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\fancyhead[CE]{H. Azizi} 
\fancyhead[CO]{ِDu Fort-Frankel scheme for the variable order time fractional diffusion equation}



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{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
\vspace*{9mm}

\begin{center}

{\Large \bf 
Du Fort-Frankel scheme for the variable order time fractional diffusion equation\\}
%{\bf Do You Have a Subtitle? \\ If so, Write It Here} 


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

{\bf H. Azizi}\vspace*{-2mm}\\
\vspace{2mm} {\small  Department of Mathematics, Taft Branch, Islamic Azad University, Taft, Iran} \vspace{2mm}

%{\bf  Second Author$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
%\vspace{2mm} {\small   Enter affiliation here} \vspace{2mm}

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{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} In this paper, we consider the variable order time fractional diffusion equations in a finite domain. A Du Fort-Frankel finite difference method is introduced to solve these equations. The stability condition for this scheme is discussed and proved via the approach of Fourier analysis. Numerical examples are prepared to illutrate that the numerical method is computationally efficient.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 65M06, 35A25

\noindent{\bf Keywords and Phrases:} Variable order time fractional diffusion equation, Du Fort-Frankel method, Stability, Fourier analysis
\end{quotation}}

\section{Introduction}
\label{intro} % It is advised to give each section and subsection a unique label.
Scientific problems related to the field of fractional calculus have become very important in recent years. Description of the memory and hereditary properties of various materials and processes are the main advantages of fractional derivatives \cite{Abu,Achar,Babaei,Ganji,Hosseini2,Jafari1,Jafari2,Kaabar1,Kaabar2,Leda Galue,Matar,Rezapour,Sadeghi,Tuana}.\\
Many dynamic processes have the behavior of time and space fractional derivatives therefore, it is crucial for researchers to develop the concept of variable-order calculus. Recently, variable-order calculus has been utilized in many various fields such as geographic data  \cite{Cooper}, viscoelastic mechanics \cite{Coimbra}, signal and confirmation \cite{Tseng}.\\
Samko and Ross \cite{Samko} firstly suggested the perception of variable order operators that allow the order to be a function of place, time, or some other variable rather than an arbitrary order constant \cite{Lorenzo, Ramirez}.\\
Compared with the fractional variable order operators and the constant order operators which based on their non-stationary power-law kernel is more careful to describe these complex physical processes and systems  \cite{Sun1, Sun2}. Afterward, Lorenzo and Hartley \cite{Lorenzo} investigated another kinds of variable order fractional operator definitions and made some theoretical studies by the iterative Laplace transform. These probes show that the variable-order operator is a new advancement of learning in science \cite{Bouazza,Ganji2,Ganji3,Hosseini1}. \\
Many various numerical methods have been suggested to solve variable order time fractional diffusion equations \cite{hajipour,Hosseini3, shen, wang1, wang2, zeng, zhao, zheng, zhuang}.\\
The aim of this article is to utilize a finite difference scheme based on Du Fort-Frankel for solving variable order time fractional diffusion equation
\begin{equation} \label{eq1}
{}_{0} D_{t}^{\alpha (x,t)} \, u(x,t)=\frac{\partial ^{2} u(x,t)}{\partial x^{2} } +p(x,t),\, \, \, x\in (0,L],\, \, t\in (0,T],
\end{equation}
with initial and boundary conditions
\begin{equation} \label{eq2}
u(x,0)=f(x),\, x\in [0,L],
\end{equation}
\begin{equation} \label{eq3}
u(0,t)=g(x),\, \, u(L,t)=h(t),\, \, t\in [0,T],
\end{equation}
where   $0<\alpha(x,t)\leq1$ , $p(x,t)$ is a source term and ${}_{0} D_{t}^{\alpha (x,t)}$ denotes the Caputo variable order time fractional derivative defined by \cite{shen}
\begin{equation} \label{eq4}
{}_{0} D_{t}^{\alpha (x,t)} \, u(x,t)=\frac{1}{\Gamma (1-\alpha (x,t))} \, \int _{0}^{t}(t-s)^{-\alpha (x,t)} \, \frac{\partial u(x,s)}{\partial s} \, ds.
\end{equation}
It should be noted that the Du Fort and Frankel scheme first published in 1953 \cite{du fort} and used for solving a lot of equations \cite{gottlieb, lai, wu}.

