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\begin{document}
\begin{frontmatter}

\author[a]{Masoud Pourhasan }
\author[b,c]{Mohammad Ramezani \footnotemark[2]}

\cortext[cor2]{Corresponding author:\texttt{mr\_63\_90@yahoo.com}}
\address[a]{Department of mathematics, Factulay Islamic Azad University Shoushtar branch, Iran.}
\address[b]{Young Researchers and Elite Club, Parand Branch, Islamic Azad University, Tehran, Iran}
\address[c]{Department of Mathematics, Imam Khomeini International, Qazvin, Iran}
\title{ AN EFFECTIVE APPROACH TO SOLVE  A MULTI-TERM TIME FRACTIONAL DIFFERENTIAL EQUATION($M-TFDE$) WITH FUNCTION SPACE APPROXIMATION }
\begin{abstract}
This paper studies a B-spline algorithm for calculating the solution
of the multi-term time-fractional diffusion equations(M-TT-FDEs).
This model describes the diffusion prossing  in the fluid mechanics
and provides valuable predictions. The solution of the M-TT-FDEs is
discretized by means of B-spline function based on the B-spline
shape technique. It is verified that the proposed strategy is more
efficient in terms of computational time and accuracy in  domain.
\end{abstract}
\begin{keyword}
 Multi-term  time fractional;  Fractional B-spline functions;
 Differential equation; Function space approximation;
\end{keyword}
\end{frontmatter}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
 A significant tool in various sciences the fractional differential equation$(FDE)$ \cite{vahid1, vahid2, Yousefi, Yousefi1}
 that with a discretization method the $FDE$ are solved by computer \cite{vahid3}. Finite difference, finite volume, finite element,
discrete element, boundary element, no mesh, or combination of these
methods are the most common methods of discretization \cite{Stynes,
Simmons, Li, Li1, vahid4, vahid5}. Most methods offer the same
solution to the original $PDEs$ in theory. In \cite{ Baleanu1}
Baleanu et al., the FDE's existence was studied using Caputo, and some analytical solutions were obtained for the hybrid differential equation \cite{Baleanu2}.\\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%end111111111111%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
Numerical methods presented to solve approximate answers to
differential equations of mathematical samples of different problems
\cite{mr1, mr2, mr3, mr4}. The collocation method solves a finite
number of nodes by solving the differential equation. The easy and
high speed is the biggest advantage of this method. The fractional
B-spline function($fBSf$) is a  smoothness to connect with the low
calculating cost of collocation.Our goal in this manuscript is to
seek the performance of $fBSf$  at the collocation method to solve
initial and boundary value problems. Our goal in this manuscript is
to seek the performance of $fBSf$  at collocation method to solve
initial and boundary value problems.

$M-TFDE$ reduced of the problem to a system of the ordinary by
Edwards et. al. \cite{Edwards}. Another method is meshless that was introduced by
Hosseini et. al.for solving $M-TFDE$ in  \cite{vahid2,vahid6,vahid7}. That
left-side caputo fractional derivative persented by Lin and Lazarov
et. al where they got the  $\mathcal {O} (h^{2}+\tau^{2-\bbalpha}) $
\cite{Liu}.
 On different intervals focus on the fractional predictor-corrector method $M-TFDE$ by Liu \cite{Meerschaert}.
 The other method, the space-time spectral scheme  presented by  Zheng et. al. was  an impressive numerical
  method  \cite{Zheng}.
Assuming the norm to be  $L^2$ the stability and convergence proved  at
finite-difference scheme leads to a lower accuracy order $ \mathcal
{O} (\tau ^\bbalpha)$. With  spectral collocation method expanded an
power accurate fractional for solving  time-dependent fractional
partial differential equations with help  new fractional Lagrange
interpolants by Zayernouri et. al  \cite{Zayernouri}.  A composition
of finite difference and matrix transfer method presented by Zhao
et. al. \cite{Zhao}.

This manuscript is formed as follows: in section 2, some basic
definitions and theorems of $fBSf$ are expressed. Section 3 is
dedicated to the solution of $M-TDFE$  using the collocation
technique with $fBSf$. Insection 4, five numerical examples are
presented.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic function}
In this section, the efficiency and usefulness of spline functions
in computers, math and Box splines have been demonstrated
in \cite{Schempp, Chui, Nürnberger, de Boor}. We will provide several definitions and theorems of \cite{Forster}.
 \begin{definition}
 Functions are called polynomial spline function of degree  $n+1$. The conditions of functions is a piece of multinomial function with degree $n$ on interval $[a,b]$ are as follows:\\
 1) The points  interpolation are ${a=t_{1}\leq t_{2}\leq t_{3}\leq \ldots\leq
t_{d}=b}$ and in amongst any $[t_{i},t_{(i+1)}]$ is one polynomials
of degree $n$ too conjunction
 $[t_{(i+1)},t_{(i+2)}]$ to another polynomials:
\begin{eqnarray}\label{IEQ2}
S^{n}(t)=\left\{ \begin{aligned}
&s_{1}(t)\quad\quad\quad ;t_{1} \leq t\leq t_{2},\\
&s_{2}(t)\quad\quad\quad ;t_{2}\leq t\leq t_{3},\\
&\quad\quad.\\
&\quad\quad.\\
&\quad\quad.\\
&s_{(d-1)}(t)\quad\quad ;t_{(d-1)} \leq t\leq t_{d}.\\
\end{aligned} \right.
\end{eqnarray}
Spline function presented $S^{n}(t)$  that on each partition
$s_{i}(t),i=1,2,\ldots,d-1$ is a polynomial of $n$ degree.\\
2)The characteristics of the $n$th derivative which are limited,
displays several isolated case that it is not continuities in points,
and they are  continuities at knots among the polynomial piece
where the continuous derivative of the order of $n-1$ is one of
the properties of $s_{i}(t),i=1,2,\ldots,d-1$ functions at
$[t_{i},t_{(i+1)}]$.
 \end{definition}
B-Splines functions($BSf$) polynomials were introduced by I. J.
Schoenberg  in \cite{Schoenberg1, Schoenberg2}  in 1946.
He   formed the basic functions for terms $BSf$ as follows:
\begin{eqnarray}
S^{n}(t)=\sum_{j\in{\mathbb{Z}}} c_{j}\beta^{n}(t-j),
\end{eqnarray}
%\binom{n+1}{j}
\begin{eqnarray}
\beta^{n}(t)=\frac{1}{n!}\sum_{j=0}^{n+1}(-1)^{j}{n+1\choose
j}(t-j)^{n}_{+}.
\end{eqnarray}
Where
\begin{eqnarray}\label{IEQ2}
(t-j)^{n}_{+}=\left\{ \begin{aligned}
&(t-j)^{n}\quad\quad\quad  t>j,\\
& t>j\quad\quad\quad\quad t\leq j .\\
\end{aligned} \right.
\end{eqnarray}
The $BSf$ with different powers:\\

%fig 1 a ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=8cm, height=4cm,angle=0]{1a.eps}
    \end{center}
    \caption{ The $BSf$ shapes with $0$ degree is really $\beta^{0}(t).$
    }
    \label{states}
\end{figure}
%fig 1b  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=8cm, height=4cm,angle=0]{1b.eps}
    \end{center}
    \caption{ The $BSf$ shapes with $1$  degree is really $\beta^{1}(t).$}
    \label{states}
\end{figure}
%fig 1c  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=8cm, height=4cm,angle=0]{1c.eps}
    \end{center}
    \caption{The $BSf$ shapes with $2$  degree is really $\beta^{2}(t).$ }
    \label{states}
\end{figure}
%fig 1d  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=8cm, height=4cm,angle=0]{1d.eps}
    \end{center}
    \caption{The $BSf$ shapes with $3$ degree is really $\beta^{3}(t).$ }
    \label{states}
\end{figure}
In $Fig. 1$,  the power $0$ for  $\beta^{0}(t)$ if constant
function, in $Fig. 2$, $\beta^{1}(t)$ called Hat function that is a
linear function, in $Fig. 3$, $\beta^{2}(t)$ of degree two and in
$Fig.4$, $\beta^{3}(t)$ called bell function that is degree tree.
These functions play essential role in the theory of defense
approximation and analysis.  The reason for using these functions in
a variety of applications and their
 widespread use is that they have desirable properties \cite{Boor, Prenter, Bartels, Unser1}.

The extension of constant's  presented by Thierry Blu and Michael
Unser  of $fBSf$\cite{Unser2}. The favorable attributes of $fBSf$
showed to transfer to the fractional case.
\begin{definition}
The $fBSf$  $\beta^{\bbalpha}(t)$ is:
\begin{eqnarray}\label{IEQ51}
\beta^{\bbalpha}(t)=\frac{1}{\Gamma(\bbalpha+1)}\sum_{k\leq0}(-1)^{k}\binom{\bbalpha+1}{k}(t-k)_{+}^{\bbalpha}
\end{eqnarray}
the $Eq. \ref{IEQ51}$ is credible point to point for everyone $t\in
\mathbb{R}$ and a well as into the $L^{2}(\mathbb{R})$.
\end{definition}
In $Figs. 5, 6, 7,$ and $Fig.8$ several samples of $fBSf$ are
introduced, it seems to be destroyed, only time the $\bbalpha$ be an
integer then the $fBSf$ are compactly supported.     in this
sample, we have covered the classical $BS$. Generally, they have an axis of asymmetric.\\
%fig 2 a------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
  \includegraphics[width=8cm, height=4cm,angle=0]{aa.eps}
    \end{center}
   \caption{The  $fBSf$ shapes with $0.1$ degree is really $\beta^{0.1}(t)$.}
   \label{states}
\end{figure}
%fig 2b  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
  \includegraphics[width=8cm, height=4cm,angle=0]{2b.eps}
    \end{center}
   \caption{The  $fBSf$ shapes with $0.3$ degree is really $\beta^{0.3}(t)$.}
   \label{states}
\end{figure}
%fig 2c  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=8cm, height=4cm,angle=0]{2c.eps}
    \end{center}
    \caption{The  $fBSf$ shapes with $0.3$ degree is really $\beta^{0.7}(t)$.}
    \label{states}
\end{figure}
%fig 2d  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
  \includegraphics[width=8cm, height=4cm,angle=0]{2d.eps}
    \end{center}
    \caption{The $fBSf$ shapes with $1.3$ degree is really $\beta^{1.3}(t)$.}
    \label{states}
\end{figure}
Functions with fractional power are well approximated by the
 $fBSf$ because they have fractional power. They have
every continuous parameter $\bbalpha>-1$. If the $\bbalpha$  is an
integer, this function interpolates the normal splines.

