\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 57, pp. 1--7.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2014/57\hfil Mixed parabolic-hyperbolic equations] {Boundary problems for mixed parabolic-hyperbolic equations with two lines of changing type and fractional derivative} \author[B. J. Kadirkulov \hfil EJDE-2014/57\hfilneg] {Bakhtiyor J. Kadirkulov} % in alphabetical order \address{Bakhtiyor J. Kadirkulov \newline Tashkent State Institute of Oriental Studies, Tashkent, Uzbekistan} \email{kadirkulovbj@gmail.com} \thanks{Submitted January 21, 2014. Published February 28, 2014.} \subjclass[2000]{35M10} \keywords{Parabolic-hyperbolic equation; Tricomi problem; \hfill\break\indent Caputo fractional derivative; diffusion equation; Green's function; Volterra integral equation} \begin{abstract} In this article, we study a boundary value problem for a parabolic-hyperbolic equation with Caputo fractional derivative. Under certain conditions, we prove its unique solvability using methods of integral equations and Green's functions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} Gelfand started the study of mixed parabolic-hyperbolic type equations in his work \cite{g1}. Later on, Tricomi and Gellerstedt studied main boundary problems, and many authors have continued these studies as seen in the detailed bibliographies of \cite{d1,d2}. Son recent works \cite{e3,e3,k2,n2} have been devoted to the study of boundary problems for parabolic-hyperbolic equations with two or more lines of changing type. While other works have been devoted to the study of Riemann-Liouville, Caputo, Hadamard, Hadamard-Marchaud and other general fractional operators; see for example \cite{a2,b1,g1,h1,k2,k1,k5,n1,p1}. Non-local problems for parabolic-hyperbolic equations with one or two lines of changing type containing the Riemann-Liouville fractional derivative were investigated in \cite{b2,b5,e1,k3,k4}. The Caputo fractional derivative is suitable for numerical methods in the study of fractional differential equations \cite{a1,s1}, and it appears in many mathematical models of real-life processes \cite{c1,h1,m1}. In this article we study a boundary value problem for a parabolic-hyperbolic equation with the Caputo fractional derivative, and having two lines where it changes type. \section{Formulation of the problem and main result} Let $0 < \alpha$ be a real number. For a function $\varphi (t)$, given on $(0,\ell )$, $\ell < \infty$ an integral-differential operator in a sense of the Riemann-Liouville starting at $0$, is defined as follows \cite{k5,n1,p1}, \[ D_{0t}^\alpha \varphi (t) = \begin{cases} \frac{1}{{\Gamma ( - \alpha )}}\int_0^t {\frac{{\varphi (\tau )d\tau }}{{{{(t - \tau )}^{\alpha + 1}}}}} ,& \alpha < 0, \\ \varphi (t),& \alpha = 0, \\ \frac{{{d^n}}}{{d{t^n}}}D_{0t}^{\alpha - n}\varphi (t), & n - 1 < \alpha \leq n,\; n \in N. \end{cases} \] The operator \[ {}_CD_{0t}^\alpha \varphi (t) = D_{0t}^{\alpha - n}{\varphi ^{(n)}}(t),\quad n - 1 < \alpha \leq n,\; n \in N \] is called the Caputo fractional differential operator. The Riemann-Liouville