\documentclass[preprint,8pt,fleqn]{elsarticle} \setlength\hoffset{-1in} \setlength\oddsidemargin{2.5cm} \setlength\textwidth{16.1cm} \setlength\textheight{20cm} \linespread{1} \usepackage[mathscr]{eucal} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{bbm} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{mathbbold} \usepackage{stmaryrd} \usepackage{pifont} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amssymb} \usepackage{babel} \usepackage{graphicx} \usepackage{amsopn,amstext,amsfonts,amsbsy} \usepackage{times} \usepackage{lineno} \usepackage{color} \usepackage{ dsfont } \usepackage{ amssymb } \usepackage{ mathrsfs } \usepackage{ upgreek } \usepackage{ gensymb } \usepackage{ amssymb } \usepackage{booktabs} \usepackage[ruled,vlined]{algorithm2e} \usepackage[table]{xcolor} \usepackage[bbgreekl]{mathbbol} \usepackage[table]{xcolor} %\smartqed %\newtheorem{example}{\sc \bf Example}[section] \usepackage{comment} \usepackage{hyperref} \hypersetup{colorlinks,linkcolor={blue},citecolor={blue},urlcolor={red}} %\usepackage[width=15cm, left=2cm, right=2cm, vscale=0.8, nohead]{geometry} \newcommand{\doubleint}{\mathop{\int\!\!\int}} \newcommand{\bep}{\begin{proposition}} \newcommand{\eep}{\end{proposition}} \newcommand{\bet}{\begin{theorem}} \newcommand{\eet}{\end{theorem}} \newcommand{\blm}{\begin{lemma}} \newcommand{\elm}{\end{lemma}} \newcommand{\bp}{\begin{proof}} \newcommand{\ep}{\end{proof}} \newcommand{\bcor}{\begin{corollary}} \newcommand{\ecor}{\end{corollary}} \newcommand{\bex}{\begin{example}} \newcommand{\eex}{\end{example}} \newcommand{\bremark}{\begin{remark}} \newcommand{\eremark}{\end{remark}} \newcommand{\bdf}{\begin{definition}} \newcommand{\edf}{\end{definition}} %\numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{proof}{Proof}[section] \setpagewiselinenumbers \modulolinenumbers[1] \linenumbers \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}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. %%%%00000000000000000000000000000000000000000000000 %00000000000000000000000000000000000000000000000000000000000000000000000000000000000 \end{thebibliography} \end{document}