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\fancyhead[CE]{O. Ozturk} 
\fancyhead[CO]{A New Perspective via Fractional Calculus}



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{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (2017), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
‎\vspace*{9mm}
‎
\begin{center}

{\Large \bf 
A New Perspective via Fractional Calculus for the Radial Schr{\"{o}}dinger Equation\\}
 


\let\thefootnote\relax\footnote{\scriptsize Received: April 20, 2017; Accepted: XXXX (Will be inserted by editor)}

{\bf Okkes Ozturk$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small  Bitlis Eren University} \vspace{2mm}


\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} Differintegral theorems are applied to solve some ordinary differential equations and fractional differential equations. By using these theorems, we obtain different results in the fractional differintegral forms. In this paper, we aim to solve the radial Schr{\"{o}}dinger equation under the potential $ V(r)=H/r^{2}-K/r+Lr^{\kappa} $ in $ \kappa=0,-1,-2 $ cases. We also obtain the solutions in the hypergeometric form.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 26A33; 34A08

\noindent{\bf Keywords and Phrases:} Fractional calculus, differintegral theorems, fractional solutions, radial Schr{\"{o}}dinger equation
\end{quotation}}

\section{Introduction}
\label{intro} % It is advised to give each section and subsection a unique label.
As is known, the order of derivative consists of an integer in ordinary calculus; differentiation with integer order is always provided for a favorable function. What's interesting is that a function can be held differentiation with any arbitrary order in fractional calculus that is an area of mathematics that grows out of the traditional definitions of the calculus integral and derivative operators. And, same situation is valid for the integration. So, differentiation and integration are generalized such as differintegral in fractional calculus theory. This theory has an important position in areas of science and engineering such as robot technology, PID control systems, Schr{\"{o}}dinger equation, heat transfer, relativity theory, economy, filtration, controller design, mechanics, optics, modelling and so on \cite{Miller, Oldham, Podlubny, Ross, Yilmazer}.\\
Riemann-Liouville differintegral definitions are, respectively,
\begin{equation}
_aD_{t}^{-\mu}\varphi(t)=[\varphi(t)]_{-\mu}=\frac{1}{\Gamma(\mu)}\int_a^t\frac{\varphi(\omega)}{(t-\omega)^{1-\mu}}d\omega\quad(t>a,\mu>0), \label{1.1}
\end{equation}
and,
\begin{equation}
_aD_{t}^{\mu}\varphi(t)=[\varphi(t)]_{\mu}=\frac{1}{\Gamma(k-\mu)}\frac{d^{k}}{dt^{k}}\int_a^t\frac{\varphi(\omega)}{(t-\omega)^{\mu+1-k}}d\omega, \label{1.2}
\end{equation}
\[ (k-1\leq\mu<k,k\in\mathbf{N}). \]
The Schr{\"{o}}dinger equation has an important place in fractional calculus. In this contex, many scientific works was suggested. For instance, in \cite{Herrmann}, based on the Riesz definition of the fractional derivative the fractional Schr{\"{o}}dinger equation with an infinite well potential is studied by Herrmann. And, Laskin \cite{Laskin2002} presented some properties of the fractional Schr{\"{o}}dinger equation. In \cite{Laskin2000}, the path integrals over the Levy paths are defined and fractional quantum and statistical mechanics have been developed via new fractional path integrals approach. A fractional generalization of the Schr{\"{o}}dinger equation has been found by Laskin. Wang \cite{Wang} expressed fractional Schr{\"{o}}dinger equations with potential and optimal controls. The fractional Schr{\"{o}}dinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well by Luchko in \cite{Luchko}. Bayin \cite{Bayin} showed effective potential approach and presented a free particle solution for the space and time fractional Schr{\"{o}}dinger equation in general coordinates in terms of Fox's H-functions. In \cite{Rozmej}, some applications of a fractional approach to the Schr{\"{o}}dinger equation are discussed by Rozmej. Jeng \cite{Jeng} studied on the one-dimensional infinite square well and presented that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schr{\"{o}}dinger equation for the general fractional parameter. In \cite{Khan}, Khan formed approximate solutions of the time-fractional Schr{\"{o}}dinger equations, with zero and nonzero trapping potential, by homotopy analysis method HAM.\\
In present paper, we deal with radial part of the Schr{\"{o}}dinger equation given by the potential $ V(r)=H/r^{2}-K/r+Lr^{\kappa} $. The radial equation is a second-order homogeneous ordinary differential equation with variable coefficients. And, fractional calculus theorems can be apply to such equations.

