\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 97, pp. 1--17.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2014/97\hfil Well-posedness] {Well-posedness of fractional parabolic differential and difference equations \\ with Dirichlet-Neumann conditions} \author[A. Ashyralyev, N. Emirov, Z. Cakir \hfil EJDE-2014/97\hfilneg] {Allaberen Ashyralyev, Nazar Emirov, Zafer Cakir} % in alphabetical order \address{Allaberen Ashyralyev \newline Department of Mathematics, Fatih University, Buyukcekmece, Istanbul, Turkey} \email{aashyr@fatih.edu.tr} \address{Nazar Emirov \newline Department of Mathematics, Fatih University, Buyukcekmece, Istanbul, Turkey} \email{nazaremirov@gmail.com} \address{Zafer Cakir \newline Department of Mathematical Engineering, Gumushane University, Gumushane, Turkey} \email{zafer@gumushane.edu.tr} \thanks{Submitted December 26, 2013. Published April 10, 2014.} \subjclass[2000]{35R11, 35B35, 47B39, 47B48} \keywords{Fractional parabolic equations; Dirichlet-Neumann conditions; \hfill\break\indent positive operator; difference schemes; stability} \begin{abstract} We study initial-boundary value problems for fractional parabolic equations with the Dirichlet-Neumann conditions. We obtain a stable difference schemes for this problem, and obtain theorems on coercive stability estimates for the solution of the first order of accuracy difference scheme. A procedure of modified Gauss elimination method is applied for the solution of the first and second order of accuracy difference schemes of one-dimensional fractional parabolic differential equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} Theory, applications and methods of solutions of problems for fractional differential equations have been studied extensively by many researchers (see, e.g., \cite{4}--\cite{7}, \cite{34}--\cite{5}, \cite{21}, \cite{27}, \cite{44}, \cite{41}--\cite{8}, \cite{10}--\cite{1}, \cite{43}--\cite{35} and the references given therein). In this article, we study the initial-boundary value problem \begin{equation} \begin{gathered} D_t^{\alpha }u(t,x)-a(x)u_{xx}(t,x)+\sigma u(t,x)=f(t,x),\quad 00$, $\sigma >0$. Theorem on coercive stability estimates for the solution of the initial-boundary value problem \eqref{1} is established. Stable difference schemes for the approximate solution of problem \eqref{1} are considered. Theorem on coercive stability estimates for the solution of the first order of accuracy in $t$ difference scheme is proved. A procedure of modified Gauss elimination method is applied for the solution of the first and second order of accuracy difference schemes for the fractional parabolic equations. The organization of the present paper as follows. The first section is introduction where we provide the history and formulation of the problem. In Section 2, theorem on coercivity stability of problem \eqref{1} is established. In Section 3, stable difference schemes for the approximate solution of problem \eqref{1} are considered. Theorem on coercivity stability for the first order of accuracy in $t$ difference scheme is proved. In Section 4, the numerical application is given. Finally, Section 5 is conclusion. \section{Theorems on coercive stability} We will give some statements