\section{ Description of the method}
\label{sec:2}
In this section, the process of solving the variable order time fractional diffusion equations is considered. For describing the Du Fort-Frankel method we suppose that
$$ x_{i}=ih, i=0, 1, ..., N, $$
$$ t_{n}=nk, n=0,1, ..., M. $$
The second-order spatial derivative can be approximated by the Du Fort-Frankel finite difference:

\begin{equation} \label{eq5}
\frac{\partial ^{2} u(x_{i} ,t_{n} )}{\partial x^{2} } =\frac{u(x_{i+1} ,t_{n-1} )-u(x_{i} ,t_{n} )-u(x_{i} ,t_{n-2} )+u(x_{i-1} ,t_{n-1} )}{h^{2} }+O(h^{2})+O(k^{2})+O(\frac{k^{2}}{h^{2}}).
\end{equation}
Now we discretize the variable order time fractional derivative as
\begin{equation} \label{eq6}
D_{t}^{\alpha (x_{i},t_{n})} \, u(x_{i},t_{n})=\frac{k^{-\alpha (x_{i} ,t_{n} )} }{\Gamma (2-\alpha (x_{i} ,t_{n} ))} \sum _{j=1}^{n}b_{j} (x_{i} ,t_{n} )\left[u(x_{i} ,t_{n-j+1} )-u(x_{i} ,t_{n-j} )\right],
\end{equation}
where $b_{j}(x_{i},t_{n})=j^{1-\alpha (x_{i},t_{n})}-(j-1)^{1-\alpha (x_{i},t_{n})}$.\\
We supposed that $u(x_{i},t_{n})=u^{n}_{i}$, $\alpha(x_{i} ,t_{n})=\alpha^{n}_{i}$, $b_{j}(x_{i} ,t_{n})=b_{j}^{i,n}$ and $p(x_{i} ,t_{n})=p_{i}^{n}$ therefore, from (\ref{eq5}) and (\ref{eq6}) we obtain the following approximate method for Eq. (\ref{eq1}):
\begin{equation} \label{eq7}
\frac{k^{-\alpha_{i}^{n}}}{\Gamma(2-\alpha_{i}^{n})}\lbrace (u_{i}^{n} -u_{i}^{n-1} )+\sum _{j=2}^{n}b_{j}^{i,n} \left[u_{i}^{n-j+1} -u_{i}^{n-j} \right] \rbrace =\frac{u_{i+1}^{n-1} -u_{i}^{n} -u_{i}^{n-2} +u_{i-1}^{n-1} }{h^{2} } +p_{i}^{n},
\end{equation}
if put $\gamma _{i}^{n} =1+\frac{h^{2} }{k^{\alpha _{i}^{n} } \, \Gamma (2-\alpha _{i}^{n} )}$ hence we have 
\begin{equation} \label{eq8}
\gamma _{i}^{n} u_{i}^{n} =(\gamma _{i}^{n} -1)u_{i}^{n-1} +(1-\gamma _{i}^{n} )\sum _{j=2}^{n}b_{j}^{i,n} \left[u_{i}^{n-j+1} -u_{i}^{n-j} \right] +u_{i+1}^{n-1} -u_{i}^{n-2} +u_{i-1}^{n-1} +h^{2} \, p_{i}^{n},
\end{equation}
for $i=1,2,...,N-1$ and $n=1,2,...,M$.\\
Expression $\sum _{j=2}^{n}b_{j}^{i,n} \left[u_{i}^{n-j+1} -u_{i}^{n-j} \right]$ in (\ref{eq8}) can be expand and rework by the following approch:
\begin{equation}\label{eq9}
\begin{array}{l} {\sum _{j=2}^{n}b_{j}^{i,n} \left[u_{i}^{n-j+1} -u_{i}^{n-j} \right] } \\ {=b_{2}^{i,n} (u_{i}^{n-1} -u_{i}^{n-2} )+b_{3}^{i,n} (u_{i}^{n-2} -u_{i}^{n-3} )+\ldots +b_{n-1}^{i,n} (u_{i}^{2} -u_{i}^{1} )+b_{n}^{i,n} (u_{i}^{1} -u_{i}^{0} )} \\ {=b_{2}^{i,n} \, u_{i}^{n-1} +(b_{3}^{i,n} -b_{2}^{i,n} )u_{i}^{n-2} +(b_{4}^{i,n} -b_{3}^{i,n} )u_{i}^{n-3} +...+(b_{n-1}^{i,n} -b_{n-2}^{i,n} )u_{i}^{2} +(b_{n}^{i,n} -b_{n}^{i,n} )u_{i}^{1} -b_{n}^{i,n} \, u_{i}^{0} } \\ {=b_{2}^{i,n} \, u_{i}^{n-1} -b_{n}^{i,n} \, u_{i}^{0} +\sum _{j=2}^{n-1}d_{j+1}^{i,n} \, u_{i}^{n-j}  }, \end{array}\ 
\end{equation}
where $d_{i}^{i,n}=b_{nj}^{i,n}-b_{j-1}^{i,n}$. Now from (\ref{eq8}) and (\ref{eq9}) we obtain
\begin{equation}\label{eq10}
\begin{array}{l} {\gamma _{i}^{n} u_{i}^{n} =(b_{2}^{i,n} -1)(1-\gamma _{i}^{n} )\, u_{i}^{n-1} +u_{i+1}^{n-1} +u_{i-1}^{n-1} -u_{i}^{n-2} } \\ {\, \, \, \, \, \, \, \, \, \, \, \,\,\,\,
+(1-\gamma _{i}^{n} )\sum _{j=2}^{n-1}d_{j+1}^{i,n} \, u_{i}^{n-j}  +(\gamma _{i}^{n} -1)b_{n}^{i,n} \, u_{i}^{0} +h^{2} \, p_{i}^{n} }, \end{array}\
\end{equation}
for $i=1,2,...,N-1$ and $n=1,2,...,M$.
Finally (\ref{eq10}) can be rewrite by the following expression:
\begin{equation}\label{eq11}
\left\{\begin{array}{cc} {\gamma _{i}^{1} u_{i}^{1} =(b_{2}^{i,1} -1)(1-\gamma _{i}^{1} )\, u_{i}^{0} +u_{i+1}^{0} +u_{i-1}^{0} -u_{i}^{-1} +h^{2} \, p_{i}^{1} }, & {n=1}, \\ {\begin{array}{l} {\gamma _{i}^{n} u_{i}^{n} =(b_{2}^{i,n} -1)(1-\gamma _{i}^{n} )\, u_{i}^{n-1} +u_{i+1}^{n-1} +u_{i-1}^{n-1} -u_{i}^{n-2} } \\ {\, \, \, \, \, \, \, \, \, \, \,\,\,\,\, +(1-\gamma _{i}^{n} )\sum _{j=2}^{n-1}d_{j+1}^{i,n} \, u_{i}^{n-j}  +(\gamma _{i}^{n} -1)b_{n}^{i,n} \, u_{i}^{0} +h^{2} \, p_{i}^{n} ,} \end{array}} & {2\le n\le M}. \end{array}\right.
\end{equation}