\section{Preliminaries}

\begin{definition}
	\label{def1} If $ \varphi(z) $ is analytic and has no branch point inside and on $ C $, where $ C:=\{C^{-},C^{+}\} $, $ C^{-} $ is a contour along the cut joining the points $ z $ and $ -\infty+iIm(z) $, which starts from the point at $ -\infty $, encircles the point $ z $ once counter-clockwise, and returns to the point $ -\infty $, and $ C^{+} $ is a contour along the cut joining the points $ z $ and $ \infty+iIm(z) $, which starts from the point at $ \infty $, encircles the point $ z $ once counter-clockwise, and returns to the point at $ \infty $,
	\[ \varphi_{\mu}(z):=\frac{\Gamma(\mu+1)}{2\pi{i}}\int_{C}\frac{\varphi(t)dt}{(t-z)^{\mu+1}}\quad(\mu\notin\mathbf{Z^{-}}), \]
	and,
	\[ \varphi_{-k}(z):=\lim_{\mu \to -k}\varphi_{\mu}(z)\quad(k\in\mathbf{Z^{+}}), \]
	where $ t\neq{z} $, $ -\pi\leq\arg(t-z)\leq\pi $ for $ C^{-} $ and, $ 0\leq\arg(t-z)\leq2\pi $ for $ C^{+} $, then $ \varphi_{\mu}(z)~~(\mu>0) $ is said to be the fractional derivative of  $ \varphi(z) $ of order $ \mu $ and, $ \varphi_{\mu}(z)~~(\mu<0) $
	is said to be fractional integral of $ \varphi(z) $ of order $ -\mu $, provided that $ \mid \varphi_{\mu}(z) \mid<\infty~~(\mu\in\mathbf{R}) $ \rm\cite{Campos, Nishimoto, Ortigueira}.
\end{definition}

\begin{lemma}[Linearity]
	Let $ \varphi(z) $ and $ \psi(z) $ be analytic and single-valued functions. If $ \varphi_{\mu}(z) $ and $ \psi_{\mu}(z) $ exist, then
	\begin{equation}
	[\alpha\varphi(z)+\beta\psi(z)]_{\mu}=\alpha\varphi_{\mu}(z)+\beta\psi_{\mu}(z),
	\end{equation}
	where $ \alpha $ and $ \beta $ are constants and, $ \mu\in\mathbf{R}, z\in\mathbf{C} $ \rm\cite{Yilmazer}.
\end{lemma}

\begin{lemma}[Index law]
	Let $ \varphi(z) $ be an analytic and single-valued function. If $ (\varphi_{\eta})_{\mu}(z) $ and $ (\varphi_{\mu})_{\eta}(z) $ exist, then
	\begin{equation}
	(\varphi_{\eta})_{\mu}(z)=(\varphi_{\eta+\mu})(z)=(\varphi_{\mu})_{\eta}(z),
	\end{equation}
	where $ \mu,\eta\in\mathbf{R}, z\in\mathbf{C} $ and $ \Big| \frac{\Gamma(\mu+\eta+1)}{\Gamma(\mu+1)\Gamma(\eta+1)} \Big|<\infty $ \rm\cite{Yilmazer}.
\end{lemma}