 First of all,  investigated a rather forced adjust univariate analysis with spaced
points; for making multiresolution wavelet bases their monotonous
net in special is needed.
 Second,  these functions can be used in many numerical methods, and also the $fBSf$ have the characteristics
 of a type the $BS$ such as the support domain of the $BS$ for nonintegral where
$\bbalpha$ is no longer compact. Particulary,  functions were dense
in $L^{2}$ with condition $\bbalpha
  >\frac{-1}{2}$.\\
The definition of $fBSf$ spaces on the $a$ scale is
as follows:\\
\begin{eqnarray}\label{IEQ42}
S_{a}^{\bbalpha}=\{s_{a}:\exists c\in
l^{2},s_{a}(x)=\sum_{k\in\mathbb{Z}}c_{k}\beta^{\bbalpha}(\frac{x}{a}-k)\}
\end{eqnarray}
We assess its least squares approximation in $S_{a}^{\bbalpha}$ for an arbitrary function $f \in L^{2}(\mathbb{R})$. \\

\begin{theorem}
The $fBSf$ has a fractional order of approximation $\bbalpha+1$. In
particular, the least-squares approximation error is limited by
\begin{eqnarray}
\forall f\in W_{2}^{\bbalpha+1},\|f-P_a f\|_{L^{2}} \leq
a^{\bbalpha+1} \|\mathcal{D}^{\bbalpha+1}f\|_{L^{2}}
\frac{\sqrt{2\xi (\bbalpha+2)-\frac{1}{2}}}{\Pi^{\bbalpha+1}} ;
a\rightarrow 0
\end{eqnarray}
\textbf{Proof.}  The proofs in \cite{Unser2}, (Theorem 4.1).
\end{theorem}
In this theorem, $P_a f$ is an interpolation function of function
$f$. The $fBSf$ produces credible multiresolution analysis of
$L^{2}$ for $\bbalpha>-\frac{1}{2}$. The $fBSf$ can be a scheme to
have an optional order of smooth. These functions produce a sequence
of space flow as:
\begin{eqnarray}
{0}\subset ...\subset \mathcal {X}_{-1}\subset \mathcal {X}_{0}
\subset \mathcal {X}_{1}\subset ... \subset L^{2}(\mathbb{R})
\end{eqnarray}
they have properties:\\
 a) $\bigcap_{i\in\mathbb{Z}} \mathcal
{X}_{i}={0}$ and $\overline{\bigcup_{i\in\mathbb{Z}} \mathcal
{X}_{i}}=L^{2}(\mathbb{R})$.\\
 b) $f(*)\in \mathcal {X}_{i}$ if and
only if $f(2^{-i}*)\in \mathcal {X}_{0}$\\
 c) $f(*)\in \mathcal
{X}_{0}$ if and only if $f(*-k) \in \mathcal {X}_{0}$ for each $k\in
\mathbb{Z}$ and there be a function $\varphi\in \mathcal {X}_{0}$,
called a scale factor, such a way that
$\varphi(*-k)_{k\in\mathbb{Z}}$ format an orthonormal foundations of
$\mathcal {X}_{0}$. The spaces $fBSf$ produce $\mathcal {X}_{n}$ are
of order $\bbalpha\in\mathbb{R}$ with points $k\times2^{n},
k\in\mathbb{Z}$  where the forms spaces are:\\
\begin{eqnarray}\label{IEQ33}
\mathcal
{X}_{n}=\overline{span\{\beta^{\bbalpha}(\frac{x-2^{n}k}{2^{n}})^{L^{2}(\mathbb{R})}\}}
;\bbalpha\geq-\frac{1}{2},n\in\mathbb{Z},
\end{eqnarray}
That $\beta^{\bbalpha}$ produces a multiresolution analysis. Let's
take, $a=2^{i}$, then several sample of multiresolution and shift
$fBSf$ $\beta^{\bbalpha}$ as illustrated  below:\\
%fig 3a ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=8cm, height=4cm,angle=0]{3a.eps}
    \end{center}
    \caption{The one
 degree of $BSf$ shape are by $i=0$ i.e. $a=1$ and several various $k$ of $Eq.\ref{IEQ42}$ really $\beta^{1}(t)$, $\beta^{1}(t-1)$,
 $\beta^{1}(t+1)$, $\beta^{1}(t+2)$.}
    \label{states}
\end{figure}
%fig 3b  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
   \includegraphics[width=8cm, height=4cm,angle=0]{3b.eps}
    \end{center}
    \caption{The two
 degree of $BSf$ shape  are  by $i=0$ i.e. $a=1$ and several various $k$ of $Eq.\ref{IEQ42}$ really $\beta^{2}(t)$, $\beta^{2}(t-1)$,
 $\beta^{2}(t+1)$.}
    \label{states}
\end{figure}
%fig 3c  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=8cm, height=4cm,angle=0]{3d.eps}
    \end{center}
    \caption{The  one
 degree  of $BSf$ shape are  by $i=-1$ i.e. $a=\frac{1}{2}$ and several various $k$ of $Eq.\ref{IEQ42}$ really $\beta^{1}(2t)$, $\beta^{1}(2t-1)$,
 $\beta^{1}(2t+1)$, $\beta^{1}(2t+2)$, $\beta^{2}(2t)$.}
    \label{states}
\end{figure}
%fig 3d  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
   \includegraphics[width=8cm, height=4cm,angle=0]{3c.eps}
    \end{center}
    \caption{The two
 degree $BSf$ shape are by $i=-1$ i.e. $a=\frac{1}{2}$ and several various $k$ of$ Eq.\ref{IEQ42}
 $ really $\beta^{2}(2t-2)$, $\beta^{2}(2t-1)$,
 $\beta^{2}(2t+1)$.}
    \label{states}
\end{figure}
$Figs. 9, 10 , 11$, and $Fig. 12$ are some shift
$\beta^{1}(t-k)$,$\beta^{2}(t)$, $\beta^{1}(2t)$ and $\beta^{2}(2t)$
,respectively. In our methods numerical analysis basic functions are
 those functions.


%fig 4 a ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=8cm, height=4cm,angle=0]{4a.eps}
    \end{center}
    \caption{The diagram of the $\bbalpha=0.3$
 degree are  by $i=0$ i.e. $a=1$ and several  $k$ of $Eq.\ref{IEQ42}$ really $\beta^{0.3}(t)$,
 $\beta^{0.3}(t-1)$.
, $\beta^{0.3}(t-2)$, $\beta^{0.3}(t-3)$ for $fBSf$.}
    \label{states}
\end{figure}
%fig 4b  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
   \includegraphics[width=8cm, height=4cm,angle=0]{4b.eps}
    \end{center}
     \caption{The diagram of the $\bbalpha=0.3$
 degree are  by $a=-1$ and several $k$ of $Eq.\ref{IEQ42}$ really $\beta^{0.3}(2t)$, $\beta^{0.3}(2t-1)$,
$\beta^{0.3}(2t+1)$, $\beta^{0.3}(2t-2)$  for $fBSf$.}
    \label{states}
\end{figure}

Several shift $fBSf$ of the $\bbalpha=0.3$  with $a=2^{0}$ and
$a=2^{-1}$ and several different $k$ of conforming to
$Eq.\ref{IEQ42}$ in actuality $\beta^{0.3}(t)$ and $\beta^{0.3}(2t)$
are shown in $Fig. 13$ and $Fig. 14$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5%%%%%%%%%%%%%%%%%%%%%%%%%
\section {$M-T FDE$}
 With $M-TFDE$ of diffusion-wave time equations  a lot work extensions have been conducted.
  We are using base $fBSf$ in the collocation method on approximation. In this article, we discuss
 Caputo time derivative in one and two dimensions:
\begin{eqnarray}\label{IEQ1}
\left\{ \begin{aligned}
&\mathbb{P}(\mathcal{D}_{t})(\overline{\textbf{X}},t)-\Delta \mathcal {U}(\overline{\textbf{X}},t)=\mathbb{F}(\overline{\textbf{X}},t) \quad \quad \quad (\overline{\textbf{X}},t)\in \Omega\times(0, T],\\
& \mathcal {U}(\overline{\textbf{X}},0)=\psi_{1}(\overline{\textbf{X}}), \quad \quad\quad \quad \quad  \quad \quad  \quad \quad \quad\overline{\textbf{X}}\in \Omega \\
& \mathcal
{U}(\overline{\textbf{X}},t)=\Phi(\overline{\textbf{X}},t), ~~~\quad
\quad \quad \quad \quad \quad \quad  \quad \quad
\overline{\textbf{X}}\in\partial\Omega,
\end{aligned} \right.
\end{eqnarray}
where $\Omega$  is domain and $\partial\Omega$ is a boundary.\\
 The $\mathbb{F}$ is the source term in equation above, issued to the suitable
  initial and boundary condition, respectively. Condition $\psi_{1}$ and $\Phi$ are presented functions on $\Omega$.

Then, the $\mathbb{P}(\mathcal {D}_{t})$ is fractional operator  to
form under:
\begin{eqnarray}
\mathbb{P}(\mathcal{D}_{t})=\mathcal
{D}_{t}+\sum_{i=1}^{\mathbbm{m}} r_{i}\mathcal
{D}^{\bbalpha_{i}}_{t},
\end{eqnarray}
where the $\mathbbm{m}\in \mathds{N}$ and $\mathcal
{D}^{\bbalpha_{i}}_{t}$ represents the Caputo fractional
derivative of order $\bbalpha_{i}\in (0,1)$, is defined by
\begin{eqnarray}\label{IEQ2}
\mathcal {D}^{\bbalpha_{i}}_{t}\mathcal {U}(t)=\left\{
\begin{aligned}
&\frac{1}{\Gamma(k-\bbalpha_{i})}\int^{t}_{0}(t-\xi)^{k-\bbalpha_{i}-1}\mathcal {U}^{k}(\xi)d\xi\quad k-1<\bbalpha_{i}<k,\quad t>0,\\
&\mathcal {U}^{k}(t) \quad\quad\quad\quad\quad \quad\quad
\quad\quad\quad\quad \quad \quad\quad \quad\quad \bbalpha_{i}=k.
\end{aligned} \right.
\end{eqnarray}
the  $\Gamma(.)$ is a usual Gamma function. The $fBSf$ does not have
compact support but it decays toward infinity as:
 $$\beta^{\bbalpha}(t)=\frac{1}{|t|^{-2-\bbalpha}},$$
 moreover however, $\beta^{\bbalpha}$ is $\bbalpha$-$H\ddot{o}lder$ continuous, belonging to $L^{2}(\mathbb{R})$ and reproducing
 polynomials up to degree $[\bbalpha]$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection {Collocation technique $fBSf$ with one variable for unknown function}
First, we want to explain the method with a variable one dimension
for unknown function, from $Eq.\ref{IEQ1}$

