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\fancyhead[CE]{S. Sheykhi, M. Matinfar, M.A. Firoozjaee} 
\fancyhead[CO]{Solving a class of variable-order  differential equations via Ritz-approximation method and Genocchi polynomials}



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

\begin{center}

{\Large \bf 
Solving a class of variable-order  differential equations via Ritz-approximation method and Genocchi polynomials\\}
%{\bf Do You Have a Subtitle? \\ If so, Write It Here} 


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

{\bf  S. Sheykhi}\vspace*{-2mm}\\
\vspace{2mm} {\small  Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran} \vspace{2mm}

{\bf  M. Matinfar}\vspace*{-2mm}\\
\vspace{2mm} {\small  Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran} \vspace{2mm}

{\bf  M.A. Firoozjaee$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small   Department of Mathematics, University of Science and Technology of  Mazandaran, P.O. Box 48518-78195, Behshahr, Iran} \vspace{2mm}

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{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} 
In this article, a  class of variable-order differential equations by the Ritz-approximation is solved. Firstly, the unknown function is estimated using the Ritz-approximation via Genocchi polynomials as the basis functions. Then, by collocation method and preference of Genocchi roots as the collocation points, a set of algebraic equations is obtained. This system of nonlinear equations is solved by \textit{Mathematica 10} software. Finally, by solving some numerical examples and comparing the achieved results with other methods, the validity and efficiency of the presented method are shown.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 34A08; 65L60

\noindent{\bf Keywords and Phrases:} 
Ritz-approximation; Caputo variable-order derivative; Genocchi polynomials; satisfier function.
\end{quotation}}

\section{ Introduction}
\quad Fractional calculus indicates the differential and integral with fractional order. In recent years, many phenomena in real life such as finance \cite{5.s}, medical \cite{2.s}, heat transfer \cite{3.s}, Geo-Hydrology \cite{atan}, are modeled by fractional calculus.\\
Fractional equations can be classified into fractional differential equations, optimal control problems, initial and boundary value problems, fractional partial differential, and partial integro-differential equations.\\
In recent years it has been observed that many phenomena in dynamic processes can not be characterized by constant fractional operators \cite{7.jj4, 8.jj4}. This means  the order of differential operator possesses a dynamic matter that can  vary a function of time, space, or parameters of the system.  In 1993, Samko and Ross introduced the concept of variable-order(VO) fractional operators and studied their properties \cite{10.jj4}. After that many scholars generalize the theory of VO fractional calculus and investigated its applications in various fields such as in   viscoelasticity oscillators \cite{21.s}, petroleum engineering \cite{22.s}, signal processing \cite{24.s} and engineering \cite{25.s}. The variable exponent kernel of the VO operators  causes achieving  the analytical solutions to be difficult and in many cases impossible. So, many researchers investigate numerical methods. In \cite{21.43, 22.43} the finite difference method is used for VO differential equations. Bhrawey and Zaky \cite{27.s} used the shifted Legendre polynomials for solving VO differential cable equation. Tavares et al. \cite{30.43} applied a numerical method for solving VO  partial differential equations. The authors in \cite{31.s}  solved the VO boundary value problems with the kernel method.
\par 
In this study, a class of VO differential eqations is studied as: 
\begin{equation}\label{1}
\begin{cases}
D^{\alpha(x)}u(x)+a(x)u'(x)+b(x)u(x)+c(x)u(\tau(x))=f(x),& x\in[0,1],\\
u(0)=\lambda_{0}, ~~u(1)=\lambda_{1},&
\end{cases}
\end{equation} 
where $a(x),b(x),c(x)\in C^2[0,1],\alpha(x),\tau(x),f(x)\in C[0,1], 0\leq\tau(x)\leq1$ and $ D^\alpha(x)$ represents the VO Caputo derivative that, $n-1\leq\alpha(x)<n$.
\par
The present article is arranged as follows: In Sect. 2, some preliminary definitions of VO calculus are introduced.
In Sect. 3, the Genocchi polynomials and function approximation are expressed. In Sect. 4, the numerical method for solving the VO fractional differential equations is explained.  In Sect. 5, the error bound is showed. In Sect. 6, the numerical conclusions achieved by this method are announced. Results display  the method  is very efficient for finding the numerical solution of VO differential equations.