\begin{lemma}[Generalized Leibniz rule]
	Let $ \varphi(z) $ and $ \psi(z) $ be analytic and single-valued functions. If $ \varphi_{\mu}(z) $ and $ \psi_{\mu}(z) $ exist, then
	\begin{equation}
	[\varphi(z)\psi(z)]_{\mu}=\sum_{k=0}^{\infty}\frac{\Gamma(\mu+1)}{\Gamma(\mu+1-n)\Gamma(k+1)}\varphi_{\mu-k}(z)\psi_{k}(z), \label{2.3}
	\end{equation}
	where $ \mu\in\mathbf{R}, z\in\mathbf{C} $ and $ \Big| \frac{\Gamma(\mu+1)}{\Gamma(\mu+1-n)\Gamma(n+1)} \Big|<\infty $ \rm\cite{Yilmazer}.
\end{lemma}

\begin{remark}
	Let $ \vartheta $ be a constant as $ \vartheta\neq0 $. Then \rm\cite{Yilmazer},
	\begin{equation}
	(e^{\vartheta{z}})_{\mu}=\vartheta^{\mu}e^{\vartheta{z}}\quad(\mu\in\mathbf{R}, z\in\mathbf{C}), \label{2.4}
	\end{equation}
	\begin{equation}
	(e^{-\vartheta{z}})_{\mu}=e^{-i\pi{\mu}}\vartheta^{\mu}e^{-\vartheta{z}}\quad(\mu\in\mathbf{R}, z\in\mathbf{C}), \label{2.5}
	\end{equation}
	\begin{equation}
	(z^{\vartheta})_{\mu}=e^{-i\pi{\mu}}\frac{\Gamma(\mu-\vartheta)}{\Gamma(-\vartheta)}z^{\vartheta-\mu}\quad\Big(\mu\in\mathbf{R}, z\in\mathbf{C}, \Big| \frac{\Gamma(\mu-\vartheta)}{\Gamma(-\vartheta)} \Big|<\infty\Big). \label{2.6}
	\end{equation}
\end{remark}

\begin{remark}
	\begin{equation}
	\Gamma(\mu-k)=(-1)^{k}\frac{\Gamma(\mu)\Gamma(1-\mu)}{\Gamma(k+1-\mu)}\quad(k\in\mathbf{Z}^{+}\cup\{0\}, \mu\in\mathbf{R}). \label{2.7}
	\end{equation}
\end{remark}

\begin{theorem}\label{myth1} 
	Let $ \varphi_{-\mu}\neq0 $ where $ \varphi $ is a given function and, $ M(z;m) $ and $ N(z;n) $ be polynomials in $ z $ of degrees $ m $ and $ n $, respectively, given by
	\begin{equation}
	M(z;m)=\sum_{k=0}^{m}a_{k}z^{m-k}=a_{0}\prod_{j=1}^{m}(z-z_{j})\quad(a_{0}\neq0, m\in\mathbf{N}), \label{2.8}
	\end{equation}
	and,
	\begin{equation}
	N(z;n)=\sum_{k=0}^{n}b_{k}z^{n-k}\quad(b_{0}\neq0, n\in\mathbf{N}). \label{2.9}
	\end{equation}
	Thus, the nonhomogeneous linear ordinary fractional differintegral equation
	\[ M(z;m)\chi_{\eta}(z)+\Bigg[\sum_{k=1}^{m}{\mu \choose k}M_{k}(z;m)+\sum_{k=1}^{n}{\mu \choose k-1}N_{k-1}(z;n)\Bigg]\chi_{\eta-k}(z) \]
	\begin{equation}
	+{\mu \choose k}n!b_{0}\chi_{\eta-n-1}(z)=\varphi(z)\quad(m,n\in\mathbf{N}, \mu,\eta\in\mathbf{R}), \label{2.10}
	\end{equation}
	has a particular solution as follows
	\begin{equation}
	\chi(z)=\Bigg[\Bigg(\frac{\varphi_{-\mu}(z)}{M(z;m)}e^{\sigma(z;m,n)}\Bigg)_{-1}e^{-\sigma(z;m,n)}\Bigg]_{\mu-\eta+1}, \label{2.11}
	\end{equation}
	\[ (z\in\mathbf{C}\backslash\{z_{1},...,z_{m}\}), \]
	where for suitable condition,
	\begin{equation}
	\sigma(z;m,n)=\int^{z}\frac{N(\zeta;n)}{M(\zeta;m)}d\zeta\quad(z\in\mathbf{C}\backslash\{z_{1},...,z_{m}\}), \label{2.12}
	\end{equation}
	confirmed that the second component of Equ. (\ref{2.11}) exists. Moreover, the homogeneous linear ordinary fractional differintegral equation
	\[ M(z;m)\chi_{\eta}(z)+\Bigg[\sum_{k=1}^{m}{\mu \choose k}M_{k}(z;m)+\sum_{k=1}^{n}{\mu \choose k-1}N_{k-1}(z;n)\Bigg]\chi_{\eta-k}(z) \]
	\begin{equation}
	+{\mu \choose k}n!b_{0}\chi_{\eta-n-1}(z)=0\quad(m,n\in\mathbf{N}, \mu,\eta\in\mathbf{R}), \label{2.13}
	\end{equation}
	has solutions as follows
	\begin{equation}
	\chi(z)=\alpha[e^{-\sigma(z;m,n)}]_{\mu-\eta+1}, \label{2.14}
	\end{equation}
	where $ \sigma(z;m,n) $ is given by Equ. (\ref{2.12}) and, $ \alpha $ is an arbitrary constant \rm\cite{Lin}.
\end{theorem}

