\documentclass[a4paper,12pt]{article}
\usepackage{mathspec}
\usepackage{csquotes}
\usepackage{xltxtra}
\usepackage{amsmath,amsthm,amssymb}
%\usepackage{setspace}
\usepackage[margin=1in]{geometry}
\XeTeXlinebreakskip = 0pt plus 1pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{definition}{Definition}
\newtheorem{theorem}{Theorem}
\newtheorem{pproposition}{Proposition}
%\newtheorem{corollary}[Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\newtheorem{example}{Example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\defaultfontfeatures{Scale=1.5}
%If you don't have TH SarabunPSK, you have to use TH Sarabun New
%\setmainfont{Times New Roman}
%\setstretch{1.5}
%----------------------------------------------------------------
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{}
\renewcommand{\headrulewidth}{0pt}
\fancyhead[R]{\thepage}
%------------------------change academic year------------------------
\fancyfoot[CE,CO]{30269159 Seminar, Academic Year 2018}
%\renewcommand{\footrulewidth}{1pt}
%---------------------------------------------------------------
\begin{document}
%------------------------Your information------------------------
\begin{flushleft}{
\begin{tabular}{lll}
Title & : & A New Definition of Fractional Derivative \\
		
Author(s) & : & R. Khalil, M. Al Horani, A. Yousef and M. Sababheh\\
Journal & : & Journal of Mathematics, Year 2014, Vol. 264, pp. 65 – 70  \\
Student & : & Miss Sirunya Thanompolkrang    \quad Student ID: 59910295  \\
Advisor & : & Assistant Professor Dr.Duangkamol Poltem\\
\end{tabular}
}\end{flushleft}
\rule{6.27in}{0.75pt}

%=========(Abstract)============
\begin{center}{\subsection*{Abstract}}\end{center}
\hspace*{0.5in}In this work, a new definition of fractional derivative and fractional integral are proposed. The authors obtain many useful theorems, and they also give some applications to fractional differential equations.\\
\\
%=========(Kewwords)============
{\bf Keywords:}  Conformable, Fractional derivative, Fractional integral\\

%=========(Introduction)============
\subsection*{Introduction}
Fractional derivative is as old as calculus. The most popular of fractional derivative are Riemann-Liourille and Caputo [1-4].
In this work , the authors propose a new definition of fractional derivative and fractional integral as follows.
\\\\
\begin{definition} Given a function  $ {f}  :  [0_,\hspace{0.5ex}\infty)\rightarrow \mathbb{R}  $. Then the \enquote{conformable fractional derivative} of $ {f} $ of order $ \alpha $ is defined by 
\begin{equation*}\label{MainEq}
T_{ \alpha }(f)(t)=\lim_{\varepsilon\to 0}\dfrac{f(t+\varepsilon t^{1-\alpha})-f(t)}{\varepsilon} ,\hspace{2ex}\forall t>0,\alpha\in(0_,1).
\end{equation*}
\end{definition} 