which will be useful in the sequel. Let $E$ be a Banach space, and $A:D(A)\subset E\to E$ be a linear unbounded operator densely defined in $E$. We call $A$ $strongly$ $positive$ in the Banach space $E$, if its spectrum $\sigma _{A}$ lies in the interior of the sector of angle $\phi $, $0<2\phi <\pi $, symmetric with respect to the real axis, and if on the edges of this sector, $S_1(\phi )=\{\rho e^{i\phi }:0\leq \rho \leq \infty \}$ and $S_2(\phi )=\{\rho e^{-i\phi }:0\leq \rho \leq \infty \}$, and outside of the sector the resolvent $(\lambda -A)^{-1}$ is subject to the bound \begin{equation} \| (\lambda -A)^{-1}\| _{E\to E}\leq \frac{M}{1+|\lambda |}. \label{2} \end{equation} The infimum of such angles is called spectral angle $\varphi (A,E)$ of $A$. Throughout this article, positive constants have different values in time and they will be indicated with $M$ On the other hand $ M\left( \alpha ,\beta ,\cdots \right) $ is used to focus on the fact that the constant depends only on $\alpha ,\beta ,\cdots $. For a positive operator $A$ in the Banach space $E$, let us introduce the fractional spaces $E_{\beta }=E_{\beta }(E,A)(0<\beta <1)$ consisting of those $\nu \in E$ for which the norm \[ \| \nu \| _{E_{\beta }}=\sup_{\lambda >0}\lambda ^{\beta} \| A(\lambda +A)^{-1}\nu \| _{E}+\| \nu\| _{E} \] is finite. \begin{theorem}[\cite{18,13}]\label{thm1} Let $A$ and $B$ be two commutative positive operators with $\varphi(A,E)+\varphi(B,E)<\pi$. Then it follows that there exists the bounded operator $(A+B)^{-1}$ defined on whole space $E$. Moreover, for every $\beta\in(0,1)$ and $f$, there exists a unique solution $u=u(f)$ of the problem \[ Au+Bu=f \] and the following estimates hold \begin{gather*} \| Au \|_{E_{\beta}(E,B)}+\| Bu \|_{E_{\beta}(E,B)}+\| Bu \|_{E_{\beta}(E,A)} \leq M(\beta)\| f \|_{E_{\beta}(E,B)}, \\ \| Au \|_{E_{\beta}(E,A)}+\| Bu \|_{E_{\beta}(E,A)}+\| Au \|_{E_{\beta}(E,B)}\leq M(\beta)\| f \|_{E_{\beta}(E,A)}. \end{gather*} \end{theorem} \begin{theorem}[\cite{13}]\label{thm2} Let $A$ be the positive operator with $\varphi(A,E)<\pi$. Then for $\beta\leq\frac{1}{2},A^{\beta}$ is a positive operator with $\varphi(A^{\beta},E)<\frac{\pi}{2}$. \end{theorem} \begin{theorem}[\cite{12}]\label{thm3} Let $A$ be the operator acting in $E=C[0,T]$ defined by the formula $Av(t)=v'(t)$, with the domain $D(A)=\{v(t):v'(t)\in C[0,T], v(0)=0\}$. Then $A$ is a positive operator in the Banach space $E=C[0,T]$ and \[ A^{\beta}f(t)=D_t^{\beta}f(t) \] for all $f(t)\in D(A)$. \end{theorem} From the above theorems it follows the following theorem. \begin{theorem}\label{thm4} Let $A$ and $B$ be the positive operators with $\varphi(A,E)<\pi$ and $\varphi(B,E)\leq\frac{\pi}{2}$. Then for $\beta\leq\frac{1}{2}$ it follows that there exists bounded $(D^{\beta}+B)^{-1}$ defined on whole space $E$. Moreover, for every $f$, there exists a unique solution $u=u(f)$ of the problem \[ D^{\beta}u+Bu=f \] and the following estimate holds \[ \| D^{\beta}u \|_{E_{\beta}(E,B)}+\| Bu \|_{E_{\beta}(E,B)} \leq M(\beta)\| f \|_{E_{\beta}(E,B)}. \] \end{theorem} Now, we consider the second order differential operator \begin{equation} B^{x}u(x)=-a(x)u_{xx}(x)+\sigma u(x) \label{3} \end{equation} with the domain $D(B^{x})=\{u;u,u',u''\in