$$\overline{\mathbf{X}}\in \mathcal{X}_{N}\subseteq\mathcal {X}$$ concerning  $Eq.\ref{IEQ33}$ since
 $\mathcal {X}$ to $\mathcal {X}$.
The $\tilde{\mathcal {U}}_{N}(\overline{\mathbf{X}},t)$ is
approximate of $\mathcal {U}_{N}(\overline{\mathbf{X}},t)$ that we
select a limited family of functions. The $\overline{\mathbf{X}}$ is
single variable thus $\overline{\mathbf{X}}=x$, the $\mathcal
{X}_{N}$ is a series of dimensional subspace that $\mathcal
{X}_{N}\subset \mathcal {X}; N\geq 0$ that $\mathcal {X}_{N}$  have
a basis $\beta^{r}(\frac{x-2^{N}k}{2^{N}})$ and
$\beta^{p}(\frac{t-2^{N}l}{2^{N}})$. We search a function
$\tilde{\mathcal {U}}_{N}(x,t)\in \mathcal {X}_{N} \times \mathcal
{X}_{N} $ that it can be written as:
\begin{eqnarray}\label{IEQ3}
&&\tilde{\mathcal
{U}}_{N}(x,t)=\sum_{k,l=1}^{d,d}\mathbbm{c}_{kl}\beta^{r}(\frac{x-2^{N}k}{2^{N}})\beta^{p}(\frac{t-2^{N}l}{2^{N}}).
\end{eqnarray}
We sub $\tilde{\mathcal {U}}_{N}(x,t)$ to $\mathcal {U}_{N}(x,t)$ in
the $Eq.\ref{IEQ1} $ and dissolve it. then, assume considerate
$(x,t)\in [a,b]\times[c,d]$, which the numbers $k,l$ in
$Eq.\ref{IEQ3}$ is confined on $[a,b]$. We search knots
$(x_{i},t_{i}), i=1,...,d$, so that $(x,t)\in [a,b]\times[c,d]$ and
${\mathbbm{c}_{11},...,\mathbbm{c}_{dd}}$ are assess by dissolving
linear system:
\begin{eqnarray}
\nonumber R_{N}(x_{i},t_{j})&=&\sum_{i=1}^{\mathbbm{m}}
r_{i}\mathcal{D}^{\bbalpha_{i}}_{t}\sum_{k,l=1}^{d,d}
\mathbbm{c}_{kl}\beta^{r}(\frac{x_{i}-2^{N}k}{2^{N}})\beta^{p}(\frac{t_{j}-2^{N}l}{2^{N}})\\
&-&\sum_{k,l=1}^{d,d}
\mathbbm{c}_{kl}\Delta\beta^{r}(\frac{x_{i}-2^{N}k}{2^{N}})\beta^{p}(\frac{t_{j}-2^{N}l}{2^{N}})
-\sum_{j,i=1}^{d,d}\mathbb{F}(x_{i},t_{j})=0,
\end{eqnarray}
next we utilization of $Eq.\ref{IEQ51}$ at up equation, which is
obtained:
\begin{eqnarray}\label{IEQ4}
&&\nonumber R_{N}(x_{i},t_{j})=\\
\nonumber&&\sum_{k,l=1}^{d,d}\mathbbm{c}_{kl}\left(\sum_{s\geq
0}(-1)^{s}{r+1\choose s}\frac{(\frac{x_{i}-2^{N}k}{2^{N}}-s)^{r}_{t}}{\Gamma(r+1)}\right)
\left(\sum_{i=1}^{\mathbbm{m}} r_{i}\mathcal
{D}^{\bbalpha_{i}}_{t}\sum_{h\geq 0}(-1)^{s}{p+1\choose
h}\frac{(\frac{t_{j}-2^{N}l}{2^{N}}-s)^{p}_{t}}{\Gamma(p+1)}\right)\\
\nonumber&&-\sum_{k,l=1}^{d,d}\mathbbm{c}_{kl}\Delta\left(\sum_{s\geq
0}(-1)^{s}{r+1\choose s}\frac{(\frac{x_{i}-2^{N}k}{2^{N}}
-s)^{r}_{t}}{\Gamma(r+1)} \right) \left(\sum_{h\geq
0}(-1)^{s}{p+1\choose h}\frac{(\frac{t_{j}-2^{N}l}{2^{N}}-s)^{p}_{t}}{\Gamma(p+1)}\right)\\
&=&\sum_{j,i=1}^{d,d}\mathbb{F}(x_{i},t_{j}),i,j=0,...,d-1.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection {Collocation method $fBSf$ with two variable for unknown function}
In the second case, we tend to explain the method with a variable
two dimension for unknown function, from $Eq.\ref{IEQ1}$ , we assume
$\overline{\mathbf{X}}\in \mathbb{R}^{2}$ i.e.
$(\overline{\mathbf{X}},t)=(x,y,t)$ then like the mode of a variable
we select  a series of dimensional subspace $\mathcal{X}_{N}\subset
\mathcal{X}; N\geq 0$ that $\mathcal{X}_{N}$ have a basis
$\beta^{r}(\frac{x-2^{N}i}{2^{N}}),\beta^{q}(\frac{y-2^{N}j}{2^{N}})$
and $\beta^{p}(\frac{t-2^{N}k}{2^{N}})$. We seek a function
$\tilde{\mathcal{U}}_{N}(x,y,t)\in \mathcal{X}_{N} \times
\mathcal{X}_{N} \times \mathcal{X}_{N} $ that  can be written as:
\begin{eqnarray}\label{IEQ5}
&&\tilde{\mathcal {U}}_{N}(x,y,t)=\sum_{i,j,k\in
\mathbb{N}}\mathbbm{c}_{ijk}\beta^{r}(\frac{x-2^{N}i}{2^{N}})\beta^{q}(\frac{y-2^{N}j}{2^{N}})\beta^{p}(\frac{t-2^{N}k}{2^{N}}).
\end{eqnarray}
next change $\tilde{\mathcal {U}}_{N}(x,y,t)$ with $\mathcal
{U}(x,y,t)$ in the $Eq.\ref{IEQ1}$ and dissolving it. Next, we
assume by considering  $(x,y,t)\in [c,d]\times[e,f]\times[a,b]$,
with this
$i,j,k$  in $Eq.\ref{IEQ5}$ is limited on $[a,b]$. \\
Now we search knots $(x_{i},y_{j},t_{k}),i,j,k=1,...,d$ where
$(x,y,t)\in [a,b]\times[c,d]\times[e,f]$ and $\mathbbm{c}_{111}$,
$\mathbbm{c}_{211}$, ..., $\mathbbm{c}_{ddd}$ are assess by dissolve
linear system below:
\begin{eqnarray}\label{IEQ6}
 R_{N}(x_{w},y_{v},t_{z})&=& \sum_{i=1}^{\mathbbm{m}}
r_{i}\mathcal {D}^{\bbalpha_{i}}_{t}\sum_{i,j,k=1}^{d,d,d}
\nonumber\mathbbm{c}_{ijk}\beta^{r}(\frac{x_{w}-2^{N}i}{2^{N}})\beta^{p}(\frac{y_{v}-2^{N}j}{2^{N}})\beta^{q}(\frac{t_{z}-2^{N}k}{2^{N}})\\
&-&\Delta \sum_{i,j,k=1}^{d,d,d}
\nonumber\mathbbm{c}_{ijk}\beta^{r}(\frac{x_{w}-2^{N}i}{2^{N}})\beta^{p}(\frac{y_{v}-2^{N}j}{2^{N}})\beta^{q}(\frac{t_{z}-2^{N}k}{2^{N}})\\
&-&\sum_{i,j,k=1}^{d,d,d}\mathbb{F}(x_{w},y_{v},t_{z})=0,w,v,z=0,...,d-1.
\end{eqnarray}
Similar previous case, putting $Eq.\ref{IEQ51}$ can obtain the
unknown factors. With Placement points in two modes are mentioned,
two matrices are created. we solve $Eq.\ref{IEQ1}$  with collocation
technique by usage of $fBSf$. we assume  $P_{n}$ that maps
$\mathcal{X}$ onto $\mathcal{X}_{n}$, define $P_{n}\mathcal {U}
(\overline{\mathbf{x}},t)$ to be that atom of $\mathcal{X}_{n}$ that
approximate $\mathcal{X}$ at the knots used at $Eq.\ref{IEQ3}$ and
$Eq.\ref{IEQ5}$.
 We can found following relation:
$$P_{n}\mathcal {U}(\overline{\mathbf{X}},t)=\tilde{\mathcal {U}}_{N}(\overline{\mathbf{X}},t)$$
with the factors $\mathbbm{c}_{ij}$ with one variable and
$\mathbbm{c}_{ijk}$ with two variable specified dissolving the
linear system $Eq.\ref{IEQ4}$ and $Eq.\ref{IEQ51}$ next our problem
has a alone one answer if
$$det(R_{N}(x_{i},t_{j}))\neq0$$
or
$$det(R_{N}(x_{w},y_{v},t_{z}))\neq0.$$
The convergence of this method is guaranteed by means of Theorem
2.3.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Applications and Results}
Now, we present the conclusions made for several samples using our
method with $fBSf$ for $Eq.\ref{IEQ1}$ explained previously. At
 samples, the precision of the methods, and  we
compare with the suggested technique two types of error measures,
$\varepsilon_\infty$ that is a maximum absolute error and $RMS$
$\varepsilon_R$:
 \begin{eqnarray}\label{IEQ7}
Error=\Big\|{{\mathcal{\widetilde{U}}_{N}}}(\overline{\textbf{X}}_i,t)-\mathcal{U}(\overline{\textbf{X}}_i,t)\Big\|_{\infty},\
\ \ 0\leq t\leq T
\end{eqnarray}
\begin{eqnarray}\label{IEQ8}
RMS=\sqrt{\frac{\sum_{i=1}^n\left(\mathcal{\widetilde{U}}_{N}(\overline{\textbf{X}}_i,t)-\mathcal{U}(\overline{\textbf{X}}_i,t)\right)^2}{n}},
\end{eqnarray}
are employed, which  the $\mathcal{U}(\overline{\textbf{X}}_i,t)$ is
exact answers and
$\mathcal{\widetilde{U}}_{N}(\overline{\textbf{X}}_i,t)$ is
approximate answers, $N$ is dimension of $fBSf$ and $n$ is number
knots for plot shape and compute error  between exact and
approximate answers in order. At every example, we  are assume
regular node be regular partition next by solve $Eq.\ref{IEQ4}$ or
$(18)$ and obtain $\mathbbm{c}_{kl}$ or $\mathbbm{c}_{ijk}$ for
 $Eq.\ref{IEQ3}$ and $Eq.\ref{IEQ5}$ that it is approximate answers
 then we divide to $n$ of the equal part the scope of the answer and
by using $Eq.\ref{IEQ7$ to calculate error and draw it. For one and
two dimensions of $fBSf$ and $\bbalpha$ with attention example , we are considering error $Eq.\ref{IEQ8}$. \\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{Example 1.  } \\
First example, we discuss the $Eq.\ref{IEQ1}$ with different
$\bbalpha_{1}$, $\bbalpha_{2}$ and$t\in [0,1]$ and  $\triangle
t^{i}=t^{i}-t^{i-1}=0.01$ in partition  $\Omega=[0,0.5]$. The
$\mathcal{U}(x,t)=x^{3}(t^{1+\bbalpha_{1}+\bbalpha_{2}})$ is exact
solution too
\begin{eqnarray}
    \nonumber\mathbb{F}(x,t)&=&-6t^{2+\bbalpha_{1}+\bbalpha_{2}}x\\
    &+& x^{3}\Gamma{(1+\bbalpha_{1}+\bbalpha_{2})}(1+\bbalpha_{1}+\bbalpha_{2})
    \left[\frac{(t^{1+\bbalpha_{1}})\Gamma{(2-\bbalpha_{1})}}{\Gamma{(3+\bbalpha_{1})}\Gamma{(1-\bbalpha_{2})}}
    +\frac{(t^{1+\bbalpha_{2}})\Gamma{(2-\bbalpha_{2})}}{\Gamma{(3+\bbalpha_{2})}\Gamma{(1-\bbalpha_{1})}}\right]
\end{eqnarray}
and tree term fractal  $\bbalpha_{i},i=1,2,3$,
$$\mathcal{U}(x,t)=x^{3}(t^{1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3}})$$
also
\begin{eqnarray}
\nonumber\mathbb{F}(x,t)&=&-6t^{2+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3}}x\\
    \nonumber&+&x^{3}\Gamma{(1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3})}(1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3})\\
    &&\left[\frac{(t^{1+\bbalpha_{1}+\bbalpha_{2}})\Gamma{(2-\bbalpha_{3})}}{\Gamma{(3+\bbalpha_{1}+\bbalpha_{2})}\Gamma{(1-\bbalpha_{3})}}
    +\frac{(t^{1+\bbalpha_{1}+\bbalpha_{3}})\Gamma{(2-\bbalpha_{2})}}{\Gamma{(3+\bbalpha_{2}+\bbalpha_{3})}\Gamma{(1-\bbalpha_{2})}}
    +\frac{(t^{1+\bbalpha_{2}+\bbalpha_{3}})\Gamma{(2-\bbalpha_{1})}}{\Gamma{(3+\bbalpha_{2}+\bbalpha_{3})}\Gamma{(1-\bbalpha_{1})}}\right]
\end{eqnarray}