\section{Preliminaries}
In this section, some required  definitions of VO operators will be mentioned.
\begin{definition}
	The VO Riemann-Liouville integral operator of order $ \alpha(x)  \geq0 $ of  function $ u(x) $ is defined as \cite{re}
	\begin{equation*}
	I^{\alpha(x)}u(x)=\frac{1}{\Gamma{(\alpha(x))}}\int_{0}^{x}\frac{u(t)}{(x-t)^{1-\alpha(x)}}\mathrm{d}t~~\alpha(x)>0,
	\end{equation*}
	where $ \Gamma{()}$,  is the Gamma function.
\end{definition}
\begin{definition}
	The VO Caputo derivative operator for a differentiable and continuous function $u(x)$, is defined as \cite{33.s}
	\begin{equation}\label{c}
	^{C}D^{\alpha(x)}u(x) =\frac{1}{\Gamma(n-\alpha(x))}\int_{0}^{x}(x-t)^{n-\alpha(x)-1}\frac{{d}^{n} u(t)}{dt^{n}}\mathrm{d}t,  
	\end{equation}
	where $n-1<\alpha(x)<n,~n\in\mathbb{N}$.
\end{definition}
If the values ​​of $\alpha(x)$, are an integer then the VO Caputo derivative  becomes identical to classical derivative.\\
\begin{corollary}
	As a direct conclusion from \eqref{c}, the  VO Caputo derivative is a linear operator such as the fractional
	and integer-order derivative.
	\[D^{\alpha(x)}(\lambda f(x)+\mu g(x))= \lambda D^{\alpha(x)} f(x)+\mu D^{\alpha(x)} g(x),\]
	where $\lambda$ and $\mu$ are constants.
\end{corollary}
\begin{corollary}
	Let $u(x)=x^k,k> 1$, then \cite{34.s} 
	\[ I^{\alpha(x)}u(x)=\frac{\Gamma(k+1)}{\Gamma(k+1+\alpha(x))}x^{k+\alpha(x)},~~ k>1,\]
	and
	\begin{equation}
	D^{\alpha(x)} u(x) =\left\{
	\begin{array}{ll}
	\frac{\Gamma(k+1)}{\Gamma(k+1-\alpha(x))}x^{k-\alpha(x)}, &k\in \mathbb{N}, k\geq n,\\
	0, &\text{otherwise,}\\
	\end{array}%
	\right.
	\end{equation}
	that $n=[\alpha(x)]$. Furthermore
	\[D^{\alpha(x)} C=0\quad (C \;\text{is a constant}).\]
\end{corollary}
%@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