\section{Main Results}

The radial Schr{\"{o}}dinger equation under the potential $ V(r)=H/r^{2}-K/r+Lr^{\kappa} $ is defined as \cite{Aygun}
\begin{equation}
R_{2}(r)+\frac{2m}{\hbar^{2}}\Big[E-\frac{H}{r^{2}}+\frac{K}{r}-Lr^{\kappa}-\frac{l(l+1)\hbar^{2}}{2mr^{2}}\Big]R(r)=0, \label{3.1}
\end{equation}
where $ H,K $ and $ L $ are positive constants. Now, we get
\begin{equation}
\begin{aligned}
& -\lambda^{2}=\frac{2mE}{\hbar^{2}},\quad \textbf{H}=\frac{2mH}{\hbar^{2}},\quad \textbf{K}=\frac{2mK}{\hbar^{2}}, \\
& \textbf{L}=\frac{2mL}{\hbar^{2}},\quad \rho=l(l+1),\quad \tau=\textbf{H}+\rho. \label{3.2}
\end{aligned}
\end{equation}
And, by substituting (\ref{3.2}) into (\ref{3.1}), we have
\begin{equation}
r^{2}R_{2}(r)-(\lambda^{2}r^{2}-\textbf{K}{r}+\textbf{L}{r^{\kappa+2}}+\tau)R(r)=0. \label{3.3}
\end{equation}
According to the values of $ \kappa=0,-1,-2 $, Equ. (\ref{3.3}) is given by
\begin{equation}
r^{2}R_{2}(r)-[(\lambda^{2}+\textbf{L})r^{2}-\textbf{K}{r}+\tau]R(r)=0\quad(\kappa=0), \label{3.4}
\end{equation}
\begin{equation}
r^{2}R_{2}(r)-[\lambda^{2}r^{2}+(\textbf{L}-\textbf{K})r+\tau]R(r)=0\quad(\kappa=-1), \label{3.5}
\end{equation}
\begin{equation}
r^{2}R_{2}(r)-[\lambda^{2}r^{2}-\textbf{K}{r}+(\textbf{L}+\tau)]R(r)=0\quad(\kappa=-2). \label{3.6}
\end{equation}
In this paper, we use the fractional calculus theorems for (\ref{3.4}), (\ref{3.5}) and (\ref{3.6}) and so, we find particular solutions of the radial Schr{\"{o}dinger equation in the fractional differintegral forms.
	