%Table1--------------------------------------------------------------------------------------
\begin{flushleft}
\begin{table}[ht!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1}, \bbalpha_{2}$ have tree variable $t$, $x$, $n$ . }
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
 &\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
\rowcolor{lightgray}
 \hline \hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.4$&$1.37691715\times10^{-4}$&$1.36817007\times10^{-4}$&$1.36784227\times10^{-4}$\\
 \hline
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.4$&$1.31697622\times10^{-4}$&$1.31062956\times10^{-4}$&$1.31000040\times10^{-4}$\\
\rowcolor{lightgray}  \hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.6$&$1.28816508\times10^{-4}$&$1.28369975\times10^{-4}$&$1.27977642\times10^{-4}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.9$&$2.44772992\times10^{-4}$&$2.12264571\times10^{-5}$&$4.87391324\times10^{-6}$\\
\hline \rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.8$&$3.03165220\times10^{-5}$&$1.34647287\times10^{-5}$&$4.79382664\times10^{-6}$\\
\hline
\end{tabular}
\end{scriptsize}
}
\end{table}
\end{flushleft}

%Table2--------------------------------------------------------------------------------
\begin{flushleft}
\begin{table}[ht!]
\caption{ Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and
the  $\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ have tree variable
$t$, $x$, $n$.}
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.2,\bbalpha_{3}=0.3$&$1.35596505\times10^{-4}$&$1.34395454\times10^{-4}$&$1.34377414\times10^{-4}$\\
 \hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.3,\bbalpha_{3}=0.4$&$1.27265808\times10^{-4}$&$1.26561905\times10^{-4}$&$1.25629737\times10^{-4}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.4,\bbalpha_{3}=0.5$&$1.20259940\times10^{-5}$&$1.19793031\times10^{-4}$&$1.16883116\times10^{-4}$\\
 \hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.3,\bbalpha_{3}=0.8$&$2.99362980\times10^{-5}$&$1.32748298\times10^{-5}$&$4.65066226\times10^{-6}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.3,\bbalpha_{3}=0.9$&$2.88319590\times10^{-5}$&$1.27763920\times10^{-5}$&$4.65066226\times10^{-6}$\\
\hline
\end{tabular}
\end{scriptsize}
      }
\end{table}
\end{flushleft}
At our tables, we obtain $RMS$ of $Eq.\ref{IEQ8}$ for several
$\bbalpha$'s. The $RMS$ solutions is not much more than $10^{-4}$.
The table 1 with $\bbalpha_{1},  \bbalpha_{2}$  and the table 2 with
$\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$, shows the $RMS$ produced
using with $n=500$ and several of $\bbalpha$ and $\Delta t$. When
the $N$ grow, the $RMS$ is reducing slowly and decreasing the error
by grow the $X$ to little by little in $ Fig.15$, and $Fig.16$.

%fig11----------------------------------------------------------------------------------------
\begin{flushleft}
\begin{center}
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=14cm, height=7cm, angle=0]{example1_1.eps}
    \end{center}
    \caption{The shape $RMS$ for $\bbalpha_{1}$, $\bbalpha_{2}$ that are
$\bbalpha_{1}=0.1$, $\bbalpha_{2}=0.4$ of  $Eq.\ref{IEQ1}$ and error
$Eq.\ref{IEQ7}$ .}
    \label{states}
\end{figure}
\end{center}
\end{flushleft}
%fig12----------------------------------------------------------------------------------------
\begin{flushleft}
\begin{center}
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=14cm, height=7cm,angle=0]{example1_2.eps}
    \end{center}
    \caption{The shape $RMS$ for $\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ that are
$\bbalpha_{1}=0.1$, $\bbalpha_{2}=0.2$, $\bbalpha_{3}=0.3$ of $
Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
\end{center}
\end{flushleft}
 We are displaying the $Error$ of $Eq.\ref{IEQ7}$ that
estimate answers with $\bbalpha_{1}=0.1$, $\bbalpha_{2}=0.4$ and
$\bbalpha_{1}=0.1$, $\bbalpha_{2}=0.2$, $\bbalpha_{3}=0.3$, the $N$
is number of variable of $fBSf$ at $Fig .15$ and $Fig.16$. We view in
the  $Fig.15$ and $Fig.16$, $Error$ in axis $X$ is not decrease until
$10^{-3}$ by attention to that in $N=2$ it is $10^{-4}$, it is
manner is not fast, it is not t rapidity increase  tangible .\\
%example 2- 2 fractional---------------------------------------------------------------------
\textbf{Example 2} \\
We discuss the $Eq.\ref{IEQ1}$ with two variable $x,y$ that is mean
$\overline{\mathbf{X}}\in \mathbb{R}^{2}$ and several amounts for
$\bbalpha$ and $\triangle t^{i} =0.01$ and $t\in [0,1]$ in partition
$\Omega = [0,0.5]\times[0,0.5]$ . The
$\mathcal{U}(x,y,t)=t^{1+\bbalpha_{1}+\bbalpha_{2}}x^{2}y^{2}$ is
solution, and force term can expreesed as follows
\begin{eqnarray}
\nonumber\mathbb{F}(x,y,t)&=&-2t^{2+\bbalpha_{1}+\bbalpha_{2}}(x^{2}+y^{2})x^{2}y^{2}
+\Gamma{(1+\bbalpha_{1}+\bbalpha_{2})}(1+\bbalpha_{1}+\bbalpha_{2})\\
&&\nonumber\left[\frac{(t^{2+\bbalpha_{1}})\Gamma{(2-\bbalpha_{1})}}{\Gamma{(3+\bbalpha_{1})}\Gamma{(1-\bbalpha_{2})}}
+\frac{(t^{2+\bbalpha_{2}})\Gamma{(2-\bbalpha_{2})}}{\Gamma{(3+\bbalpha_{2})}\Gamma{(1-\bbalpha_{1})}}
\right]
\end{eqnarray}
and tree term  fractional $\bbalpha_{i},i=1,2,3$
$$\mathcal{U}(x,y,t)=t^{1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3}}x^{2}y^{2}$$
also
\begin{eqnarray}
\nonumber\mathbb{F}(x,y,t)&=&-2t^{2+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3}}(x^{2}+y^{2})+
x^{2}y^{2}\Gamma{(1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3})}(1+\bbalpha_{1}+\bbalpha_{2})\\
&&\nonumber\left[\frac{(t^{1+\bbalpha_{1}+\bbalpha_{2}})\Gamma{(2-\bbalpha_{3})}}{\Gamma{(3+\bbalpha_{1}+\bbalpha_{2})}\Gamma{(1-\bbalpha_{3})}}
+\frac{(t^{1+\bbalpha_{1}+\bbalpha_{3}})\Gamma{(2-\bbalpha_{2})}}{\Gamma{(3+\bbalpha_{2}+\bbalpha_{3})}\Gamma{(1-\bbalpha_{2})}}
+\frac{(t^{1+\bbalpha_{2}+\bbalpha_{3}})\Gamma{(2-\bbalpha_{1})}}{\Gamma{(3+\bbalpha_{2}+\bbalpha_{3})}\Gamma{(1-\bbalpha_{1})}}\right]
\end{eqnarray}
 In this sample plotting the error of obtained answers by
amounts of Degree of fraction, assume one of the variables  the
variable $X$ or $Y$to be constant then  we calculate the $RMS$.We
assume amounts fixed away from knots primary. Anew the $N$ is
dimension of $fBSf$ and the $N$ is grow $Error$ isn't increase. The
$Fig. 17$, $Fig.18$, $Fig.19$ and $Fig.20$ are answers
 at several time surfaces for $\bbalpha$ have been presented.
%Table2-1-x-------------------------------------------------------------------------------
\begin{flushleft}
\begin{table}[ht!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1}, \bbalpha_{2}$ have tree variable $t$, $x$, $n$, that
$y$ is fixed.}
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.2$&$3.94497585\times10^{-4}$&$9.15524676\times10^{-5}$&$1.59141638\times10^{-5}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.8$&$2.48475179\times10^{-4}$&$4.72961107\times10^{-5}$&$1.25629737\times10^{-5}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.5 , \bbalpha_{2}=0.6$&$2.17263429\times10^{-4}$&$3.81143002\times10^{-5}$&$3.81143002\times10^{-5}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.6$&$3.17518103\times10^{-5}$&$1.93898497\times10^{-6}$&$1.41841301\times10^{-7}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.7$&$2.85753808\times10^{-5}$&$1.56945742\times10^{-6}$&$1.13979494\times10^{-7}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%Table2-2-x-------------------------------------------------------------------------------
\begin{flushleft}
\begin{table}[ht!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ have tree variable $t$,
$x$, $n$, that $y$ is fixed.}
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.2,\bbalpha_{3}=0.3$&$1.35596505\times10^{-4}$&$1.34395454\times10^{-4}$&$1.34377414\times10^{-4}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4,\bbalpha_{3}=0.6$&$1.27265808\times10^{-4}$&$1.26561905\times10^{-4}$&$1.25629737\times10^{-4}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.6,\bbalpha_{3}=0.7$&$1.20259940\times10^{-4}$&$1.19793031\times10^{-4}$&$1.16883116\times10^{-4}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.8$&$2.99362980\times10^{-5}$&$1.32748298\times10^{-5}$&$4.65066226\times10^{-6}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.4 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.6$&$2.88319590\times10^{-5}$&$1.27763920\times10^{-5}$&$4.65066226\times10^{-6}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%Table2-3-y-------------------------------------------------------------------------------
\begin{flushleft}
\begin{table}[ht!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1}, \bbalpha_{2}$ have tree variable $t$, $y$, $n$, that
$x$ is fixed.}
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.2$&$6.54169632\times10^{-4}$&$1.382539696\times10^{-4}$&$3.93536798\times10^{-5}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.8$&$4.82846136\times10^{-4}$&$1.999813782\times10^{-5}$&$7.21156527\times10^{-5}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.5 , \bbalpha_{2}=0.6$&$4.55836128\times10^{-4}$&$5.821545927\times10^{-5}$&$1.60713243\times10^{-5}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.6$&$5.75138282\times10^{-5}$&$2.944033108\times10^{-6}$&$1.78113445\times10^{-7}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.7$&$5.68271757\times10^{-5}$&$2.393451379\times10^{-6}$&$1.43214173\times10^{-7}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%Table2-4-y-------------------------------------------------------------------------------
\begin{flushleft}
\begin{table}[h!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ have tree variable $t$,
$y$, $n$, that $x$ is fixed. }
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.2,\bbalpha_{3}=0.3$&$2.33138317\times10^{-3}$&$1.56846535\times10^{-4}$&$3.18440163\times10^{-5}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4,\bbalpha_{3}=0.6$&$1.92621300\times10^{-3}$&$8.62977077\times10^{-5}$&$1.32199024\times10^{-5}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.6,\bbalpha_{3}=0.7$&$1.28240167\times10^{-3}$&$5.23971866\times10^{-5}$&$1.01573409\times10^{-5}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.8$&$1.46864232\times10^{-4}$&$2.23977676\times10^{-6}$&$9.71019231\times10^{-8}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.4 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.6$&$1.79950021\times10^{-4}$&$2.21143135\times10^{-6}$&$9.21224652\times10^{-8}$\\
\hline \hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}