\section{The properties of the Genocchi polynomials}
In this section, first, the Genocchi function is introduced  then  the function approximation is described.
\subsection{Genocchi polynomials}
The Genocchi polynomials of degree $n$ can be defined with generating function \cite{15.33}
\begin{equation}\label{G}
\frac{2t e^{xt}}{e^t+1}=\sum_{n=0}^{\infty}G_{n}(x). \frac{t^n}{n!},~~~~(\vert t \vert < \pi).
\end{equation}
Some of the first values of Genocchi polynomials are
\begin{align}
&G_1(x)=1,~~~G_2(x)=2x-1,~~~G_3(x)=3x^2-3x,& \notag\\
&G_4(x)=4x^3-6x^2+1,~~G_5(x)=5x^4-10x^3+5x.& \notag
\end{align}
Also, the Genocchi numbers $g_n:=G_n(0)$ are defined with the generating function 
\begin{equation}
\frac{2t }{e^t+1}=\sum_{n=0}^{\infty}g_{n}. \frac{t^n}{n!},~~~~(\vert t \vert < \pi).
\end{equation}
The Genocchi polynomial in \eqref{G} can be presented by the Genocchi number as follows
\[G_n(x)=\sum_{k=0}^{n}\binom{n}{k}g_{n-k}x^{k}.\]
In the following, some properties of the Genocchi polynomials are recalled
\begin{align*}
&G_n(1)+g_n=0,\quad n>1,\\
&\frac{dG_n(x)}{d x}=n G_{n-1}(x),\quad  n\geq 1,\\
&\int_{0}^{1} G_n(x) G_m(x) \mathrm{d}x=\frac{2(-1)^n m! n!}{(m+n)!}g_{m+n}(x), \quad m, n\geq 1.
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Function approximation}
Assume $H=span\lbrace G_1(x),\dots,G_N(x)\rbrace\subset L^2[0,1]$ is a finite-dimensional space. Any arbitrary $u(x)\in  L^2[0,1]$ has a unique best approximation $\tilde{u}(x)$ in $H$ such that \cite{14.s}
\[ \exists \tilde{u}(x)\in H;\quad \forall y(x)\in H \quad\quad \Vert u(x)-\tilde{u}(x) \Vert \leq  \Vert u(x)-y(x) \Vert.\]
Since $H \subset L^2$ is close subspace, then $L^2$ can be decomposed as  $L^2=H \oplus H^\bot$. Also, we have $u(x)=y(x)+y^\bot (x)$, therefore $u(x)-y(x) \in H^ \bot$ then
\[\forall y(x)\in H,\quad  \left\langle u(x)-\tilde{u}(x), y(x) \right\rangle =0,\]
where $\left\langle .,. \right\rangle$ showes the inner product. On the other hand, as $ \tilde{u}\in H$ then  $\lbrace c_i \rbrace_{i=0,\dots, n}$ are unique coeffficients that
\begin{equation}\label{d} 
u(x)\cong \tilde{u}(x)=\displaystyle\sum_{n=0}^{N}c_n G_n(x).
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Describtion the method}
Ritz-approximation is a simple and effective method first introduced by Ritz in 1908 to solve the initial and boundary differential equations. Now we apply this approximation for VO differential equations.\\
Suppose $u(x) \in L^2[0,1]$, is estimated using the Ritz-approximation as
\begin{equation}\label{R}
u(x)\cong \tilde{u}(x)=\displaystyle\sum_{n=0}^{N}c_n \phi(x) G_n(x)
\end{equation}
where $\phi(x)$ is a non-unique function that must be chosen in some way that $\phi (x_n)$ is not zero. Meanwhile, the  \eqref{R}  satisfy  the
homogenous initial problem \eqref{1}. Also, $w(x)$ that is called  ''\textit {satisfier function}'' must be satisfied in boundary conditions as
\[w(0)=u(0),~~~w(1)=u(1).\]
Several  ways  are to choose the satisfyer function but regularly interpolation is applied. Experience perceived that when  selected  satisfier function  is closer to the exact solution, obtained numerical results are  cost-effective computational \cite{16.b2017}.\\
Concerning boundary conditions \eqref{1}, the $\phi(x)$ is considered as 
\[ \phi(x)=x(x-1).\]
Now the unknown function $u(x)$ is estimated as
\begin{equation}
u(x)\cong u_N(x)=\displaystyle\sum_{n=0}^{N}c_n x(x-1)G_n(x)+w(x).
\end{equation}
This approximation is substituted in \eqref{1} and yields
\begin{equation}\label{7}
D^{\alpha(x)}u_k(x)+a(x)u'_k(x)+b(x)u_k(x)+c(x)u_k(\tau(x))=f(x),
\end{equation}
and  using the collocation method for \eqref{7} in $ x_i=\frac{i}{N+1},i=1,2\cdots,N$ points, a system of algebraic equations is achieved. This resulting system is solved with \textit{Mathematica 10} software and unknown coefficients $c_n$ are obtained.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Error analysis}
In the previous section, we consider $H$ is the set of Geocchi functions on $[0,1]$. Suppose $u(x)\in C^{N+1}[0,1]$ and $\tilde{u}(x)$ is the best approximation for $u(x)$. Also, $u_N(x) \in H$ in $[0,1]$ is defined as
\[ u_N(x)=\sum_{n=1}^N \frac{1}{n!}u^{(n)}(\xi)(x-\xi)^n,~~\xi \in[0,1].\]
Utilizing Taylor polynomials is infered that
\begin{equation}\label{27}
\vert u(x)-u_N(x) \vert \leq \frac{(x- \xi)^{N+1}}{(N+1)!}  \sup\limits_{x \in[0,1]}\vert u^{N+1}(x) \vert.
\end{equation} 
Consider $M=\sup_{x \in[0,1]} \vert u^{N+1}(x) \vert$ and using \eqref{27}, we have
\begin{align*}
\Vert u-\tilde{u}\Vert ^2_{L^2[0,1]} &\leq \Vert u-u_N \Vert^{2}_{L^2[0,1]}\\
&=\int_{0}^{1} \vert u(x)-u_N(x) \vert^2\mathrm{d}x\\
&\leq \int_{0}^{1} \left| u^{n+1}(\xi) \frac{(x-\xi)^{(n+1)}}{(n+1)!}\right|  ^{2} \mathrm{d}x\\
&=\frac{M^2}{[(N+1)!]^2} \int_{0}^{1} \vert (x- \xi )\vert^{2N+1},
\end{align*}
therefore,
\[ \int_{0}^{1} \vert (x- \xi) \vert ^{2N+2} \mathrm{d}x \leq \frac{2}{2N+3}.\]
Finally, we conclude that
\begin{equation}\label{28}
\parallel u-u_N\parallel_{L^2[0,1]} \leq \frac{M}{N+1} \sqrt{\frac{2}{2N+3}},
\end{equation}
when $N$  (the number of Genocchi functions)  is increasing, the right value of \eqref{28} inclines to zero, that means the approximation converges to $u(x)$.