C[0,l],u(0)=0,u'(l)=0\}$. Let us introduce the Banach space $C^{\gamma }[0,l]$, $\gamma \in (0,1] $ of all continuous function $\varphi (x)$ defined on $[0,l]$ and satisfying a H\"{o}lder condition for which the following norm is finite \[ \| \varphi \| _{C^{\gamma }[0,l]}=\| \varphi \| _{C[0,l]}+\sup_{x_1\neq x_2}\frac{|\varphi (x_1)-\varphi (x_2)|}{ |x_1-x_2|^{\gamma }}, \] where $C[0,l]$ is the Banach space of all continuous functions $\varphi (x)$ defined on $[0,l]$ with the norm \[ \| \varphi \| _{C[0,l]}=\max_{x\in [ 0,l]}|\varphi (x)|. \] The positivity of the operator $B^{x}$ in the Banach space $C[0,l]$ was established (see, \cite{45,46}). Moreover, we have that for any $\beta \in (0,1/2)$ the norms in the spaces $E_{\beta }(E,B)$ and $C^{2\beta }[0,l]$ are equivalent. \begin{theorem}\label{thm5} For $\beta \in (0,1/2)$, the norms of the space $E_{\beta}(C[0,l],B^{x})$ and the H\"{o}lder space $C^{2\beta}[0,l]$ are equivalent. \end{theorem} The proof of Theorem \ref{thm5} is based on the following estimates \[ |G^{x}(x,x_0;\lambda )|\leq \frac{M(\sigma ,a)}{\sqrt{\sigma +\lambda }} \begin{cases} e^{-\frac{1}{2}\sqrt{\frac{\sigma +\lambda }{a}}(x-s)}, & 0\leq x_0 \leq x, \\ e^{-\frac{1}{2}\sqrt{\frac{\sigma +\lambda }{a}}(x_0-x)}, & x\leq x_0 \leq l , \end{cases} \] \[ |G_{x}^{x}(x,x_0;\lambda )|\leq M(\sigma ,a) \begin{cases} e^{-\frac{1}{2}\sqrt{\frac{\sigma +\lambda }{a}}(x-x_0)}, & 0\leq x_0 \leq x, \\ e^{-\frac{1}{2}\sqrt{\frac{\sigma +\lambda }{a}}(x_0-x)}, & x\leq x_0 \leq l \end{cases} \] for the Green's function of the differential operator $B^{x}$ defined by the formula \eqref{3} and it follows the scheme of the proof of the Theorem of paper \cite{20}. \begin{theorem}\label{thm6} For the solution of problem \eqref{1} the coercive stability estimate \[ \max_{0\leq t \leq T} \|u_{xx}(t,.)\|_{C^{\beta}[0,l]}\leq M(\beta) \| f(t,.)\|_{C^{\beta}[0,l]} \] holds, where $ M(\beta) $ does not depend on $ f(t,x)$ $(0\leq t\leq T$, $x\in [0,l])$ and $0<\beta<1$. \end{theorem} The proof of Theorem \ref{thm6} is based on the positivity of differential operator $B^{x}$ defined by formula \eqref{3}, on the Theorem \ref{thm3} on connection of fractional derivatives with fractional powers of positive operators, on the Theorem \ref{thm2} on spectral angle of fractional powers of positive operators, and on the Theorem \ref{thm4} on fractional powers of coercively positive sums two operators. \section{Difference schemes and stability estimates} The discretization of problem \eqref{1} is carried out in two steps. In the first step, let us define the grid space \[ [ 0,l]_h=(x_{n}=nh,\; 0\leq n\leq M,\; Mh=l) \] To the differential space operator $B^{x}$ generated by formula \eqref{3}, we assign the difference operator $B_h^{x}$ by the formula \begin{equation} B_h^{x}u^{h}=-a(x)u_{{x}_{n}\bar{x}_{n}}^{h}+\sigma u(x)^{h} \label{3a} \end{equation} acting in the space of grid functions $u^{h}(x)$, satisfying the conditions $u^{h}(x)=0$ for all $x=0$ and $D^{h}u^{h}(x)=0$ for $x=l$. Here $D^{h}u^{h}(x)$ is the approximation of $u_{x}$. With the help of $B_h^{x}$ we arrive at the initial boundary value problem \begin{equation} \begin{gathered} D_t^{\alpha }v^{h}(t,x)+B_h^{x}v^{h}(t,x)=f^{h}(t,x),\quad 