\section{Stability analysis of the scheme}
In this section, the technique of Fourier analysis utilizes for discussing the stability of the approximate method (\ref{eq11}).
Let
$$\hspace{-70mm} w_{l}^{n}=u_{l}^{n}-U_{l}^{n}, $$ where $U_{l}^{n} $ is approximate solution in $(x_{l},t_{n})$ for $n=1,2,...,M$, $l=1,2,...,N-1$ and
$$\hspace{-50mm} w^{n}=\left[  w_{1}^{n},w_{2}^{n},...,w_{N-1}^{n} \right]^{T}.  $$
Consider the following equation
\begin{equation}\label{eq12}
 \left\{\begin{array}{cc} {\gamma _{l}^{1} w_{l}^{1} =(b_{2}^{l,1} -1)(1-\gamma _{l}^{1} )\, w_{l}^{0} +w_{l+1}^{0} +w_{l-1}^{0} -w_{l}^{-1}  }, & {n=1}, \\ {\begin{array}{l} {\gamma _{l}^{n} w_{l}^{n} =(b_{2}^{l,n} -1)(1-\gamma _{l}^{n} )\, w_{l}^{n-1} +w_{l+1}^{n-1} +w_{l-1}^{n-1} -w_{l}^{n-2} } \\ {\, \, \, \, \, \, \, \, \, \, \, \,\,\,\, +(1-\gamma _{l}^{n} )\sum _{j=2}^{n-1}d_{j+1}^{l,n} \, w_{l}^{n-j}  +(\gamma _{l}^{n} -1)b_{n}^{l,n} \, w_{l}^{0}  } \end{array}}, & {2\le n\le M} \end{array}\right.
\end{equation}