In our tables, we obtain $RMS$ of $Eq.\ref{IEQ8}$ for several
$\bbalpha$'s. The $RMS$ solutions isn't much more than $10^{-4}$.
With $n=500$, several amounts  $\bbalpha_{1}$, $\bbalpha_{2}$ and
$\Delta t$ with $y=0.5$, Beginning The $RMS$ is of $10^{-4}$ until
to $10^{-7}$ that the outcomes and the answers are accord and
variable time at has nearly effectless when it is tiny enough at
tables 3 and  the table 4  we have tree fractional the
$\bbalpha_{i},i=1,2,3$ that have been illustrated for two term
$\bbalpha_{1}$, $\bbalpha_{2}$ and tree term $\bbalpha_{1}$,
$\bbalpha_{2}$, $\bbalpha_{3}$ with $x=0.5$, the $RMS$ is among
$10^{-4}$ until $10^{-6}$ and  $10^{-3}$ to $10^{-8}$ respectively.
 When the $N$ grow, the $RMS$ is reducing slowly and decreasing the error by grow
the $X$ to little by little in $Fig.15$ and $Fig.16$.
\\
%fig example2-1x  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{example2_dx2a.eps}
    \end{center}
    \caption{The shape $RMS$ for $u(x,0.5,t)$  with $\bbalpha_{1}$, $\bbalpha_{2}$ that are
    $\bbalpha_{1}=0.2$, $\bbalpha_{2}=0.6$ of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$. }
    \label{states}
\end{figure}
%fig example2-2x  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{example2_dx3a.eps}
    \end{center}
    \caption{The shape $RMS$ for $u(x,0.5,t)$  with  $\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ that are $\bbalpha_{1}=0.1$,
   $\bbalpha_{2}=0.5$, $\bbalpha_{3}=0.8$ of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$. }
    \label{states}
\end{figure}
%fig example2-3y  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example2_X_3.eps}
    \end{center}
    \caption{ The shape $RMS$ for $u(0.5,y,t)$  with  $\bbalpha_{1}, \bbalpha_{2}$ that are $\bbalpha_{1}=0.5$,
   $\bbalpha_{2}=0.6$ of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$. }
    \label{states}
\end{figure}
%fig example2-4y  ------------------------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{example2_Y_3.eps}
    \end{center}
    \caption{The shape $RMS$ for $u(0.5,y,t)$  with $\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ that are $\bbalpha_{1}=0.1$,
  $\bbalpha_{2}=0.5$, $\bbalpha_{3}=0.8$ of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
It is in the above figures$\Delta t=0.01$ and  $n=500$. For
approximate answers  with $y=0.5$ that in $Fig.17$ in fact displays
the $Error$ of $Eq.\ref{IEQ7}$ and we considered $\bbalpha_{1}=0.2$,
$\bbalpha_{2}=0.6$  in $Fig.18$ we considered $\bbalpha_{1}=0.1$,
$\bbalpha_{2}=0.5$, $\bbalpha_{3}=0.8$, the $N$ is dimensions of
$fBSf$. we look in the shapes $RMS$ in axis $X$ isn't decrease than
$10^{-3}$ by notice with $N=2$ it is $10^{-4}$, at in $Fig.19$ and
$Fig.20$ the powers factional are look to $Fig.17$ and $Fig.18$ in
order only  $x =0.5$ instead $y=0.5$.
It is manner is not  fast it is not  rapidity increase  tangible .\\\\
%example 3- 2 fractional----------------------------------------------------------------
\textbf{Example 3} \\
The third example, we discuss the $Eq.\ref{IEQ1}$ with two variable
$x,y$ that's mean $\overline{\mathbf{X}}\in \mathbb{R}^{2}$ and
several amounts for $\bbalpha$ and $t\in [0,1]$ and $\triangle t^{i}
= 0.01$ in partition $\Omega = [0,0.5]\times[0,0.5]$. The
$\mathcal{U}(x,y,t)=t^{1+\bbalpha_{1}+\bbalpha_{2}}x^{2}e^{y}$ is
solution
\begin{eqnarray}
\nonumber\mathbb{F}(x,y,t)&=&-2t^{1+\bbalpha_{1}+\bbalpha_{2}}e^{y}+
x^{2}e^{y}\Gamma{(1+\bbalpha_{1}+\bbalpha_{2})}(1+\bbalpha_{1}+\bbalpha_{2})
\left[\frac{(t^{2+\bbalpha_{1}})\Gamma{(2-\bbalpha_{1})}}{\Gamma{(3+\bbalpha_{1})}\Gamma{(1-\bbalpha_{2})}}
+\frac{(t^{2+\bbalpha_{2}})\Gamma{(2-\bbalpha_{2})}}{\Gamma{(3+\bbalpha_{2})}\Gamma{(1-\bbalpha_{1})}}\right]
\end{eqnarray}
and tree term  fractional  $\bbalpha_{i},i=1,2,3$
$$\mathcal{U}(x,y,t)=t^{1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3}}x^{2}e^{y}$$
also
\begin{eqnarray}
\nonumber\mathbb{F}(x,y,t)&=&-2t^{2+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3}}(x^{2}+y^{2})+
x^{2}e^{y}\Gamma{(1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3})}(1+\bbalpha_{1}+\bbalpha_{2})\\
&&\nonumber\left[\frac{(t^{1+\bbalpha_{1}+\bbalpha_{2}})\Gamma{(2-\bbalpha_{3})}}{\Gamma{(3+\bbalpha_{1}+\bbalpha_{2})}\Gamma{(1-\bbalpha_{3})}}
+\frac{(t^{1+\bbalpha_{1}+\bbalpha_{3}})\Gamma{(2-\bbalpha_{2})}}{\Gamma{(3+\bbalpha_{2}+\bbalpha_{3})}\Gamma{(1-\bbalpha_{2})}}
+\frac{(t^{1+\bbalpha_{2}+\bbalpha_{3}})\Gamma{(2-\bbalpha_{1})}}{\Gamma{(3+\bbalpha_{2}+\bbalpha_{3})}\Gamma{(1-\bbalpha_{1})}}\right]
\end{eqnarray}
In this sample  the exact answers is one exponent function in $x$
variable for plot the $Error$ of obtained answers by amounts of
Degree of fraction, assume one of the variables  the variable $X$ or
$Y$to be constant then  we calculate the $RMS$.We assume amounts
fixed away from knots primary. Anew the $N$ is dimension of $fBSf$
and the $N$ is grow $Error$ is not  increase. The $Fig. 21$, $Fig.22$,
$Fig.23$ and $Fig.24$ are answers at several time surfaces for
$\bbalpha$ have been presented.

%Table3-1------is table y is constant and t,x,n are variable------------------------------------------------------
\begin{flushleft}
\begin{table}[ht!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1}, \bbalpha_{2}$ have tree variable $t, x, n$, that $y$
is fixed.}
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.5 , \bbalpha_{2}=0.6$&$9.04541182\times10^{-5}$&$1.41615859\times10^{-6}$&$6.03249119\times10^{-7}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.7$&$4.16261408\times10^{-5}$&$1.93217574\times10^{-6}$&$8.35037092\times10^{-7}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.6$&$8.58065467\times10^{-5}$&$1.73144761\times10^{-6}$&$7.46330818\times10^{-7}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4$&$4.56260027\times10^{-5}$&$3.62032205\times10^{-6}$&$6.53930371\times10^{-7}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.7 , \bbalpha_{2}=0.8$&$1.80267851\times10^{-5}$&$1.36214067\times10^{-6}$&$2.43485256\times10^{-7}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%Table3-2------is table x is constant and t,y,n are variable-----------------------------
\begin{flushleft}
\begin{table}[h!]
\caption{ Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and
the $\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ have tree variable
$t, x, n$, that $y$ is fixed. }
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.6$&$5.19353341\times10^{-4}$&$3.80155456\times10^{-5}$&$9.73121322\times10^{-6}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.7$&$4.80850444\times10^{-4}$&$3.78465263\times10^{-5}$&$9.69569172\times10^{-6}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.3,\bbalpha_{3}=0.8$&$4.68682804\times10^{-4}$&$3.43935168\times10^{-5}$&$8.59295668\times10^{-6}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4,\bbalpha_{3}=0.6$&$4.04031852\times10^{-4}$&$2.32012072\times10^{-5}$&$1.42442171\times10^{-6}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.4,\bbalpha_{3}=0.9$&$3.09153935\times10^{-4}$&$1.74275616\times10^{-5}$&$1.04006198\times10^{-6}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{flushleft}
\begin{table}[ht!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq\ref{IEQ8}$ and the
$\bbalpha_{1}, \bbalpha_{2}$ have tree variable $t, y, n$, that $x$
is fixed. }
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.5 , \bbalpha_{2}=0.6$&$9.04541182\times10^{-5}$&$1.24974484\times10^{-6}$&$4.05615235\times10^{-7}$\\
\hline
$\bbalpha_{1}=0.7 , \bbalpha_{2}=0.1$&$9.90638751\times10^{-5}$&$1.71281036\times10^{-6}$&$5.62713624\times10^{-7}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.6 , \bbalpha_{2}=0.3$&$9.26941318\times10^{-5}$&$1.05348294\times10^{-6}$&$5.03312048\times10^{-7}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4$&$5.54470808\times10^{-5}$&$6.02212710\times10^{-6}$&$3.83331255\times10^{-7}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.7 , \bbalpha_{2}=0.8$&$2.16824420\times10^{-5}$&$2.26147866\times10^{-6}$&$1.43730085\times10^{-7}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Table3-4---This is table x is constant and t,y,n are variable.-------------------------------------------------
\begin{flushleft}
\begin{table}[ht!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ have tree variable $t, y,
n$, that $x$ is fixed. }
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.6$&$7.50950353e\times10^{-5}$&$9.16838821\times10^{-6}$&$3.04125495\times10^{-7}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.7$&$6.99727485\times10^{-5}$&$9.13493187\times10^{-6}$&$3.03772247\times10^{-7}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.3,\bbalpha_{3}=0.8$&$6.64170418\times10^{-5}$&$8.22103967\times10^{-6}$&$2.73865500\times10^{-7}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4,\bbalpha_{3}=0.6$&$5.59944023\times10^{-5}$&$4.34802764\times10^{-6}$&$3.44168282\times10^{-7}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.4,\bbalpha_{3}=0.9$&$3.74528508\times10^{-5}$&$2.84022165\times10^{-6}$&$4.65066226\times10^{-6}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
At Our tables, we obtain $RMS$ of $Eq.\ref{IEQ8}$ for several
$\bbalpha$'s. The $RMS$ solutions is not  much more than $10^{-4}$.
With $n=1000$, several amounts  $\bbalpha_{1}, \bbalpha_{2}$ and
$\Delta t$ with $y=0.5$ at tables 7 and 8, Beginning The $RMS$ is of
$10^{-5}$ until to $10^{-7}$ that the outcomes and the answers are
accord and  variable time at has nearly effectless when it is
tiny enough at tables 9 and 10  we have tree fractional the
$\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ that have been
illustrated  for two term $\bbalpha_{1}$, $\bbalpha_{2}$ and tree
term $\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ with $x=0.5$, the
$RMS$ is among $10^{-4}$ until $10^{-6}$.
 %----------------------------------------------------------------------------------------
%example 3- 3 fractional---------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example3_X_2}
    \end{center}
    \caption{Example of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$ and
in diagram of absolute error of $u(x,0.5,t)$ at with $\bbalpha_{1},
\bbalpha_{2}$ that are $\bbalpha_{1}=0.3$, $\bbalpha_{2}=0.6$.}
    \label{states}
\end{figure}
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example3_X_3}
    \end{center}
    \caption{The shape $RMS$ for $u(x,0.5,t)$  with $\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ that are
    $\bbalpha_{1}=0.3, \bbalpha_{2}=0.4, \bbalpha_{3}=0.9$. of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example3_Y_2}
    \end{center}
    \caption{The shape $RMS$ for $u(0.5,y,t)$  with $\bbalpha_{1}, \bbalpha_{2}$ that are
    $\bbalpha_{1}=0.3, \bbalpha_{2}=0.6$ of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example3_Y_3}
    \end{center}
    \caption{    The shape $RMS$ for $u(0.5,y,t)$  with $\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ that are
    $\bbalpha_{1}=0.01, \bbalpha_{2}=0.4, \bbalpha_{3}=0.9$. of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
It is in the above figures $\Delta t=0.01$ and  $n=500$. For
approximate answers  with $y=0.5$ that in $Fig.21$ in fact displays
the $Error$ of $Eq.\ref{IEQ7}$ and we considered $\bbalpha_{1}=0.3,
\bbalpha_{2}=0.6$  in $Fig.22$ we considered $\bbalpha_{1}=0.3,
\bbalpha_{2}=0.4, \bbalpha_{3}=0.9$, the $N$ is dimensions of
$fBSf$. we look in the shapes $RMS$ in axis $X$ is not decrease than
$10^{-3}$ by notice with $N=2$ it is $10^{-4}$, at in $Fig.23$ and
$Fig.24$ the powers factional are look to $Fig.21$ and $Fig.22$ in
order only  $x =0.5$ instead $y=0.5$. It is manner is not
fast  it is not  rapidity increase  tangible.\\
%example 4- 2 fractional---------------------------------------------------------------------
\textbf{Example 4}
We discuss the $Eq.(\ref{IEQ1})$ with two
variable $x,y$ that's mean $\overline{\mathbf{X}}\in \mathbb{R}^{2}$
and several amounts for $\bbalpha$ and $\triangle t^{i} =
t^{i}-t^{i-1}=0.01$ in partition $\Omega = [0,0.5]\times[0,0.5]$ and
$t\in [0,1]$. The
$\mathcal{U}(x,y,t)=t^{1+\bbalpha_{1}+\bbalpha_{2}}x^{2}\sin{(\pi
y)}$ is solution