%@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
\section{Illustrative examples}
The proposed numerical approach is used for solving two examples. The fractional order is $1\leq\alpha(x)<2$. The simulation is performed for $N = 5$ ($N $ is the number of terms in the truncated Genocchi series). The collocation points are $ x=\frac{i}{N+1},~i=1,\cdots,k$. The absolute error  is calculated with $ E_ {N} (x) = \vert u (x) -u_ N (x) \vert $. The result is compared with the previous work and tables, graphs indicate that with a low volume of conclusions,  high accuracy is obtained.
%     @@@@@@@@@    Examplae1       @@@@@@@@@
\begin{example}\label{ex1}
	Assume the VO differential equation with boundary value conditions as
	\begin{equation}\label{8}
	\begin{cases}
	D^{\alpha(x)}u(x)+cos(x) u'(x)+4u(x)+5u(x^{2})=f(x),& x\in[0,1],\\
	u(0)=0,~~ u(1)=1,&
	\end{cases}
	\end{equation}
	where
	$f(x)=\frac{2x^{2-\alpha(x)}}{\Gamma(3-\alpha(x))}+5x^{4}+4x^{2}+2x cos(x)$ and $\alpha(x)= \frac{5+sinx}{4}$. 
	The exact solution is $u(x)=x^{2}$.\\
	According to the previous section, firstly, the $u(x)$ is estimated by the Ritz-approximation. The \textit{satisfier function} that applies to boundary conditions in \eqref{8}  be as follows:
	\[ w(x)=x. \]
	Then, the unknown $c_k$ coefficients  are calculated with the collocation method and $x$ points:
	\begin{eqnarray*}
		\begin{matrix}
			c_0&=&1, &c_1=&-5.0445 \times 10^{-15}, &c_2=&-1.8983\times 10^{-14},\\
			c_3&=&-3.7484 \times 10^{-15},&c_4=&-7.37413\times 10^{-15}.
		\end{matrix}
	\end{eqnarray*}
	These Coefficients specify the $\tilde{u}(x)$.  The error of the method is drawn in Fig 1. and  the error of this method  is compared with RKSM (reproducing kernel splines method) \cite{2} in Tab 1.
	