For $n=1,2,...,M$ we define the following grid function:\\ \\
\noindent  $ w^{n} (x)=\left\{\begin{array}{cc} {w_{l}^{n} }, & {x_{l} -\frac{h}{2} <x\le x_{l} +\frac{h}{2} }, \\ \\ {0}, & {0\le x\le \frac{h}{2} \, or\, \, L-\frac{h}{2} <x\le L}. \\ \end{array}\right. $\\ \\
Therefore, $w^{n} (x)$ can be expanded in a Fourier series:\\
$$\hspace{-18mm} w^{n} (x)=\sum _{m=-\infty }^{+\infty }\delta _{n} (m)\, e^{\frac{i2\pi mx}{L} } \, \, \,  ,\, n=1,2,...,M, \\ \\$$
where\\
 $$\hspace{-40mm} \delta _{n} (m)=\frac{1}{L} \int _{0}^{L}w^{n} (x) \, \, e^{\frac{i2\pi mx}{L} } dx.$$
It can be proved that \cite{chen}
\begin{equation}\label{norm2}
 \left\| w^{n} \right\| _{2}^{2} =\sum _{m=-\infty }^{+\infty }\left|\delta _{n} (m)\right|^{2}.
\end{equation}
Suppose that the solution of the (\ref{eq12}) has the form\\
$w_{l}^{n}=\delta_{n}e^{i\sigma lh},$
where $\sigma=\frac{2\pi m}{L}$.\\
Substituting the above expression into (\ref{eq12}) gives\\
for $n=1$:
\begin{equation}\label{eq13}
(\gamma _{l}^{1}+1)\delta_{1}e^{i\sigma lh}=(b_{2}^{l,1} -1)(3-\gamma _{l}^{1})\delta_{0}e^{i\sigma lh}+\delta_{0}e^{i\sigma (l+1)h}+\delta_{0}e^{i\sigma (l-1)h},        
\end{equation}
and for $n=2,3,..,M$
\begin{equation}\label{eq14}
\begin{array}{l} {\gamma _{l}^{n} \, \delta _{n} \, e^{i\sigma lh} =(b_{2}^{l,n} -1)(1-\gamma _{l}^{n} )\delta _{n-1} \, e^{i\sigma lh} +\delta _{n-1} \, e^{i\sigma (l+1)h} +\delta _{n-1} e^{i\sigma (l-1)h} } \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, -\delta _{n-2} \, e^{i\sigma lh} +(1-\gamma _{l}^{n} )\sum _{j=2}^{n-1}d_{j+1}^{l,n} \, \delta _{n-j} \, e^{i\sigma lh}  +(\gamma _{l}^{n} -1)b_{n}^{l,n} \delta _{0} e^{i\sigma lh} }. \end{array}
\end{equation}

\begin{lemma}\label{lem1}
Let $\delta_{n}$ be the solution of (\ref{eq13}, \ref{eq14}), then
$$\vert \delta _{n} \vert \leq \vert \delta _{0} \vert, \ \ \ n=1,2,..,M.$$
\end{lemma}
\begin{proof}
For $n=1$, in view of (\ref{eq13}) , we have
$$\delta _{1} =\frac{(b_{2}^{l,1} -1)(3-\gamma _{l}^{1} )+2\cos (\sigma h)}{1+\gamma _{l}^{1} } \, \delta _{0}.
$$
Therefore, we obtain
\begin{equation*}
\vert    \delta _{1} \vert =\vert \frac{(b_{2}^{l,1} -1)(3-\gamma _{l}^{1} )+2\cos (\sigma h)}{1+\gamma _{l}^{1} } \, \delta _{0}      \vert \leq \delta _{0}.
\end{equation*}
Now suppose that
$$  \vert       \delta _{n}    \vert  \leq     \vert     \delta _{0}     \vert , \ \ \ \ \  m=2,3,...,n-1, 
$$
by  mathematical induction and view of (\ref{eq14}) show that $  \vert       \delta _{n}    \vert  \leq     \vert     \delta _{0}     \vert ,   
$ for $m=n.$
\begin{equation*}
\begin{array}{l} {\left|\gamma _{l}^{n} \, \delta _{n} \right|\, =\left|\begin{array}{l} {(b_{2}^{l,n} -1)(1-\gamma _{l}^{n} )\delta _{n-1} \, +\delta _{n-1} \, e^{i\sigma h} +\delta _{n-1} e^{-i\sigma h} } \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, -\delta _{n-2} \, +(1-\gamma _{l}^{n} )\sum _{j=2}^{n-1}d_{j+1}^{l,n} \, \delta _{n-j} \,  +(\gamma _{l}^{n} -1)b_{n}^{l,n} \delta _{0} } \end{array}\right|} \\ \\ { \le \left|(b_{2}^{l,n} -1)(1-\gamma _{l}^{n} )+2\cos (\sigma h)-1+(1-\gamma _{l}^{n} )\sum _{j=2}^{n-1}d_{j+1}^{l,n} \,  +(\gamma _{l}^{n} -1)b_{n}^{l,n} \right|\left|\delta _{0} \right|}, \end{array}
\end{equation*}
then we obtain
\begin{equation*}
\begin{array}{l} {\left|\delta _{n} \right|\le } \\ {\, \, \, \, \, \, \, \, \, \, \, \, \left|\frac{(1-\gamma _{l}^{n} )\overbrace{\left[(b_{2}^{l,n} -1)+\sum _{j=2}^{n-1}d_{j+1}^{l,n} \,  -b_{n}^{l,n} \right]}^{=1}-\sin ^{2} (\frac{\sigma h}{2} )}{\gamma _{l}^{n} } \right|\le \left|\delta _{0} \right|}. \end{array}
\end{equation*}
The proof of Lemma (\ref{lem1}) is completed.
\end{proof}\\
Pursuant to  Lemma  (\ref{lem1}) and (\ref{norm2}), it can be found that the solution of Eq. (\ref{eq12}) satisfies
$$  \Vert w^{n} \Vert _{2} \leq \Vert  w^{0}  \Vert _{2}, \ \ \ \ n=1,2,...,M.
 $$
Hence, we obtain the following result:
\begin{theorem}
The approximate method (\ref{eq11}) is unconditionally stable.
\end{theorem}