\begin{eqnarray}
\nonumber\mathbb{F}(x,y,t)&=&(t^{1+\bbalpha_{1}+\bbalpha_{2}}\sin{(\pi
y))}(-2+\pi^{2}x^{2}) + x^{2}\sin{\pi
y}\Gamma{(1+\bbalpha_{1}+\bbalpha_{2})}(1+\bbalpha_{1}+\bbalpha_{2})\\
&&\nonumber\left[\frac{(t^{2+\bbalpha_{1}})\Gamma{(2-\bbalpha_{1})}}{\Gamma{(3+\bbalpha_{1})}\Gamma{(1-\bbalpha_{2})}}
+\frac{(t^{2+\bbalpha_{2}})\Gamma{(2-\bbalpha_{2})}}{\Gamma{(3+\bbalpha_{2})}\Gamma{(1-\bbalpha_{1})}}\right]
\end{eqnarray}
and tree term  fractional  $\bbalpha_{i},i=1,2,3$
$$\mathcal{U}(x,y,t)=t^{1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3}}x^{2}\sin{(\pi y)}$$
also
\begin{eqnarray}
\nonumber\mathbb{F}(x,y,t)&=&(t^{2+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3}})(-2+(x^{2}\sin{(\pi
y)})+ x^{2}\sin{(\pi
y)}\Gamma{(1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3})}(1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3})\\
&&\nonumber\left[\frac{(t^{1+\bbalpha_{1}+\bbalpha_{2}})\Gamma{(2-\bbalpha_{3})}}{\Gamma{(3+\bbalpha_{1}+\bbalpha_{2})}\Gamma{(1-\bbalpha_{3})}}
+\frac{(t^{1+\bbalpha_{1}+\bbalpha_{3}})\Gamma{(2-\bbalpha_{2})}}{\Gamma{(3+\bbalpha_{2}+\bbalpha_{3})}\Gamma{(1-\bbalpha_{2})}}
+\frac{(t^{1+\bbalpha_{2}+\bbalpha_{3}})\Gamma{(2-\bbalpha_{1})}}{\Gamma{(3+\bbalpha_{2}+\bbalpha_{3})}\Gamma{(1-\bbalpha_{1})}}\right]
\end{eqnarray}
In this sample  the exact answers is one $\sin(x)$ function in $x$
variable for plot the $Error$ of obtained answers by amounts of
Degree of fraction, assume one of the variables  the variable $X$ or
$Y$to be constant then  we calculate the $RMS$.We assume amounts
fixed away from knots primary. Anew the $N$ is dimension of $fBSf$
and the $N$ is grow $Error$ is not increase. The $Fig. 25$,
$Fig.262$, $Fig.27$ and $Fig.28$ are answers at several time
surfaces for $\bbalpha$ have been presented.

%Table4-1-------------This is table y is constant and t,x,n are variable-------------------------------------------
\begin{flushleft}
\begin{table}[h!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1}, \bbalpha_{2}$ have tree variable $t, x, n$, that $y$
is fixed. }
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.2$&$2.48704511\times10^{-5}$&$2.48680178\times10^{-5}$&$2.50683895\times10^{-6}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.4$&$2.11915060\times10^{-6}$&$2.11905899\times10^{-6}$&$2.11839033\times10^{-6}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.6$&$1.47744861\times10^{-6}$&$1.47738445\times10^{-6}$&$1.47691804\times10^{-6}$\\
\hline
$\bbalpha_{1}=0.5 , \bbalpha_{2}=0.7$&$4.45767624\times10^{-8}$&$1.32072454\times10^{-8}$&$2.85215545\times10^{-9}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.4 , \bbalpha_{2}=0.8$&$4.45767614\times10^{-8}$&$1.32072443\times10^{-8}$&$2.85215514\times10^{-9}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%Table4-2----------------------This is table x is constant and t,y,n are variable.------------------------------------
\begin{flushleft}
\begin{table}[h!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ have tree variable $t, x,
n$, that $y$ is fixed. }
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.2,\bbalpha_{3}=0.3$&$1.93352892\times10^{-9}$&$1.93352789\times10^{-10}$&$1.93351972\times10^{-10}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4,\bbalpha_{3}=0.5$&$1.24062859\times10^{-9}$&$1.24062783\times10^{-10}$&$1.24062144\times10^{-10}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.5 , \bbalpha_{2}=0.6,\bbalpha_{3}=0.7$&$6.87005350\times10^{-9}$&$6.87004855\times10^{-10}$&$6.87000483\times10^{-10}$\\
\hline
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.9$&$2.61481782\times10^{-9}$&$7.74760432\times10^{-10}$&$1.67347895\times10^{-10}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.7 , \bbalpha_{2}=0.8,\bbalpha_{3}=0.9$&$2.61481782\times10^{-9}$&$7.74760433\times10^{-10}$&$1.67347895\times10^{-10}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%Table4-3-------------This is table x is constant and t,y,n are variable-------------------------------------------
\begin{flushleft}
\begin{table}[h!]
\caption{Sample of $Eq.
\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1},\bbalpha_{2}$ have tree variable $t, y, n$, that $x$
is fixed. }
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.2$&$8.31787593\times10^{-13}$&$7.27500489\times10^{-13}$&$3.77477483\times10^{-13}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.4$&$6.89621726\times10^{-13}$&$6.02980391\times10^{-13}$&$3.12902522\times10^{-13}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.6$&$4.80796722\times10^{-13}$&$4.20121940\times10^{-13}$&$2.18027408\times10^{-13}$\\
\hline
$\bbalpha_{1}=0.5 , \bbalpha_{2}=0.7$&$2.49135913\times10^{-14}$&$6.70824908\times10^{-15}$&$8.76460781\times10^{-16}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.4 , \bbalpha_{2}=0.8$&$2.49126107\times10^{-14}$&$6.70727405\times10^{-15}$&$8.76281172\times10^{-16}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%Table4-4----------------------This is table y is constant and t,x,n are variable.------------------------------------
\begin{flushleft}
\begin{table}[h!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ have tree variable $t, y,
n$, that $x$ is fixed. }
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.2,\bbalpha_{3}=0.3$&$6.29148619\times10^{-14}$&$5.33007986\times10^{-15}$&$6.79864808\times10^{-16}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4,\bbalpha_{3}=0.5$&$4.03686627\times10^{-14}$&$3.32564899\times10^{-15}$&$1.51765516\times10^{-16}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.5 , \bbalpha_{2}=0.6,\bbalpha_{3}=0.7$&$2.23543833\times10^{-14}$&$1.77788355\times10^{-15}$&$1.13809033\times10^{-16}$\\
\hline
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.9$&$1.46147562\times10^{-14}$&$3.93388530\times10^{-15}$&$5.13326847\times10^{-16}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.7 , \bbalpha_{2}=0.8,\bbalpha_{3}=0.9$&$1.46149330\times10^{-14}$&$3.93429190\times10^{-15}$&$5.13422205\times10^{-16}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
In our tables, we obtain $RMS$ of $Eq.\ref{IEQ8}$ for several
$\bbalpha$'s. The $RMS$ solutions is not much more than $10^{-4}$.
With $n=1000$, several amounts  $\bbalpha_{1}$, $\bbalpha_{2}$ and
$\Delta t$ with $y=0.5$ at tables 11 and 12, Beginning The $RMS$ is
of $10^{-5}$ until to $10^{-7}$ that the outcomes and the answers
are accord and  variable time at has nearly effectless when it is
tiny enough at tables 13 and 14  we have tree fractional the
$\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ that have been
illustrated  for two term $\bbalpha_{1}$, $\bbalpha_{2}$ and tree
term $\bbalpha_{1}$, $\bbalpha_{2}$, $\bbalpha_{3}$ with $x=0.5$,
the $RMS$ is among $10^{-4}$ until $10^{-6}$.

