\documentclass[11pt,twoside]{article} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks]{hyperref} \usepackage{float} \usepackage{lipsum} \usepackage{afterpage} \usepackage[labelfont=bf]{caption} \usepackage[nottoc,notlof,notlot]{tocbibind} %\renewcommand\bibname{References} \def\bibname{\Large \bf References} \usepackage{lipsum} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \renewcommand{\headrulewidth}{0pt} \fancyhead[LE,RO]{\thepage} \thispagestyle{empty} %\afterpage{\lhead{new value}} \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} %\topmargin=-1.6cm \textheight 17.5cm% \textwidth 12cm % \topmargin 8mm % \oddsidemargin 20mm % \evensidemargin 20mm % \footskip=24pt % \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} %\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \renewenvironment{proof}{{\bfseries \noindent Proof.}}{~~~~$\square$} \makeatletter \def\th@newremark{\th@remark\thm@headfont{\bfseries}} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\long\def\symbolfootnote[#1]#2{\begingroup% %\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup} \def \thesection{\arabic{section}} \begin{document} %\baselineskip 9mm %\setcounter{page}{} \thispagestyle{plain} {\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} \end{center} \vspace{4mm} {\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)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 $n-1<\alpha(x) 1$, 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. 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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}