\section{Numerical examples}
\label{sec:4}
In this section of the paper, we present two examples in order to show the ability and
efficiency of the proposed scheme. Also, the results of the suggested method and other schemes are compared.  

\begin{example}
Consider the following variable order time fractional diffusion equation: \cite{Sun3}
\begin{equation}\label{eq16}
\left\{\begin{array}{c} {{}_{0} D_{t}^{\alpha (x,t)} \, u(x,t)=K\frac{\partial ^{2} u(x,t)}{\partial x^{2} } +p(x,t),\, \, \, x\in (0,L],\, \, t\in (0,T],} \\ {u(x,0)=0\, ,\, \, \, \, \, \, \, \, x\in [0,L],\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {u(0,t)=u(L,t)=0,\, \, \, \, \, \, \, t\in [0,T],\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \end{array}\right.
\end{equation}
where $0<\alpha (x,t) \leq 1$ for $\forall (x,t)$ and 
\begin{equation}\label{eq17}
p(x,t)=\frac{2}{\Gamma (3-\alpha (x,t))} t^{2-\alpha (x,t)} sin(\frac{x\pi}{L})+\frac{K\pi^{2}t^{2}}{L^{2}} sin(\frac{x\pi}{L}).
\end{equation}
The exact solution of this equation can be stated as
\begin{equation}\label{eq18}
u(x,t)=t^{2}sin(\frac{x\pi}{L}).
\end{equation}
In this example we supposed that $k=0.01$, $L=10$ and $T=0.5$. Comparison of the absolute errors of the proposed method and the method of \cite{Sun3} for $\alpha (x,t) =0.8$, $\alpha (x,t)=0.8+\frac{0.2t}{T}$ and $\alpha (x,t)=0.8+\frac{0.2tx}{LT}$ is provided in Table \ref{tab1}-\ref{tab3} respectively.

\begin{table}
\centering
% table caption is above the table
\caption{Absolute errors of the proposed method and method of \cite{Sun3} for constant-order
fractional diffusion equation at $x = 5$. The time step length
$k=0.01$, the space step size $h=0.1$ and the time fractional
$\alpha (x,t) =0.8$.       }
\label{tab1}

\begin{tabular}{|p{0.4in}|p{0.9in}|p{0.9in}|p{1.3in}|p{1.0in}|} \hline 
\textbf{Time} & \textbf{Explicit method} & \textbf{Implicit method} & \textbf{Crank-Nicholson method} & \textbf{Proposed method} \\ \hline 
t=0.1 & 0.007E-2 & 0.526E-3 & 0.465E--3 & 3.224E-5 \\ \hline 
t=0.2 & 0.314E-2 & 1.080E-3 & 0.919E--3 & 5.640E-5 \\ \hline 
t=0.3 & 1.114E-2 & 1.836E-3 & 1.167E--3 & 7.811E-5 \\ \hline 
t=0.4 & 2.556E-2 & 2.932E-3 & 1.109E--3 & 9.848E-5 \\ \hline 
t=0.5 & 4.766E-2 & 4.434E-3 & 0.620E--3 & 1.783E-5 \\ \hline 
\end{tabular}

\end{table} 

\begin{table}
\centering
\caption{Absolute errors of the proposed method and method of \cite{Sun3} for variable order time
fractional diffusion equation at $x = 5$. The time step length
$k=0.01$, the space step size $h=0.1$ and 
$\alpha (x,t)=0.8+\frac{0.2t}{T}$.       }
\label{tab2}

\begin{tabular}{|p{0.4in}|p{0.9in}|p{0.9in}|p{1.3in}|p{1.0in}|} \hline 
\textbf{Time} & \textbf{Explicit method} & \textbf{Implicit method} & \textbf{Crank-Nicholson method} & \textbf{Proposed method} \\ \hline 
t=0.1 & 0.012E-2 & 0.574E-3 & 0.418E-3 & 3.775E-5 \\ \hline 
t=0.2 & 0.356E-2 & 1.304E-3 & 0.698E-3 & 7.623E-5 \\ \hline 
t=0.3 & 1.257E-2 & 2.453E-3 & 0.568E-3 & 1.279E-5 \\ \hline 
t=0.4 & 2.898E-2 & 4.282E-3 & 0.233E-3 & 1.937E-4 \\ \hline 
t=0.5 & 5.425E-2 & 7.118E-3 & 2.027E-3 & 2.821E-4 \\ \hline 
\end{tabular}