00$. Then \eqref{1aa} is uniquely solvable, and the formula \begin{equation}\label{100} u^{h}=(B_h^{x}+\lambda)^{-1}f^{h}=\Bigl\{\sum_{i=1}^{M-1} G(k,i;\lambda+\sigma)f_ih\Bigr\}_0^{M} \end{equation} is valid, where \[ G(k,i;\lambda+\sigma)=\frac{h(R^{M-i}-R^{M+i})(R^{M-k}-R^{M+k})} {(1-R^2)(1+R^{2M-1})}+\frac{h(R^{|k-i|+1}-R^{k+i+1})}{(1-R^2)} \] for $1\leq i\leq M-1$, and $1\leq k\leq M$, \[ R=(1+\delta h)^{-1}, \delta=\frac{1}{2} \bigl(h(\lambda+\sigma)+\sqrt{(\lambda+\sigma)(4+h^2(\lambda+\sigma))}\bigr). \] \end{lemma} The grid function $G(k,i;\lambda +\sigma )$ is called the Green function of equation \eqref{1aa} and by the formulas for $R$ and $\delta $, we get \begin{equation} \sum_{i=1}^{M-1}G(k,i;\lambda +\sigma )h=\frac{1}{\lambda +\sigma }- \frac{1}{\lambda +\sigma }\frac{R^{k}+R^{2M-k-1}}{1+R^{2M-1}},\quad 1\leq k\leq M. \label{ssss} \end{equation} To prove the positivity on $B_h^{x}$ in the Banach space $C_h$, we need the following auxiliary lemmas \cite{19}. \begin{lemma} \label{8} The following estimate holds \begin{equation}\label{120} |\delta|\geq \max\Bigl\{\frac{|\lambda+\sigma|h}{2}, \sqrt{|\lambda+\sigma|}\Bigr\}. \end{equation} \end{lemma} \begin{lemma}\label{9} The following estimate \begin{equation}\label{130} |R|\leq \frac{1}{1+\sqrt{|\lambda+\sigma|}h\cos\theta}<1 \end{equation} is valid, where $|\theta|<\pi/2$. \end{lemma} \begin{theorem} \label{thm10} For all $\lambda$ in the sector $\Sigma_{\theta}=\{\lambda:|\arg\lambda|\leq\theta, 0\leq\theta<\pi/2\}$ the resolvent $(\lambda I+B_h^{x})^{-1}$ defined by \eqref{100} satisfies the following estimate \begin{equation}\label{140} \|(\lambda I+B_h^{x})^{-1}\|_{C_h\to C_h} \leq \frac{M(\mu,\theta,\sigma)}{1+|\lambda|}. \end{equation} \end{theorem} \begin{proof} First, we consider the operator $B_h^{x}$ defined by formula \eqref{3a} in the case $a(x)=1$. Let us set $k=M$. Since \begin{align*} u_{M} &=\frac{h^2R(1-R^{M-1})(1+R^{M-1})}{(1-R)(1+R^{2M-1})}f_{M-1} \\ &\quad +\frac{1}{(1-R)(1+R^{2M-1})}\sum_{i=1}^{M-2} \Bigl(R^{M-i}-R^{M+i}\Bigr)h^2f_i, \end{align*} we have that \begin{align*} \bigl|u_{M} \bigr| &\leq 2\Bigl|\frac{R}{1-R}\Bigr|h^2|f_{M-1}|+\frac{1}{\bigl(1-|R|\bigr)}\sum_{i=1}^{M-2} \Bigl(|R|^{M-i}+|R|^{M+i}\Bigr)h^2\bigl|f_i\bigr| \\ &\leq 2h^2\|f^{h}\|_{C_h}\Bigl\{|\frac{R}{1-R}| +\frac{|R^2|}{\bigl(1-|R|\bigr)^2}\Bigr\}. \end{align*} Now, let us $1\leq k\leq M-1$. Then by formula \eqref{100} and the triangle inequality, we obtain \begin{align*} |u_k| &\leq \frac{\bigl(|R|^{M-k} +|R|^{M+k}\bigr)}{|1-R^2| \bigl|1+R^{2M-1}\bigr|}\sum_{i=1}^{M-1}\Bigl(|R|^{M-i} +|R|^{M+i}\Bigr)h^2\bigl|f_i\bigr| \\ &\quad +\frac{1}{|1-R^2|}\sum_{i=1}^{M-1}\Bigl (|R|^{|k-i|+1}+|R|^{k+i+1}\Bigr)h^2\bigl|f_i\bigr| \\ &\leq \frac{2}{|1-R^2|} \sum_{i=1}^{M-1}\Bigl(|R|^{M-i+1} +|R|^{M+i+1}\Bigr)h^2\bigl|f_i\bigr| \\ &\quad +\frac{1}{|1-R^2|}\sum_{i=1}^{M-1} \Bigl(|R|^{|k-i|+1}+|R|^{k+i+1} \Bigr) h^2\bigl|f_i\bigr| \\ &\leq \frac{4h^2}{|1-R^2|}\|f^{h}\|_{C_h} \sum_{i=1}^{M-1}|R|^{M-i+1} \\ &\quad +\frac{2h^2}{|1-R^2|}\|f^{h}\|_{C_h}\Bigl \{\sum_{i=1}^{k-1}|R|^{k-i+1}+|R| +\sum_{i=k+1}^{M-1}|R|^{i-k+1}\Bigr\} \\ &\leq \frac{2h^2}{|1-R^2|}\|f^{h}\|_{C_h}\Bigl \{\frac{2|R|^2}{1-|R|} +\frac{2|R|^2}{1-|R|}+ |R|\Bigr\} \\ &\leq