	\begin{theorem}\label{myth2}
		If $ \mid \varphi_{\mu}(z) \mid\infty~~(\mu\in\mathbf{R}) $ and $ \varphi_{-\mu}\neq0 $, then
		\begin{equation}
		Az^{2}\chi_{2}+Bz\chi_{1}+(Dz^{2}+Ez+F)\chi=\varphi, \label{3.7}
		\end{equation}
		\[ (A,D\neq0, z\in\mathbf{C}\backslash\{0\}, \chi=\chi(z)), \]
		has a particular solution such as:\\
		$ \chi=z^{\nu}e^{\vartheta z}\Bigg \{\Bigg[A^{-1}z^{-(\mu+1)+\frac{2A\nu+B}{A}}e^{2\vartheta z} $ 
		\begin{equation}
		.\Big(z^{-(\nu+1)}e^{-\vartheta z}\varphi \Big)_{-\mu}\Bigg]_{-1}z^{\mu-\frac{2A\nu+B}{A}}e^{-2\vartheta z}\Bigg\}_{\mu-1}, \label{3.8}
		\end{equation}
		where $ \nu, \vartheta $ and $ \mu $ are in the form:
		\begin{equation}
		\nu=\frac{A-B\pm\sqrt{(A-B)^{2}-4AF}}{2A},\quad\vartheta=\pm i\sqrt{\frac{D}{A}}, \label{3.9}
		\end{equation}
		and,
		\begin{equation}
		\mu=\frac{(2A\nu+B)\vartheta+E}{2A\vartheta}. \label{3.10}
		\end{equation}
		Moreover,
		\begin{equation}
		Az^{2}\chi_{2}+Bz\chi_{1}+(Dz^{2}+Ez+F)\chi=0, \label{3.11}
		\end{equation}
		\[ (A,D\neq0, z\in\mathbf{C}\backslash\{0\}, \chi=\chi(z)), \]
		has the particular solution in the form:
		\begin{equation}
		\chi=\alpha z^{\nu}e^{\vartheta z}\Big(z^{\mu-\frac{2A\nu+B}{A}}e^{-2\vartheta z}\Big)_{\mu-1}
		\end{equation}
		where $ \nu $ and $ \vartheta $ are given by Equ. (\ref{3.9}), and $ \mu $ is given by (\ref{3.10}) and, $ \alpha $ is an arbitrary constant \rm\cite{Lin}.
	\end{theorem}
	
	\begin{theorem}
		According to the expression of Theorem \ref{myth2}, radial Schr{\"{o}}dinger equation in Equ. (\ref{3.4}) has the particular solution in the form:
		\begin{equation}
		R(r)=\alpha r^{\nu}e^{\vartheta r}\Big(r^{\mu-2\nu}e^{-2\vartheta r}\Big)_{\mu-1}, \label{3.13}
		\end{equation}
		where
		\[ \nu=\frac{1\pm\sqrt{1+4\tau}}{2},\quad\vartheta=\pm\sqrt{\lambda^{2}+\textbf{L}}, \]
		and,
		\[ \mu=\nu+\frac{\textbf{K}}{2\vartheta}. \]
	\end{theorem}
	
	\begin{theorem}
		According to the expression of Theorem \ref{myth2}, radial Schr{\"{o}}dinger equation in Equ. (\ref{3.5}) has the particular solution in the form:
		\begin{equation}
		R(r)=\alpha r^{\nu}e^{\vartheta r}\Big(r^{\mu-2\nu}e^{-2\vartheta r}\Big)_{\mu-1}, \label{3.14}
		\end{equation}
		where
		\[ \nu=\frac{1\pm\sqrt{1+4\tau}}{2},\quad\vartheta=\pm\lambda, \]
		and,
		\[ \mu=\nu+\frac{\textbf{K}-\textbf{L}}{2\vartheta}. \]
	\end{theorem}
	