\end{table} 

\begin{table}
\centering
\caption{Absolute errors of the proposed method and method of \cite{Sun3} for variable order time
fractional diffusion equation at $x = 5$. The time step length
$k=0.01$, the space step size $h=0.1$ and 
$\alpha (x,t)=0.8+\frac{0.2tx}{LT}$.       }
\label{tab3}

\begin{tabular}{|p{0.4in}|p{0.9in}|p{0.9in}|p{1.3in}|p{1.0in}|} \hline 
\textbf{Time} & \textbf{Explicit method} & \textbf{Implicit method} & \textbf{Crank-Nicholson method} & \textbf{Proposed method} \\ \hline 
t=0.1 & 0.009E-2 & 0.550E-3 & 0.442E-3 & 3.469E-5 \\ \hline 
t=0.2 & 0.335E-2 & 1.188E-3 & 0.813E-3 & 6.571E-5 \\ \hline 
t=0.3 & 1.184E-2 & 2.132E-3 & 0.885E-3 & 9.984E-5 \\ \hline 
t=0.4 & 2.722E-2 & 3.556E-3 & 0.484E-3 & 1.346E-4 \\ \hline 
t=0.5 & 5.085E-2 & 5.650E-3 & 0.578E-3 & 1.812E-4 \\ \hline 
 
\end{tabular}
\end{table} 

\end{example}




\begin{example}
Consider the following variable order time fractional diffusion equation: \cite{shen}
\begin{equation}\label{eq16}
\left\{\begin{array}{c} {{}_{0} D_{t}^{\alpha (x,t)} \, u(x,t)=\frac{\partial ^{2} u(x,t)}{\partial x^{2} } +p(x,t),\, \, \, x\in (0,1],\, \, t\in (0,1],} \\ {u(x,0)=10x^{2}(1-x)\, ,\, \, \, \, \, \, \, \, x\in [0,1],\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \\ {u(0,t)=u(1,t)=0,\, \, \, \, \, \, \, t\in [0,1],\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, } \end{array}\right.
\end{equation}
where $\alpha (x,t)=\frac{2+sin(xt)}{4} $ and 
\begin{equation}\label{eq17}
p(x,t)=20x^{2} (1-x)\left[\frac{t^{2-\alpha (x,t)} }{\Gamma (3-\alpha (x,t))} +\frac{t^{1-\alpha (x,t)} }{\Gamma (2-\alpha (x,t))} \right]-20(t+1)^{2} (1-3x).
\end{equation}
The exact solution is
\begin{equation}\label{eq18}
u(x,t)=10x^{2}(1-x)(t^{2}+1)^{2}.
\end{equation}
In table \ref{tab4}, comparison of the absolute errors of the proposed method and the method of \cite{shen} is provided.

\begin{table}
\centering
\caption{The error, numerical solution and exact solution, when $t=1, h=0.1, k=0.001$.       }
\label{tab4}

\begin{tabular}{|p{1.4in}|p{1.9in}|p{0.9in}|p{1.3in}|p{1.0in}|} \hline 
\textbf{Space (x${}_{i}$)} & \textbf{method of ref [12]} & \textbf{Present method}  \\ \hline 
0.1000 & 0.00002996 & 0.00004895 \\ \hline 
0.2000 & 0.00005972 & 0.00005532 \\ \hline 
0.3000 & 0.00008803 & 0.00007239 \\ \hline 
0.4000 & 0.00011251 & 0.00006591 \\ \hline 
0.5000 & 0.00012981 & 0.00001013 \\ \hline 
0.6000 & 0.00013595 & 0.00001771 \\ \hline 
0.7000 & 0.00012705 & 0.00001083 \\ \hline 
0.8000 & 0.00010048 & 0.00002839 \\ \hline 
0.9000 & 0.00005643 & 0.00008319 \\ \hline 
\end{tabular}
 \end{table} 

\end{example}

\section{Conclusion}
\label{sec:5}
In this paper, Du Fort-Frankel finite difference method for the variable order time fractional diffusion equation 
has been proposed. The stability of the numerical scheme has been considered by the technique of Fourier analysis. Two numerical examples have been given. The results of the proposed method and some other methods have been compared and the results have indicated the effectiveness of the theoretical analysis. All things considered, the proposed method can be used for other equations in the fields of physics and engineering.



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\begin{thebibliography}{99} % Enter references in alphabetical order and according to the following format.

\bibitem{Abu} M. Abu-Shady, M. K. A. Kaabar, A generalized definition of the fractional derivative with applications. Mathematical problems in engineering, 9 pages (2021).