and the Caputo differential operators are related by the equality \begin{equation} cD_{0t}^{\alpha }\varphi (t)={}_cD_{0t}^{\alpha }\varphi (t) +\sum_{k=0}^{n}{\frac{{{\varphi }^{k}}(0+)}{\Gamma (1+k-\mu )}{{t}^{k-\alpha }}}, \quad t>0. \label{e1} \end{equation} Let us consider the equation \begin{equation} \frac{\partial^2 u}{\partial x^2}-\frac{1-\operatorname{sign}(xy)}{2} \frac{\partial^2 u}{\partial y^2}-\frac{1+\operatorname{sign}(xy)}{2} \cdot {}_{C}D_{0y}^{\alpha }u=f(x,y), \,\,\alpha \in (0,1).\label{e2} \end{equation} This equation for $x>0, y>0$ is the fractional order diffusion equation \[ \frac{{{\partial }^{2}}u(x,y)}{\partial {{x}^{2}}}-{}_{C}D_{0y}^{\alpha }u(x,y) =f(x,y), \] which coincides at $\alpha = 1$ with the diffusion equation \cite{p1} \[ \frac{{{\partial ^2}u}}{{\partial {x^2}}} - \frac{{\partial u}}{{\partial y}} = f(x,y). \] Consider the \eqref{e2} in a finite domain $\Omega\subset \mathbb{R}^2$, bounded for $x>0$, $y > 0$ by segments $A_0B_0, B_0B$ of straight lines $y = 1$, $x = 1$; at $x>0$, $y<0$ by segments $AC, BC$ of characteristics $x + y = 0$, $x - y = 1$ of the \eqref{e2}; at $x<0$, $y>0$ by segments $AD, A_0D$ of characteristics $x + y = 0$, $y - x = 1$ of the \eqref{e2}. The parabolic part of the mixed domain $\Omega$ will be denoted by $\Omega_3$, and the hyperbolic part by $\Omega_1$, at $x>0$ and by $\Omega_2$ at $x<0$, respectively. The function $u(x,y)$ is called a regular solution of the \eqref{e2}, if it has necessary continuous derivatives participating in the \eqref{e2} and satisfies it in ${\Omega _1} \cup {\Omega _2} \cup {\Omega _3}$. In the domain $\Omega$ we study the following boundary problem. \noindent\textbf{Problem DS.} Find a function $u(x,y) \in C(\overline \Omega)$, such that: (1) $u$ is a regular solution of the \eqref{e2} in the domain $\Omega \backslash (AA_0 \cup AB)$; (2) satisfies boundary conditions \begin{equation} u(x,y)\big|_{BB_0 \cup DC} = 0;\label{e3} \end{equation} (3) on lines of type changing it satisfies the gluing conditions \begin{gather} u(x, - 0) = u(x, + 0), \,u( - 0,y) = u( + 0,y), \label{e4} \\ u_y(x,-0)=l_1(x)\cdot \lim_{y\to +0} y^{1-\alpha }u_y(x,y) +m_1(x)\cdot u(x,0)+n_1(x), \label{e5} \\ u_x(-0,y)=l_2(y)\cdot u_x(+0,y)+m_2(y)\cdot u(0,y)+n_2(y),\label{e6} \end{gather} where ${l_i}(t)$, $m_i(t)$, $n_i(t)$, $y \in [0,1]$, $i = 1,2$ are given functions. Note that Problem DS generalizes a problem studied in \cite{e2}. \begin{theorem} \label{thm1} Let the following conditions be fulfilled: $f(x,y) \in {C^\delta }(\overline \Omega )$, $0 < \delta < 1$, ${l_i}(t) \in {C^1}[0,1]$, $m_i(t),\,{n_i}(t) \in C[0,1]$, $l_i(t) \ne 0$, $l'_i(t) - 2{l_i}(t) \cdot {m_i}(t) \geq 0$ for all $t \in [0,1]$, $l_2(1) > 0$, $i = 1,2$. Then problem DS has a unique regular solution. \end{theorem} \subsection*{Proof Theorem \ref{thm1}} First, we find the main functional relations on $AB$, $AA_0$ deduced from the domains $\Omega_1$ and $\Omega_2$. We introduce the following notation \begin{gather} u(x,0) = {\tau _1}(x),\quad ,u(0,y) = {\tau _2}(y),\label{e7}\\ {u_y}(x, - 0) = \nu _1^ - (x),\quad \lim _{y \to + 0} {y^{1 - \alpha }}{u_y}(x,y) = \nu _1^ + (x),\label{e8} \\ {u_x}( - 0,y) = \nu _2^ - (y),\quad {u_x}( + 0,y) = \nu _2^ + (x).