	\begin{theorem}
		According to the expression of Theorem \ref{myth2}, radial Schr{\"{o}}dinger equation in Equ. (\ref{3.6}) has the particular solution in the form:
		\begin{equation}
		R(r)=\alpha r^{\nu}e^{\vartheta r}\Big(r^{\mu-2\nu}e^{-2\vartheta r}\Big)_{\mu-1}, \label{3.15}
		\end{equation}
		where
		\[ \nu=\frac{1\pm\sqrt{1+4(\textbf{L}+\tau)}}{2},\quad\vartheta=\pm\lambda, \]
		and,
		\[ \mu=\nu+\frac{\textbf{K}}{2\vartheta}. \]
	\end{theorem}
	
	\begin{theorem}
		Let $ \mid (r^{\mu-2\nu})_{k} \mid<\infty $ and $ \mid \frac{-1}{2\vartheta r} \mid<1 $~~($ r\neq0, k\in\mathbf{Z}^{+}\cup\{0\} $). The solution in the form:
		\[ R(r)=\alpha r^{\nu}e^{\vartheta r}\Big(r^{\mu-2\nu}e^{-2\vartheta r}\Big)_{\mu-1}, \]
		is written as
		\[ R(r)=r^{\mu-\nu}e^{-\vartheta r}{_{2}F_{0}\Big[1-\mu, 2\nu-\mu; \frac{-1}{2\vartheta r}\Big]} \]
		where $ _{2}F_{0} $  is the Gauss hypergeometric function.
	\end{theorem}
	
	\begin{proof}
		By means of Equ. (\ref{2.3}), we have
		\begin{equation}
		R(r)=\alpha r^{\nu}e^{\vartheta r}\sum_{k=0}^{\infty}\frac{\Gamma(\mu)}{\Gamma(\mu-k)k!}(r^{\mu-2\nu})_{k}(e^{-2\vartheta r})_{\mu-1-k}. \label{3.16}
		\end{equation}
		By using (\ref{2.5}), (\ref{2.6}) and (\ref{2.7}), we rewrite Equ. (\ref{3.16}) as follows:
\begin{equation*}
\begin{aligned}
R(r)=\alpha r^{\mu-\nu}e^{-\vartheta r}(2\vartheta e^{-i\pi})^{\mu-1}\\
& \times\sum_{k=0}^{\infty}\frac{\Gamma(k+1-\mu)}{\Gamma(1-\mu)}\frac{\Gamma(k+2\nu-\mu)}{\Gamma(2\nu-\mu)}\frac{1}{k!}\Big(\frac{-1}{2\vartheta r}\Big)^{k}.
\end{aligned}
\end{equation*}
		Then,
		\[ R(r)=r^{\mu-\nu}e^{-\vartheta r}\sum_{k=0}^{\infty}(1-\mu)_{k}(2\nu-\mu)_{k}\frac{1}{k!}\Big(\frac{-1}{2\vartheta r}\Big)^{k}. \]
		where $ 1/\alpha=(2\vartheta e^{-i\pi})^{\mu-1} $.\\
		Finally, we obtain
		\[ R(r)=r^{\mu-\nu}e^{-\vartheta r}{_{2}F_{0}\Big[1-\mu, 2\nu-\mu; \frac{-1}{2\vartheta r}\Big]}. \]
	\end{proof}
	
	\section*{Conclusion}
	In this study, we used fractional calculus theorems for the radial Schr{\"{o}}dinger equation given by the potential $ V(r)=H/r^{2}-K/r+Lr^{\kappa} $. And, we obtained the hypergeometric forms of the fractional solutions.
	
\begin{center}
\begin{thebibliography}{99} % Enter references in alphabetical order and according to the following format.
	
\bibitem{Aygun} M. Aygun, O. Bayrak and I. Boztosun, Solution of the radial Schr{\"{o}}dinger equation for the potential family $ V(r)=A/r^{2}-B/r+Cr^{\kappa} $ using the asymptotic iteration method, \textit{J. Phys. B: At. Mol. Opt. Phys.}, 40 (2007), 537-539.