\bibitem{Achar} S. J. Achar, C. Baishya, M. K. A. Kaabar, Dynamics of the worm transmission in wireless sensor network in the framework of fractional derivatives, Math. Methods Appl. Sci, (2021). DOI: 10.1002/mma.8039.

\bibitem{Babaei} A. Babaei, H. Jafari, A. Liya, Mathematical models of HIV/AIDS and drug addiction in prisons. Eur. Phys. J. Plus, 135, 395 (2020).

\bibitem{Bouazza} Z. Bouazza, S. Etemad, M. S. Souid, S. Rezapour, F. Martínez, M. K. A. Kaabar, A Study on the Solutions of a Multiterm FBVP of variable order, Journal of Function Spaces, 2021, 9 pages (2021).

\bibitem{chen} C. M. Chen, F. Liu, I. Turner, V. Anh , A Fourier method for the fractional diffusion equation
describing sub-diffusion, J. Comput. Phys. 227, 886-897 (2007)

\bibitem{Cooper} G. R. J. Cooper, D. R. Cowan. Filtering using variable order vertical derivatives, Computers and Geosciences 30, no. 5 (2004) 455-459.

\bibitem{Coimbra} Carlos FM. Coimbra, Mechanics with variableorder differential operators, Annalen der Physik 12, no. 1112 (2003) 692-703.

\bibitem{du fort} E. C. Du Fort and S. P. Frankel. Stability conditions in the numerical treatment of parabolic differential equations. Mathematical tables and other aids to computation, 7(43) 135–152, (1953).

\bibitem{Ganji2} R.M. Ganji, H. Jafari, D. Baleanu, A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel, Chaos, Solitons and fractal, 130, 109405 (2020).

\bibitem{Ganji} R.M. Ganji, H. Jafari, M. Kgarose, A. Mohammadia, Numerical solutions of time-fractional Klein-Gordon equations by clique polynomials, Alexandria Engineering Journal, 60,  4563-4571 (2021).

\bibitem{Ganji3} R.M. Ganji, H. Jafari, S. Nemati, A new approach for solving integro-differential equations of variable order, Journal of Computational and Applied Mathematics, 379, 112946 (2020).

\bibitem{gottlieb}	D. Gottlieb and B. Gustafsson, Generalized du fort-frankel methods for parabolic initial-boundary value problems. SIAM Journal on Numerical Analysis, 13(1) 129-144, (1976).


\bibitem{hajipour} M. Hajipour, A. Jajarmi, D. Baleanu, H. Sun, On an accurate discretization of a variable-order fractional reaction-diffusion equation. Commun. Nonlinear Sci. Numer. Simulat. 69, 119-133 (2019).

\bibitem{Hosseini1} V.R. Hosseini, M. Koushki, WN. Zou, The meshless approach for solving 2D variable-order time-fractional advection–diffusion equation arising in anomalous transport, Engineering with Computers (2021). https://doi.org/10.1007/s00366-021-01379-7.

\bibitem{Hosseini2} V.R. Hosseini, M. Remazani, W. Zou, S. Banihashemi, Stochastic model for multi-term time-fractional diffusion equations with noise. Thermal Science, 25, 287-293 (2021).

\bibitem{Hosseini3} V.R. Hosseini, F. Yousefi. WN. Zou,  The numerical solution of high dimensional variable-order time fractional diffusion equation via the singular boundary method, Journal of Advanced Research, 32, 73-84 (2021).

\bibitem{Jafari1} H. Jafari, A new general integral transform for solving integral equations, Journal of Advanced Research, 32, 133-138 (2021).

\bibitem{Jafari2} H. Jafari, A. Babaei, S. Banihashemi, A Novel Approach for Solving an Inverse Reaction–Diffusion–Convection Problem, J Optim Theory Appl, 183, 688–704 (2019).

\bibitem{Kaabar1} M. K. A. Kaabar, F. Martínez, J. Francisco Gómez-Aguilar, B.  Ghanbari, M. Kaplan, H. Günerhan, New approximate analytical solutions for the nonlinear fractional Schrödinger equation with second‐order spatio‐temporal dispersion via double Laplace transform method, Mathematical Methods in the Applied Sciences, 44, 11138-11156 (2021).

\bibitem{Kaabar2} M. K. A. Kaabar, M. Shabibi, J. Alzabut, S. Etemad, W. Sudsutad, F. Martínez, S. Rezapour, Investigation of the fractional strongly singular thermostat model via fixed point techniques. Mathematics, 9, 2298 (2021).

\bibitem{Leda Galue} S.L. Kalla Leda Galue, B.N. Al-Saqabi, Fractional extensions of the temperature field problems in oil strata, Appl. Math. Comput. 186 (2007) 35-44.