%example 4- 3 fractional---------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example4_X_2.eps}
    \end{center}
    \caption{ The shape $RMS$ for $u(x,0.5,t)$ with $\bbalpha_{1}, \bbalpha_{2}$ that are
   $\bbalpha_{1}=0.5, \bbalpha_{2}=0.7$ of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example4_X_3.eps}
    \end{center}
    \caption{The shape $RMS$ for $u(x,0.5,t)$ with  with $\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ that are
   $\bbalpha_{1}=0.3$, $\bbalpha_{2}=0.5$, $\bbalpha_{3}=0.9$. of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example4_Y_2.eps}
    \end{center}
    \caption{The shape $RMS$ for  $u(0.5,y,t)$  with $\bbalpha_{1}, \bbalpha_{2}$ that are
   $\bbalpha_{1}=0.5$, $\bbalpha_{2}=0.7$, $\bbalpha_{3}=0.9$. of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example4_Y_3.eps}
    \end{center}
    \caption{The shape $RMS$ for $u(0.5,y,t)$ with  $\bbalpha_{1}, \bbalpha_{2}, \bbalpha_{3}$ that are
  $\bbalpha_{1}=0.3$, $\bbalpha_{2}=0.5$, $\bbalpha_{3}=0.9$ of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
From the above  figures $\Delta t=0.01$ and  $n=1000$. For
approximate answers  with $y=0.5$ that in $Fig.25$ in fact displays
the $Error$ of $Eq.\ref{IEQ7}$ and we considered $\bbalpha_{1}=0.5,
\bbalpha_{2}=0.7$ in $Fig.26$ we considered $\bbalpha_{1}=0.3,
\bbalpha_{2}=0.5, \bbalpha_{3}=0.9$, the $N$ is dimensions of
$fBSf$. we look in the shapes $RMS$ in axis $X$ is not decrease than
$10^{-3}$ by notice with $N=2$ it is $10^{-4}$, at in $Fig.27$ and
$Fig.28$ the powers factional are look to $Fig.25$ and $Fig.26$ in
order only  $x =0.5$ instead $y=0.5$. It is manner is not fast  it
is not  rapidity increase  tangible.


%example 5- 2 fractional---------------------------------------------------------------------
\textbf{Example 5} \\
The fifth sample, we discuss the $Eq.\ref{IEQ1}$ with two variable
$x,y$ that's mean $\overline{\mathbf{X}}\in \mathbb{R}^{2}$ and
several amounts for $\bbalpha$ and $t\in [0,1]$ and $\triangle
t^{i}=0.01$ in partition $\Omega = [0,1]\times[0,0.5]$. The
$\mathcal{U}(x,y,t)=t^{1+\bbalpha{1}+\bbalpha{2}}\cos{(\pi
x)}\sin{(\pi y)}$ is solution
$\mathcal{U}(x,y,t)=t^{1+\bbalpha{1}+\bbalpha{2}}\cos{(\pi
x)}\sin{(\pi y)}$ also
\begin{eqnarray}
\nonumber\mathbb{F}(x,y,t)&=&(\cos{(\pi x)}\sin{(\pi y)})
[(2\pi^{2})(t^{1+\bbalpha_{1}+\bbalpha_{2}}+
\Gamma{(1+\bbalpha_{1}+\bbalpha_{2})}(1+\bbalpha_{1}+\bbalpha_{2})\\
&&\nonumber\left[\frac{(t^{2+\bbalpha_{1}})\Gamma{(2-\bbalpha_{1})}}{\Gamma{(3+\bbalpha_{1})}\Gamma{(1-\bbalpha_{2})}}
+\frac{(t^{2+\bbalpha_{2}})\Gamma{(2-\bbalpha_{2})}}{\Gamma{(3+\bbalpha_{2})}\Gamma{(1-\bbalpha_{1})}}\right]
\end{eqnarray}
and tree term  fractional $\bbalpha_{i},i=1,2,3$
$\mathcal{U}(x,y,t)=t^{1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3}}x^{2}\sin{(\pi
y)}$ also
\begin{eqnarray}
\nonumber\mathbb{F}(x,y,t)&=&(\cos{(\pi x)}\sin{(\pi y)})
[(t^{2+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3}})(2\pi^{2})
+\Gamma{(1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3})}(1+\bbalpha_{1}+\bbalpha_{2}+\bbalpha_{3})\\
&&\nonumber\left[\frac{(t^{1+\bbalpha_{1}+\bbalpha_{2}})\Gamma{(2-\bbalpha_{3})}}{\Gamma{(3+\bbalpha_{1}+\bbalpha_{2})}\Gamma{(1-\bbalpha_{3})}}
+\frac{(t^{1+\bbalpha_{1}+\bbalpha_{3}})\Gamma{(2-\bbalpha_{2})}}{\Gamma{(3+\bbalpha_{2}+\bbalpha_{3})}\Gamma{(1-\bbalpha_{2})}}
+\frac{(t^{1+\bbalpha_{2}+\bbalpha_{3}})\Gamma{(2-\bbalpha_{1})}}{\Gamma{(3+\bbalpha_{2}+\bbalpha_{3})}\Gamma{(1-\bbalpha_{1})}}\right]
\end{eqnarray}

In this sample  the exact answers is one  $\cos(x)$
 multiplied by $\sin(y)$ function in $x$ variable and variable $y$for plot
the $Error$ of obtained answers by amounts of Degree of fraction,
assume one of the variables  the variable $X$ or $Y$to be constant
then  we calculate the $RMS$.We assume amounts fixed away from knots
primary. Anew the $N$ is dimension of $fBSf$ and the $N$ is grow
$Error$ is not increase. The $Fig. 29$, $Fig.30$, $Fig.31$ and
$Fig.28$ are answers at several time surfaces for $\bbalpha$ have
been presented.

%Table5-1--------------------This is table x is constant and t,y,n are variable-------------------------------------
\begin{flushleft}
\begin{table}[ht!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1},\bbalpha_{2}$ have tree variable $t, x, n$, that $y$
is fixed. }
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4$&$2.66410382\times10^{-5}$&$8.11472163\times10^{-6}$&$1.84662960\times10^{-6}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.7$&$2.12768140\times10^{-5}$&$6.48025816\times10^{-6}$&$1.47455616\times10^{-6}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.6$&$1.90424748\times10^{-5}$&$5.79995106\times10^{-6}$&$1.31984354\times10^{-6}$\\
\hline
$\bbalpha_{1}=0.5 , \bbalpha_{2}=0.9$&$1.10666715\times10^{-5}$&$3.36960185\times10^{-6}$&$7.66820560\times10^{-7}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.6 , \bbalpha_{2}=0.8$&$1.10663554\times10^{-5}$&$3.36976093\times10^{-6}$&$7.66936795\times10^{-7}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%Table5-2--------------------------------------------------------------------------------
\begin{flushleft}
\begin{table}[ht!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{i},i=1,2,3$. have tree variable $t, x, n$, that $y$ is
fixed. }
   \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.2,\bbalpha_{3}=0.3$&$2.64417141\times10^{-5}$&$7.83458201\times10^{-6}$&$1.692269728\times10^{-6}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4,\bbalpha_{3}=0.6$&$1.36523566\times10^{-5}$&$4.08011152\times10^{-6}$&$8.827629491\times10^{-7}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.6,\bbalpha_{3}=0.9$&$7.20941288\times10^{-6}$&$2.15433676\times10^{-6}$&$4.661227410\times10^{-7}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.9$&$9.89595187\times10^{-6}$&$2.95725439\times10^{-6}$&$4.6.3983023\times10^{-7}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.6 , \bbalpha_{2}=0.7,\bbalpha_{3}=0.8$&$5.27428503\times10^{-6}$&$1.57597159\times10^{-6}$&$3.409935412\times10^{-7}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%Table5-3--------------------This is table y is constant and t,x,n are variable-------------------------------------
\begin{flushleft}
\begin{table}[ht!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{1},\bbalpha_{2}$ have tree variable $t, y, n$, that $x$
is fixed. }
  \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4$&$1.12350796\times10^{-4}$&$3.70391510\times10^{-5}$&$8.11254912\times10^{-6}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.7$&$5.82415382\times10^{-5}$&$1.98429677\times10^{-5}$&$4.56905448\times10^{-6}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.6$&$3.08286709\times10^{-5}$&$1.04770384\times10^{-5}$&$2.41259187\times10^{-6}$\\
\hline
$\bbalpha_{1}=0.5 , \bbalpha_{2}=0.9$&$4.22480211\times10^{-5}$&$1.43728151\times10^{-5}$&$3.30905635\times10^{-6}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.6 , \bbalpha_{2}=0.8$&$2.26190713\times10^{-5}$&$7.67361570\times10^{-6}$&$1.76739514\times10^{-6}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
%Table5-4--------------------------------------------------------------------------------
\begin{flushleft}
\begin{table}[ht!]
\caption{Sample of $Eq.\ref{IEQ1}$ and $RMS$ $Eq.\ref{IEQ8}$ and the
$\bbalpha_{i},i=1,2,3$ have tree variable $t, y, n$, that $x$ is
fixed.  }
   \centering
  \resizebox{\textwidth}{!}{
\begin{scriptsize}
\hspace{2.4 cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l} \hline
&\multicolumn{1}{l|}{$RMS^{0}_{j}$}&\multicolumn{1}{l|}{$RMS^{1}_{j}$}&\multicolumn{1}{l|}{$RMS^{2}_{j}$}\\
 \hline \hline
\rowcolor{lightgray}
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.2,\bbalpha_{3}=0.3$&$1.35596506\times10^{-4}$&$1.34395454\times10^{-4}$&$1.34377414\times10^{-4}$\\
\hline
$\bbalpha_{1}=0.2 , \bbalpha_{2}=0.4,\bbalpha_{3}=0.6$&$1.27265809\times10^{-4}$&$1.26561905\times10^{-4}$&$1.25629738\times10^{-4}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.3 , \bbalpha_{2}=0.6,\bbalpha_{3}=0.9$&$1.20259941\times10^{-4}$&$1.19793031\times10^{-4}$&$1.16883116\times10^{-4}$\\
\hline
$\bbalpha_{1}=0.1 , \bbalpha_{2}=0.5,\bbalpha_{3}=0.9$&$2.99362980\times10^{-5}$&$1.32748298\times10^{-5}$&$4.65066226\times10^{-6}$\\
\hline\rowcolor{lightgray}
$\bbalpha_{1}=0.6, \bbalpha_{2}=0.7,\bbalpha_{3}=0.8$&$2.88319590\times10^{-5}$&$1.27763920\times10^{-5}$&$4.65066226\times10^{-6}$\\
\hline
\end{tabular}
  \end{scriptsize}
      }
\end{table}
\end{flushleft}
In our tables, we obtain $RMS$ of $Eq.\ref{IEQ8}$ for several
$\bbalpha$'s. The $RMS$ solutions is not much more than $10^{-4}$.
With $n=1000$, several amounts  $\bbalpha_{1}$, $\bbalpha_{2}$ and
$\Delta t$ with $y=0.5$ at tables 15 and 16, Beginning The $RMS$ is
of $10^{-6}$ until to $10^{-7}$ that the outcomes and the answers
are accord and variable time at has nearly effectless when it is
tiny enough at tables 17 and 18  we have tree fractional the
$\bbalpha_{i},i=1,2,3$ that have been illustrated  for two term
$\bbalpha_{1}$, $\bbalpha_{2}$ and tree term $\bbalpha_{1}$,
$\bbalpha_{2}$, $\bbalpha_{3}$ with $x=0.5$, the $RMS$ is among
$10^{-4}$ until $10^{-6}$.