\label{e9} \end{gather} The solution to problem DS in $\Omega_1$ can be represented by the D'Alembert's formula \begin{equation} u(\xi ,\eta ) = \frac{1}{2}\Big[ {{\tau _1}(\xi ) + {\tau _1}(\eta ) - \int_\xi ^\eta {\nu _1^ - (t)dt} } \Big] - \int_\xi ^\eta {dt\int_y^\eta {{f_1}( {y,\tau })d\tau } }\label{e10} \end{equation} where \[ \xi = x + y,\quad \eta = x - y, \quad {f_1}(\xi ,\eta ) = \frac{1}{4}f\big( {\frac{{\xi + \eta }}{2},\frac{{\xi - \eta }}{2}} \big). \] Then using condition \eqref{e3} from \eqref{e10} we get the following relation, reduced from the domain $\Omega_1$ to the segment $AB$, \[ {\tau '_1}(x) - \nu _1^ - (x) = 2\int_0^x {{f_1}( {t,x})dt} ,\quad x \in (0,1).\label{e11} \] Similarly, from the formula \begin{equation} u(\xi ,\eta ) = \frac{1}{2}\Big[ {{\tau _2}(\xi ) + {\tau _2}(\eta ) - \int_\xi ^\eta {\nu _2^ - (t)dt} } \Big] - \int_\xi ^\eta {dt\int_y^\eta {{f_2}( {y,\tau })d\tau } }\label{e12} \end{equation} by \eqref{e3}, from the domain $\Omega_2$ one can deduce the following relation between functions ${\tau _2}(y)$ and $\nu _2^ - (y)$: \begin{equation} {\tau '_2}(y) - \nu _2^ - (y) = 2\int_0^y {{f_2}({t,y})dt} ,\quad y \in (0,1),\label{e13} \end{equation} where \[ \xi = x + y,\quad \eta = y - x, \quad {f_2}(\xi ,\eta ) = - \frac{1}{4}f\big( {\frac{{\xi - \eta }}{2},\frac{{\xi + \eta }}{2}} \big). \] First, we prove the uniqueness of the solution of problem DS. \begin{lemma} \label{lem1} Let the conditions of the theorem be valid. Then problem DS can not have more than one regular solution. \end{lemma} \begin{proof} Suppose the opposite. Let problem DS has two different regular solutions ${u_1}(x,y)$, ${u_2}(x,y)$ and let \[ u(x,y) = {u_1}(x,y) - {u_2}(x,y). \] It is not difficult to see that $u(x,y)$ is a regular solution of the homogeneous problem DS ($f(x,y) = 0$, ${n_i}(t) = 0$, $i = 1,2$). This is why one only needs to prove that homogeneous problem has only the trivial solution. Let $u(x,y)$ be a regular solution of the homogeneous problem DS in the domain $\Omega$. Since $u{u_{xx}} = {(u \cdot {u_x})_x} - u_x^2$, then integrating the identity $u\big( {{u}_{xx}}-{}_{C}D_{0y}^{\alpha }u(x,y) \big)=0$ along the domain ${\Omega _3}$, taking \eqref{e3}, \eqref{e7}-\eqref{e9} into account, after some evaluations we deduce \begin{equation} \iint_{{{\Omega }_{3}}}{u\,\cdot {}_{C}D_{0y}^{\alpha }u(x,y)\,dx\,dy} +\int_{0}^{1}{{{\tau }_{2}}(x)\nu _{2}^{+}(x)dx} +\iint_{{{\Omega }_{3}}}{u_{x}^{2}\,dx\,dy}=0\label{e14} \end{equation} Consider the integral \[ {I_2} = \int_0^1 {{\tau _2}(y)\nu _2^ + (y)dy}. \] Taking into consideration the relation \[ \nu _2^ + (y) = \frac{1}{{{l_2}(y)}}\nu _2^ - (y) - \frac{{{m_2}(y)}}{{{l_2}(y)}}{\tau _2}(y), \] which follows from gluing conditions \eqref{e5} and notation \eqref{e9}, integral ${I_2}$ is rewritten as \[ {I_2} = \int_0^1 {\frac{1}{{{l_2}(y)}}{\tau _2}(y)\nu _2^ - (y)dy} - \int_0^1 {\frac{{{m_2}(y)}}{{{l_2}(y)}}\tau _2^2(y)dy}. \] On the other hand from \eqref{e13} it follows that \[ \int_0^1 {\frac{1}{l_2(y)}\tau_2(y)\nu_2^-(y)dy} = \frac{\tau_2^2(1)}{2l_2(1)} + \frac{1}{2}\int_0^1 {\frac{l_2'(y)}{l_2^2(y)}\tau_2^2(y)dy}. \] Hence \begin{equation} {I_2} = \frac{{\tau _2^2(1)}}{{2{l_2}(1)}} + \frac{1}{2}\int_0^1 {\frac{{l_2'(y) - {l_2}(y){m_2}(y)}}{{l_2^2(y)}}\tau _2^2(y)dy}.