M\Bigl\{\frac{|R|^2}{\bigl(1-|R|\bigr)^2} \frac{h^2}{\bigl|1+R\bigr|}+\Bigl|\frac{R}{1-R}\Bigr|\Bigl|\frac{1} {1+R}\Bigr|h^2\Bigr\}. \end{align*} From estimate \eqref{130} it follows that \begin{equation}\label{150} \frac{|R|^2}{\bigl(1-|R|\bigr)^2}\leq\Bigl(\frac{\frac{1} {1+\sqrt{|\lambda+\sigma|}hcos\theta}} {1-\frac{1}{1+\sqrt{|\lambda+\sigma|}hcos\theta}}\Bigr)^2 =\Bigl(\frac{1}{\sqrt{|\lambda+\sigma|}hcos\theta}\Bigr)^2. \end{equation} Clearly, we have that \begin{align*} |\lambda+\sigma| &=|\rho\cos\theta+i\rho\sin\theta+\sigma| &=\sqrt{\rho^2+2\rho\sigma\cos\theta+\sigma^2} \\ &\geq\sqrt{\rho^2\cos^2\theta+2\rho\sigma\cos\theta+\sigma^2} =|\lambda|\cos\theta+\sigma. \end{align*} Thus \begin{equation} \label{160} \begin{aligned} \frac{1}{|\lambda+\sigma|} &\leq\frac{1}{|\lambda|\cos\theta+\sigma}\leq\frac{1} {|\lambda|\cos\theta+\sigma\cos\theta}\\ &=\frac{\frac{1}{\cos\theta}}{|\lambda|+\sigma} =\frac{\frac{1}{\sigma\cos\theta}}{1+\frac{1}{\sigma}|\lambda|}\\ &\leq\frac{M(\sigma,\theta)}{1+|\lambda|}. \end{aligned} \end{equation} Combining estimates \eqref{150} and \eqref{160}, we obtain that \begin{equation}\label{170} \frac{h^2|R|^2}{\bigl(1-|R|\bigr)^2} \leq\frac{\frac{1}{\cos^2\theta}}{|\lambda+\sigma|} \leq\frac{M(\sigma,\theta)}{1+|\lambda|}. \end{equation} From the definition of $R$ and estimate \eqref{120}, it follows that \begin{equation}\label{180} \Bigl|\frac{R}{1-R}\Bigr|h^2=\frac{h}{|\delta|}\leq\frac{2}{1+|\lambda|}. \end{equation} Combining estimates \eqref{170} and \eqref{180}, we obtain \[ \|u^{h}\|_{C_h}\leq\frac{M(\mu,\sigma,\theta)}{1+|\lambda|}\|f^{h}\|_{C_h}. \] This concludes the proof of Theorem \ref{thm10} in the case $a(x)=1$. Second, noted that the proof of this statement is based on estimates for the Green's function. Under one more assumption that $\sigma>0$ is sufficiently large number, applying a fixed point Theorem, same estimates for the Green's function can be obtained. Therefore, this statement of theorem is true also for difference operator $B_h^{x}$ defined by formula \eqref{3a}. Theorem \ref{thm10} is proved. \end{proof} \begin{theorem}\label{thm11} Let $0<\beta<\frac{1}{2}$. Then, the norms of spaces $E_{\beta}(C_h,B_h^{x}$) and $C_h^{2\beta}$ are equivalent uniformly in $h$, $00$ and $x\in[0,l]$. Therefore, $f^{h}\in E_{\beta}(C_h,B_h^{x})$ and \[ \|f^{h}\|_{E_{\beta}(C_h,B_h^{x})}\leq M_1(\sigma)\|f^{h}\|_{C_h^{2\beta}}. \] Now, we prove the reverse inequality. For any positive operator $B_h^{x}$, we can write \[ v=\int_0^{\infty}\sum_{i=1}^{M-1}G(k,i;\lambda+\sigma)B_h^{x}(B_h^{x} +\lambda)^{-1}f_ih_1dt. \] Consequently, \[ f_{k}-f_{k+r}=\int_0^{\infty}\sum_{i=1}^{M-1}\lambda^{-\beta}[G(k+r,i;\lambda +\sigma)-G(k,i;\lambda+\sigma)] \lambda^{\beta}A_h^{x}(A_h^{x}+\lambda)^{-1}f_ih_1dt, \] hence \[ |f_{k}-f_{k+r}|\leq\int_0^{\infty}\lambda^{-\beta} \sum_{i=1}^{M-1}|G(k+r,i;\lambda +\sigma)-G(k,i;\lambda+\sigma)|h_1dt\|f^{h}\|_{E_{\beta}(C_h,B_h^{x})}. \] Let \[ T_h=|rh_1|^{-2\beta}\int_0^{\infty}\lambda^{-\beta} \sum_{i=1}^{M-1}|G(k+r,i;\lambda+\sigma)-G(k,i;\lambda+\sigma)|h_1dt. \] The proof of estimate \[ \frac{|f_{k}-f_{k+r}|}{|rh_1|^{2\beta}} \leq T_h\|f^{h}\|_{E_{\beta}(C_h,B_h^{x})} \] is based on the Lemmas \ref{8} and \ref{9}. Thus, for any $1\leq k