\bibitem{lai}	M. C. Lai, C.Y. Huang, and T.S. Lin, A simple Du fort-Frankel-type scheme for the Gross-Pitaevskii equation of BoseEinstein condensates on different geometries, Numerical Methods for Partial Differential Equations, vol.20, no. 4, 624-638, (2004).

\bibitem{Lorenzo} C. Lorenzo, T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam. 29, 57-98 (2002).


\bibitem{Matar} M.M. Matar, M.I. Abbas, J. Alzabut, M.K.A. Kaabar, S. Etemad, S. Rezapour, Investigation of the p-Laplacian nonperiodic nonlinear
boundary value problem via generalized Caputo fractional
derivatives, Advances in Difference Equations, 68, 1-18 (2021).


\bibitem{Ramirez} L.E.S. Ramirez, C.F.M. Coimbra, A variable order constitutive relation for viscoelasticity, Ann. Phys. (Leipzig) 16, 543-552 (2007). 

\bibitem{Rezapour} S. Rezapour, A. Imran, A. Hussain, F. Martínez, S. Etemad,
M. K. A. Kaabar, Condensing functions and approximate endpoint
criterion for the existence analysis of quantum integro-difference
FBVPs., Symmetry, 13, 469 (2021).

\bibitem{Sadeghi} S. Sadeghi, H. Jafari, S.Nematia, Operational matrix for Atangana–Baleanu derivative based on Genocchi polynomials for solving FDEs, Chaos, Solitons and Fractals, 135, 109736 (2020).

\bibitem{Samko}  S. Samko, B. Ross, Integration and differentiation to a variable fractional order, Integral Transforms Spec. Funct. 1, 277-300 (1993).

\bibitem{shen} S. Shen, F. Liu, J. Chen, I. Turner, V. Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput. 218, 10861-10870 (2012).

\bibitem{Sun3} H. Sun, W. Chen, C. Li, Y. Chen, Finite difference schemes for variable order time fractional diffusion equation, International Journal of Bifurcation and Chaos, Vol. 22, No. 4, 1250085 (16 pages) (2012).

\bibitem{Sun1} H. Sun, W. Chen, H. Wei, Y. Chen, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top. 193, 185-192 (2011).

\bibitem{Sun2} H. Sun, W. Chen, H. Wei, Y. Chen, Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A 388, 4586-4592 (2009).

\bibitem{Tseng} Chien-Cheng Tseng,  Design of variable and adaptive fractional order FIR differentiators, Signal Processing 86, no. 10 (2006) 2554-2566.

\bibitem{Tuana} N.H. Tuana, R.M. Ganji, H. Jafari, A numerical study of fractional rheological models and fractional Newell-Whitehead-Segel equation with non-local and non-singular kernel, Chinese Journal of Physics, 68, 308-320 (2020).



\bibitem{wang1} H. Wang, X. Zheng, Analysis and numerical solution of a nonlinear variable-order fractional differential equation, Adv. Comput. Math. 45, 2647-2675 (2019).

\bibitem{wang2} H. Wang, X. Zheng, Wellposedness and regularity of the variable-order time-fractional diffusion equation, J. Math. Anal. Appl. 475, 1778-1802 (2019).

\bibitem{wu} L. Wu, Dufort-Frankel-type methods for linear and nonlinear Schrodinger equations, SIAM Journal on Numerical Analysis, vol. 33, no. 4, 1526-1533, (1996).

\bibitem{zeng} F. Zeng, Z. Zhang, G. Karniadakis, A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations, SIAM. J. Sci. Comput. 37, A2710-A2732 (2015).

\bibitem{zhao} X. Zhao, Z. Sun, G. Karniadakis, Second-order approximations for variable order fractional derivatives: Algorithms and applications, J. Comput. Phys. 293, 184-200 (2015).

\bibitem{zheng} X. Zheng, H. Wang, Optimal-order error estimates of finite element approximations to variable order time-fractional diffusion equations without regularity assumptions of the true solutions, IMA. J. Numer. Anal. 41, 1522-1545 (2021).

\bibitem{zhuang} P. Zhuang, F. Liu, V. Anh, I. Turner, Numerical method for the variable-order fractional advection diffusion equation with a nonlinear source term, SIAM. J. Mumer. Anal. 47, 1760-1781 (2009).

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{\small

\noindent{\bf Hadi Azizi}

\noindent Department of Mathematics

\noindent Assistant Professor of Mathematics

\noindent Department of Mathematics, Taft Branch, Islamic Azad University, Taft, Iran

\noindent Taft, Iran

\noindent E-mail: Azizi@Taftiau.ac.ir, Hadiazizi1360@gmail.com}\\





\end{document}