tabular}
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\par
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\small{{\bf Fig. 1.} Bifurcation plot for a variable viscosity when $\Gamma = \Lambda$ = 1.}
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\par
{\small{
\centerline{4. CONCLUDING REMARKS}}}
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\par
\noindent %Insert your concluding remarks
Functional framework (see Evans and Alli \cite{journ2.2})has been employed to generate analytical solutions of a model reaction diffusion equation. Table 1 demonstrates the comparison of solutions. 
\par
In Fig. 1, the critical value of the temperature is plotted against Newtonian parameter $\delta$.
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{\small{
\centerline{ACKNOWLEDGEMENTS}}}
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\noindent
					% insert your thanks if any
The author would like to thank the anonymous referee whose comments improved the original version of this manuscript.
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					% Format for nomenclature and definition of symbols
{\small{
\centerline{NOMENCLATURE}}}
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\noindent 
\begin{tabular}{ll}
$\theta_{max~cr}$ & Non-dimensionless temperature of the model \\
$\epsilon$ & Activation Energy \\
\end{tabular}
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%\scriptsize{
{\small{
\centerline{REFERENCES}
}}
\scriptsize{
\begin{enumerate} %This is the format for a journal article
[leftmargin= 0.5cm]
\bibitem[1]{journ2.1}S. O. Kim, {\it Thermal stability of a reactive
non-Newtonian flow in a sphere}, Int. Commu. Heat and Mass Transfer \textbf{60} (1) 70-81, 2009. 
									% This is the format for a book
\bibitem[2]{journ2.2}L. C. Evans and P. A. Alli, {\it Partial Differential Equations}, American Mathematical Society, Providence, Rhode Island, 1998.
										% This is the format for a thesis
\bibitem[3]{journ2.3} H. A. Okoye, {\it How to define reasonably weighted Sobolev spaces}, Ph.D. thesis,  University of Uyo, Uyo, Akwa Ibom State, Nigeria, 2009.
\end{enumerate}
}
\par
\noindent % Institutional address of the first author here:
\tiny{DEPARTMENT OF MATHEMATICS,
OBAFEMI AWOLOWO UNIVERSITY, ILE - IFE, NIGERIA
\par 
\noindent %E-mail address of the first author
{\it E-mail address}: {\tt bossnms@gmail.com} 
\par
\noindent % Institutional address of the second author here:
%\scriptsize{
DEPARTMENT OF MATHEMATICAL SCIENCES,
FEDERAL UNIVERSITY OF TECHNOLOGY, MINNA, NIGERIA
\par 
\noindent %E-mail address of the second author
{\it E-mail addresses}: {\tt sspostdoc@yahoo.co.uk,firstdoc@oauife.edu.ng} 
}
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