If $ {f} $ is $ {\alpha} $ - differentiable in the interval $(0_,a)$ for some$\hspace{1ex}a>0$ and $\lim_{t\to 0^{+}}f^{(\alpha)}(t)  $ exists, then define $ f^{(\alpha)}(0)=\lim_{t\to 0^{+}}f^{(\alpha)}(t). $
\begin{definition} Let $ {\alpha}\in (n_,\hspace{0.5ex}n+1]_, $ and $ {f} $ be an $ {n} $-differentiable at $ {t} $, where $ {t}>0. $ Then the conformable fractional derivative of $ {f} $ of order $ {\alpha} $ is defined as
\begin{equation*}
T_{\alpha}(f)(t)=\lim_{\varepsilon\to 0}\dfrac{f^{(\lceil \alpha \rceil -1)}(t+\varepsilon t^{\lceil \alpha \rceil - \alpha)})-f^{(\lceil \alpha \rceil -1)}(t)}{\varepsilon},
\end{equation*}
\\where $ {\lceil \alpha \rceil} $ is the smallest integer greater than or equal to $ {\alpha} $.
\end{definition} 
\begin{definition} The $ {\alpha} $ - fractional integral of  a function $ {f} $ starting from $ {a\geq0} $ is defined by
\begin{equation*}
I_{\alpha}^{a}(f)(t)=I_{\alpha}^{a}(t^{\alpha -1}f)=\int_{a}^{t}\dfrac{f(x)}{x^{1-\alpha}}dx,
\end{equation*}
\\where the integral is the usual Riemann improper integral, and $ {\alpha\in(0_,1)} $.
\end{definition} 
%=========(Methodology)============
\subsection*{Methodology}
As a consequence of the above definitions, we obtain the following useful theorem.
\begin{theorem} Let $ {\alpha\in (0_,\hspace{0.5ex}1]} $ and $ {f,\,g} $ be $ {\alpha} $ - differentiable at $ a $ point $ t>0 $. Then
\begin{enumerate}
\item$T_{\alpha}(af+bg) = aT_{\alpha}(f)+bT_{\alpha}(g), \forall a,\,b \in \mathbb{R}.$
\item$T_{\alpha}(t^{p})=pt^{p-{\alpha}}, \forall p\in \mathbb{R}.$
\item$T_{\alpha}(\lambda)=0,$ for all constant function $f(t)=\lambda.$
\item$T_{\alpha}(fg)=fT_{\alpha}(g)+gT_{\alpha}(f).$
\item$T_{\alpha}\left( \dfrac{f}{g}\right)  =\dfrac{gT_{\alpha}(f)-fT_{\alpha}(g)}{g^2}, \hspace{1ex}g\neq 0.$
\item \textit {If \hspace{1ex}$f$\hspace{1ex} is \hspace{1ex}differentiable,\hspace{1ex} then} \hspace{1ex} $T_{\alpha}(f)(t)=t^{1-{\alpha}}\dfrac{df}{dt}(t) $.
\end{enumerate}
\end{theorem} 
\begin{theorem} 
$T_{\alpha}I_{\alpha}^{a}(f)(t)=f(t),\hspace{1ex}$ for $ t\geq a,$ where $f$ is any continuous function in the domain of $I_{\alpha}.$
\end{theorem} 
                                  
\subsection*{Results}
The definitions and theorems are applied to solve conformable fractional differential equation as follows.
\begin{example} 
Solve $ y^{(\frac{1}{2})}+y=x^{2}+2x^{(\frac{3}{2})},\,y(0)=0 $.\\The particular solution is $ y(x)=x^{2} $.
\end{example} 
\begin{example} 
Solve $ y^{(\frac{1}{2})}+\sqrt{x}y=xe^{-x} $.
\\The general solution is $ y(x)=\frac{2}{3}x^{(\frac{3}{2})}e^{-x}+ce^{-x} $.
\end{example} 


%=========References APA style===========
\begin{thebibliography}{9}

\bibitem{Miller, K.S.}Miller, K.S. (1993). An Introduction to Fractional Calculus and Fractional Differential  Eqations. New York: J. Wiley and Sons.
\bibitem{Oldham, K.}Oldham, K., \& Spanier, J. (1974). The Fractional Calculus Theory and Applications of Differentiation and Integration of Arbitrary Order. USA: Academic Press.
\bibitem{Kilbas, A.}Kilbas, A., Srivastava, H. \& Trujillo, J. (2006). Theory and Applications of Fractional Differential Equations, in: Math. Studies. New York: North-Holland.
\bibitem{Podlubny, I.}Podlubny, I. (1999). Fractional Differential Equations. USA: Academic Press.

\end{thebibliography}

%
%=========ลงชื่อ=================
\begin{center}
\null\vfill
 \begin{tabular}{ccc}
--------------------------------------------  &~~~~~~~~~~~& --------------------------------------------  \\
(Miss Sirunya Thanompolkrang) && (Assistant Professor Dr.Duangkamol Poltem)\\
Student && Advisor   \\
\end{tabular}\\	 	           
\end{center}
\end{document}


\end{document}