%example 5- 1x fractional---------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example5_X_2.eps}
    \end{center}
    \caption{The shape $RMS$ for $u(x,0.5,t)$ with $\bbalpha_{1}$, $\bbalpha_{2}$ that are
  $\bbalpha_{1}=0.3, \bbalpha_{2}=0.6$ of $Eq.\ref{IEQ1}$ and error
  $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
%example 5- 2x fractional---------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example5_X_3.eps}
    \end{center}
    \caption{The shape $RMS$ for $u(x,0.5,t)$ with $\bbalpha_{i},i=1,2,3$ that are
  $\bbalpha_{1}=0.1, \bbalpha_{2}=0.5, \bbalpha_{3}=0.9$ of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
%example 5- 3y fractional---------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example4_Y_2.eps}
    \end{center}
    \caption{ The shape $RMS$ for $u(0.5,y,t)$  with $\bbalpha_{1}$, $\bbalpha_{2}$ that are
  $\bbalpha_{1}=0.3, \bbalpha_{2}=0.6$ of $Eq.\ref{IEQ1}$ and error $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
%example 5- 4y fractional---------------------------------------------------------------------
\begin{figure}[ht!]
    \begin{center}
    \includegraphics[width=16cm, height=7cm,angle=0]{Example5_Y_3.eps}
    \end{center}
    \caption{The shape $RMS$ for $u(0.5,y,t)$ with $\bbalpha_{i},i=1,2,3$ that are
  $\bbalpha_{1}=0.1, \bbalpha_{2}=0.5, \bbalpha_{3}=0.9$ of $Eq.\ref{IEQ1}$ and error
  $Eq.\ref{IEQ7}$.}
    \label{states}
\end{figure}
From the above  figures $\Delta t=0.01$ and  $n=1000$.  For
approximate answers  with $y=0.5$ that in $Fig.29$ in fact displays
the $Error$ of $Eq.\ref{IEQ7}$ and we considered $\bbalpha_{1}=0.3,
\bbalpha_{2}=0.6$  in $Fig.30$ we considered $\bbalpha_{1}=0.1,
\bbalpha_{2}=0.5, \bbalpha_{3}=0.9$, the $N$ is dimensions of
$fBSf$. we look in the shapes $RMS$ in axis $X$ is not decrease than
$10^{-4}$ by notice with $N=2$ it is $10^{-5}$, at in $Fig.31$ and
$Fig.32$ the powers factional are look to $Fig.29$ and $Fig.30$ in
order only  $x =0.5$ instead $y=0.5$. It is manner is not fast to it
is not  rapidity increase  tangible.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{conclusions}
In our manuscript, we  have solved  multi-term time fractional
diffusion-wave equation by Collocation Method where the
$\mathcal{D}_{t})$ in this is Caputo concept for ($0<\bbalpha<1$).
 We have considered an arbitrary one- and
two-dimentional. Of $fBSf$ used
at collocation method. \\
We have examined two issues here, the first Simplicity and ease of
applying this method to multi-term time fractional
diffusion-wave equation.  Our second goal was to apply these basic functions to these types of equations.\\
The effectiveness and high accuracy of the proposed numerical
approximate scheme  provided numerical results and figures
demonstrate. To test the correctness of the method, we provided
several examples with different exact answers in the powers.
Numerical simulations were performed using Mathlab.
\\


\begin{thebibliography}{10}
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL
}\fi

\bibitem{vahid1}
Hosseini, V. R., Shivanian, E. and  Chen, W. "Local integration of
2-D fractional telegraph equation via local radial point interpolant
approximation." The European Physical Journal Plus, 130(2) (2015):
33.
\bibitem{vahid2}
Hosseini, V. R., Chen, W. and  Avazzadeh, Z.  "Numerical solution of
fractional telegraph equation by using radial basis functions."
Engineering Analysis with Boundary Elements, 38 (2014): 31-39.

\bibitem{Yousefi}
Rivaz, A. and Yousefi, F. "An extension of the singular boundary
method for solving two dimensional time fractional diffusion
equations." Engineering Analysis with Boundary Elements, 83(2017):
167-179.

\bibitem{Yousefi1}
Yousefi, F., Rivaz, A. and Chen, W. "The construction of operational
matrix of fractional integration for solving fractional differential
and integro-differential equations." Neural Computing and
Applications,(2017): 1-12.




\bibitem{vahid3}
Hosseini, V. R., Shivanian, E. and Chen, W.  "Local radial point
interpolation (MLRPI) method for solving time fractional
diffusion-wave equation with damping." Journal of Computational
Physics, 312 (2016):307-332.

\bibitem{Stynes}
Stynes, M., O'Riordan, E. and Gracia, J. L.  "Error analysis of a
finite difference method on graded meshes for a time-fractional
diffusion equation." SIAM Journal on Numerical Analysis, 55(2)
(2017): 1057-1079.

\bibitem{Simmons}
Simmons, A., Yang, Q. and Moroney, T.  "A finite volume method for
two-sided fractional diffusion equations on non-uniform meshes."
Journal of Computational Physics, 335 (2017): 747-759.

\bibitem{Li}
Li, Y., Wang, Y. and  Deng, W. "Galerkin Finite Element
Approximations for Stochastic Space-Time Fractional Wave Equations.
SIAM Journal on Numerical Analysis.", 55(6)  (2017): 3173-3202.

\bibitem{Li1}
Li,D., Wang, J. and  Zhang, J.  "Unconditionally Convergent
L1-Galerkin FEMs for Nonlinear Time-Fractional Schrodinger
Equations." SIAM Journal on Scientific Computing, 39(6) (2017):
A3067-A3088.

\bibitem{vahid4}
Avazzadeh, Z., Hosseini, V. R. and Chen, W. "Radial basis functions
and FDM for solving fractional diffusion-wave equation." Iranian
Journal of Science and Technology (Sciences), 38(3) (2014): 205-212.

\bibitem{vahid5}
 Avazzadeh, Z., Chen, W. and Hosseini, V. R. "The
Coupling of RBF and FDM for Solving Higher Order Fractional Partial
Differential Equations."  Applied Mechanics and Materials
(598)(2014): 409-413.

\bibitem{vahid6}
Hosseini, V. R., Yousefi, F. and Zou, W. N. "The numerical solution
of high dimensional variable-order time fractional diffusion
equation via the singular boundary method." Journal of Advanced
Research (2021).

\bibitem{vahid7}
Hosseini, V. R., Koushki, M. and Zou, W. N. "The meshless approach
for solving 2D variable-order time-fractional advection–diffusion
equation arising in anomalous transport." Engineering with Computers
(2021): 1-19.



\bibitem{Baleanu1}
Baleanu, D., Khan, H., Jafari, H., Khan, R. A. and Alipour, M. "On
existence results for solutions of a coupled system of hybrid
boundary value problems with hybrid conditions." Advances in
Difference Equation 2015.1, (2015): 318.

\bibitem{Baleanu2}
Baleanu, D., Jafari, H., Khan, H. and Johnston, S. J. "Results for
mild solution of fractional coupled hybrid boundary value problems."
Open Mathematics, (2015): 13(1).


\bibitem{mr1}
 Jafari, H.,  Khalique, C., Ramezani, M. and Tajadodi, H. "Numerical solution of fractional
differential equations by using fractional B-spline." Open Physics
11.10 (2013): 1372-1376.

\bibitem{mr2}
 Ramezani, M., Jafari, H., Johnston, S. J. and  Baleanu, D. "Complex B-spline Collocation method for
solving weakly singular Volterra integral equations of the second
kind." Miskolc Mathematical Notes 16.2 (2015): 1091-1103.

\bibitem{mr3}
Ramezani, M. (2019). "Numerical analysis nonlinear multi-term time
fractional differential equation with collocation method via
fractional B-spline." Mathematical Methods in the Applied Sciences,
42(14), 4640-4663.

\bibitem{mr4}
Ramezani, M. (2021). "Numerical Analysis WSGD Scheme for One-and
Two-Dimensional Distributed Order Fractional Reaction–Diffusion
Equation with Collocation Method via Fractional B-Spline."
International Journal of Applied and Computational Mathematics,
7(2), 1-29.


\bibitem{Edwards}
Edwards, J. T., Neville, J. and  Simpson, C. A. "The numerical
solution of linear multi-term fractional differential equations:
systems of equations." Journal of Computational and Applied
Mathematics 148.2 (2002): 401-418.


\bibitem{Liu}
Liu,G. and Gu, Y." An Introduction to Meshfree Methods and Their
Programing." Springer, (2005).

\bibitem{Meerschaert}
Liu, F., Meerschaert, M., McGough, R., Zhuang, P. and Liu, Q.
"Numerical methods for solving the multi-term time-fractional
wave-diffusion equation." Fractional Calculus and Applied Analysis,
16.1 (2013): 9-25.

\bibitem{Zeng}
Zeng, F. " Second-order stable finite difference schemes for the
time-fractional diffusion-wave equation." Journal of Scientific
Computing, (2014): 1-20.

\bibitem{Zheng}
Zheng, M., Liu, F., Anh, V. and Turner, I. "A high-order spectral
method for the multi-term time-fractional diffusion equations."
Applied Mathematical Modelling, 40.7 (2016): 4970-4985.


\bibitem{Zayernouri}
Zayernouri, M. and  Karniadakis, G. E.  "Fractional spectral
collocation method." SIAM Journal on Scientific Computing, 36.1
(2014): A40-A62.


\bibitem{Zhao}
Zhao, L., Liu, F. and Anh, V. V. "Numerical methods for the
two-dimensional multi-term time-fractional diffusion equations."
Computers and Mathematics with Applications, (2017):2253-2268.

\bibitem{Schempp}
Schempp, W. "Complex contour integral representation of cardinal
spline functions." American Mathematical Society, (1982).

\bibitem{Chui}
Chui, C.  "Multivariate splines." Society for industrial and applied
mathematics, (1988).

\bibitem{Nürnberger}
Nurnberger, G. "Approximation by spline functions." Springer-Verlag,
(1989).

\bibitem{de Boor}
De Boor, C., H öl~lig, K. and Riemenschneider, S. "Box Splines."
Springer Science and Business Media(1993).

\bibitem{Forster}
Forster, B., Thierry B. and  Unser, M. "Complex B-splines." Applied
and  Computational Harmonic Analysis 20.2 (2006): 261-282.

\bibitem{Schoenberg1}
Schoenberg, I. J. "Contributions to the problem of approximation of
equidistant data by analytic functions. Part B. On the problem of
osculatory interpolation. A second class of analytic approximation
formulae." Quarterly of Applied Mathematics, 4.2 (1946): 112-141.

\bibitem{Schoenberg2}
Schoenberg, I. J. "Cardinal spline interpolation." Society for
Industrial and Applied Mathematics, (1973).


\bibitem{Boor}
De Boor, C., De Boor, C., Math ém~aticien, E. U., De Boor, C. and De
Boor, C.  "A practical guide to splines."  New York:
Springer-Verlag,(1978).

\bibitem{Prenter}
 Splines, PM Prenter. "Variational methods." New York, London, Sydney, Toronto, A Wiley-Interscince
 Publ (1975).

\bibitem{Bartels}
Bartels, R. H.,  Beatty, J. C. and Barsky, B. A. "An introduction to
splines for use in computer graphics and geometric modeling." Morgan
Kaufmann, (1987).

\bibitem{Unser1}
Unser, M.,  Aldroubi, A. and Eden, M. "B-spline signal processing.
I. Theory." IEEE transactions on
 signal processing, 41.2 (1993): 821-833.

\bibitem{Unser2}
 Unser, M. and  Blu, T. "Fractional splines and wavelets." SIAM review 42.1 (2000): 43-67.

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