\bibitem{Bayin} S. Bayin, On the consistency of the solutions of the space fractional Schr{\"{o}}dinger equation, \textit{J. Math. Phys.}, (2012). doi: 10.1063/1.4705268.

\bibitem{Campos} L. M. B. C. Campos, On the solution of some simple fractional differential equations, \textit{Internat. J. Math. and Math. Sci.}, 13(3) (1990), 481-496.

\bibitem{Herrmann}	R. Herrmann, The fractional Schr{\"{o}}dinger equation and the infinite potential well-numerical results using the Riesz derivative, \textit{Gam. Ori. Chron. Phys.}, 1(1) (2013), 1-12.

\bibitem{Jeng} M. Jeng, S.-L.-Y. Xu, E. Hawkins and J. M. Schwarz, On the nonlocality of the fractional Schr{\"{o}}dinger equation, \textit{J. Math. Phys.}, (2010). doi: 10.1063/1.3430552.

\bibitem{Khan} N. A. Khan, M. Jamil and A. Ara, Approximate solutions to time-fractional Schr{\"{o}}dinger equation via homotopy analysis method, \textit{ISRN Mathematical Physics}, (2012). doi: 10.5402/2012/197068.

\bibitem{Laskin2000}	N. Laskin, Fractional quantum mechanics and Levy path integrals, \textit{Phys. Lett. A}, 268 (2000), 298-305.

\bibitem{Laskin2002}	N. Laskin, Fractional Schr{\"{o}}dinger equation, \textit{Phys. Rev.}, E (2002). doi: 10.1103/PhysRevE.66.056108.

\bibitem{Lin} S.-D. Lin, J.-C. Shyu, K. Nishimoto and H. M. Srivastava, Explicit solutions of some general families of ordinary and partial differential equations associated with the Bessel equation by means of fractional calculus, \textit{J. Fract. Calc.}, 25 (2004), 33-45.

\bibitem{Luchko} Y. Luchko, Fractional Schr{\"{o}}dinger equation for a particle moving in a potential well, \textit{J. Math. Phys.}, (2013). doi: 10.1063/1.4777472.

\bibitem{Miller} K. Miller and B. Ross, \textit{An Introduction to the Fractional Calculus and Fractional Differential Equations}, John Wiley and Sons, USA (1993).

\bibitem{Nishimoto} K. Nishimoto, \textit{Fractional Calculus}, Descartes Press, Koriyama (1984-1987-1989-1991-1996).

\bibitem{Oldham} K. Oldham and J. Spainer, \textit{The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order}, Academic Press, USA (1974). 

\bibitem{Ortigueira} M. D. Ortigueira, \textit{Fractional Calculus for Scientists and Engineers}, Springer Netherlands, Heidelberg (2011).

\bibitem{Podlubny} I. Podlubny, \textit{Fractional Differential Equations}, Academic Press, USA (1999).

\bibitem{Ross} B. Ross, \textit{Fractional Calculus and Its Applications}, Conference Proceeding held at the  University of New Haven, Springer, USA (1975).

\bibitem{Rozmej} P. Rozmej, On Fractional Schr{\"{o}}dinger Equation, \textit{CMST}, 16(2) (2010), 191-194.

\bibitem{Wang}	J. Wang, Y. Zhou and W. Wei, Fractional Schr{\"{o}}dinger equations with potential and optimal controls, \textit{Nonlinear Anal. Real World Appl.}, 13(6) (2012), 2755-2766.

\bibitem{Yilmazer} R. Yilmazer and O. Ozturk, Explicit solutions of singular differential equation by means of fractional calculus operators, \textit{Abstr. Appl. Anal.}, (2013). doi: 10.1155/2013/715258.

\end{thebibliography}
\end{center}



{\small

\noindent{\bf Okkes Ozturk}

\noindent Department of Mathematics

\noindent Assistant Professor of Mathematics

\noindent Bitlis Eren University

\noindent Bitlis, Turkey

\noindent E-mail: oozturk27@gmail.com}\\




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