%	\FloatBarrier
	\begin{figure}[h!] 
		\centering
		\includegraphics[width=8cm]{pic1.eps}
		\caption{\footnotesize{The absolute error of the proposed method  in Example 1.}}
		\label{fig1} 
	\end{figure}
%	\FloatBarrier
%	\FloatBarrier
	\begin{table}[h] 
		\caption{\footnotesize Comparsion of absolute errors in Example 1.}
		\label{tab1} 
		\begin{center}
			{\footnotesize 
				\begin{tabular}{ccc}
					\hline
					x &  $E_{20}$\cite{2}  & $E_{5}$ present method   
					\\ \hline
					$0.1$ & $2.79655E-14$ & $1.38778E-16$\\
					$0.2$ & $3.12597E-14$ & $2.22045E-16$\\
					$0.3$ & $3.50414E-14$ & $3.33067E-16$\\
					$0.4$ & $3.91076E-14$ & $3.33067E-16$\\
					$0.5$ & $4.37705E-14$ & $3.33067E-16$\\
					$0.6$ & $4.86278E-14$ & $3.05311E-16$\\
					$0.7$ & $5.37903E-14$ & $2.498E-16$\\
					$0.8$ & $5.92859E-14$ & $2.22045E-17$\\ 
					$0.9$ & $6.51701E-14$ & $2.35922E-16$\\
					\hline
			\end{tabular}}
		\end{center}
	\end{table}
%	\FloatBarrier
	
	
	This table  indicates the advantages of the proposed scheme, since  choosing  much fewer points than previous work,
	a high level of accuracy can be obtained.
\end{example} 


%      @@@@@@@@@    Examplae2       @@@@@@@@@

\begin{example}\label{ex2}
	Assume the VO fractional problem with boundary value condition of the form \cite{2}
	\begin{equation}
	\begin{cases}
	D^{\alpha(x)}u(x)+e^x u'(x)+2u(x)+8u(e^{x-1})=f(x),& x\in[0,1],\\
	u(0)=4,~~ u(1)=9,&
	\end{cases}
	\end{equation}
	where 
	$\alpha(x)=\frac{6+\cos x}{4}$
	and 
	$f(x)=
	\frac{2x^{2-\alpha(x)}}{\Gamma(3-\alpha(x))}
	+2(x^2+4x+4)+
	8(4e^{x-1}+
	e^{2x-2}+4)+
	e^x(2x+4)$. 
	The exact solution is $u(x)=x^2+4x+4$.\\
	The satisfier function is considered as $w(x)=(x+2)^2$.Using the discused method the $c_k$ are as:
	\begin{eqnarray*}
		\begin{matrix}
			c_0&=&5.4656 \times 10^{-14}, & c_1=&3.1455 \times 10^{-14}, &c_2=&1.9810\times 10^{-13},\\
			c_3&=&2.8209 \times 10^{-14},&c_4=&7.762\times 10^{-14}.
		\end{matrix}
	\end{eqnarray*}
	In  Fig.2, the absolute error is drawn on the interval $[0,1]$.

	
%	\FloatBarrier
	\begin{figure}[!h] 
		\centering
		\includegraphics[width=8cm]{pic2.eps}
		\caption{\footnotesize{The absolute error of the proposed method  in Example 2.}}
		\label{fig2} 
	\end{figure}
%	\FloatBarrier
\end{example}


\section*{Conclusion}
In the current paper,  a class of fractional VO differential equations is estimated by Genocchi polynomials via Ritz-approximation. Then the problem is solved by the collocation method. The error bound of the method is discussed and solved examples displayed the accuracy and reliability of the mentioned method. 



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\end{thebibliography}
\end{center}



{\small

\noindent{\bf  S. Sheykhi}

\noindent Department of Mathematics

\noindent PhD student in mathematics

\noindent University of Mazandaran, 

\noindent Babolsar, Iran

\noindent E-mail: smy.sheikhi93@gmail.com}\\


{\small
	
\noindent{\bf M. Matinfar}
	
\noindent Department of Mathematics
	
\noindent Assistant Professor of Mathematics
	
\noindent University of Mazandaran
	
\noindent Babolsar, Iran

\noindent E-mail: m.matinfar@umz.ac.ir}\\

{\small
\noindent{\bf  M.A. Firoozjaee  }

\noindent  Department of Mathematics

\noindent Associate Professor of Mathematics

\noindent University of Science and Technology of Mazandaran,

\noindent Behshahr, Iran

\noindent E-mail: m\_firoozjaee@mazust.ac.ir}\\



\end{document}