\label{e15} \end{equation} Using the formula [see \cite[p. 53]{p1} \[ \lim_{t \to 0} D_{0t}^{\beta - 1}\varphi (t) = \Gamma (\beta )\lim _{t \to 0} {t^{1 - \beta }}\varphi (t), {t^{1 - \beta }}\varphi (t) \in C[0,1),0 < \beta < 1 \] from the \eqref{e2} passing to the limit at $y \to + 0$, taking notations \eqref{e7} and \eqref{e8} into account, we obtain \begin{equation} {\tau ''_1}(x) - \Gamma (\alpha )\nu _1^ + (x) = 0,\,\,x \in (0,1).\label{e16} \end{equation} Considering condition \eqref{e5}, equality \eqref{e16} can be rewritten as \[ {\tau ''_1}(x) - \frac{{\Gamma (\alpha )}}{{{l_1}(x)}} [ {\nu _1^ - (x) - {m_1}(x){\tau _1}(x)}] = 0. \] From here we obtain \begin{equation} - \int_0^1 {{{[{{\tau '}_1}(x)]}^2}dx} - \Gamma (\alpha )\int_0^1 {\frac{1}{{{l_1}(x)}}{\tau _1}(x)\nu _1^ - (x)dx} + \Gamma (\alpha )\int_0^1 {\frac{{{m_1}(x)}}{{{l_1}(x)}}\tau _1^2(x)dx} = 0.\label{e17} \end{equation} On the other hand from \eqref{e11} it follows that \[ \int_0^1 {\frac{1}{{{l_1}(x)}}{\tau _1}(x)\nu _1^ - (x)dx} = \int_0^1 {\frac{{l_1'(x)}}{{2l_2^2(x)}}\tau _1^2(x)dx}. \] Then relation \eqref{e17} will have the form \[ \int_0^1 {{{[{{\tau '}_1}(x)]}^2}dx} + \Gamma (\alpha )\int_0^1 {\frac{{l_1'(x) - 2{l_1}(x){m_1}(x)}}{{2l_1^2(x)}} \tau _1^2(x)dx} = 0. \] Here considering the conditions of the theorem we have ${\tau '_1}(x) = 0$. Hence, ${\tau _1}(x) =$ const. Since, ${\tau _1}(0) = 0$, it follows that ${\tau _1}(x) = 0$, $x \in [0,1]$. Taking \eqref{e15}, ${\tau _1}(x) = 0$ and formula \eqref{e1} into account, \eqref{e14} is rewritten as \begin{align*} &\iint_{{\Omega _3}} {uD_{RL}^\alpha u(x,y)\,dx\,dy + } \frac{{\tau _2^2(1)}}{{2{l_2}(1)}} + \frac{1}{2}\int_0^1 {\frac{{l_2'(x) - 2{l_2}(x){m_2}(x)}}{{l_2^2(x)}}\tau _2^2(x)dx}\\ &+ \iint_{{\Omega _3}} {u_x^2(x,y)}\,dx\,dy = 0. \end{align*} According to \cite[Theorem 1.7.1]{n1}, from the last equality we have $u(x,y)\equiv 0$ in ${\overline \Omega_3}$. Further, from formulas \eqref{e10} and \eqref{e12}, by virtue of the uniqueness of the solution of the Cauchy problem we have that $u(x,y)\equiv 0$ in ${\overline \Omega _1} \cup {\overline \Omega_2}$. Hence, $u(x,y)\equiv 0$; i.e., $u_1(x,y)\equiv u_2(x,y)$ in $\bar \Omega$. The proof is complete \end{proof} Now we prove the existence of the solution of problem DS. From the \eqref{e2} we deduce the following functional relation between functions ${\tau _1}(x)$ and $\nu _1^ + (x)$, on $AB$ \[ {\tau ''_1}(x) - \Gamma (\alpha )\nu _1^ + (x) = f(x,0).\label{e18} \] Defining function $\nu _1^ + (x)$. from conditioin \eqref{e5} and \eqref{e11} and substituting into the above equation we obtain the following problem for the unknown ${\tau _1}(x)$: \begin{gather*} {{\tau}_1}''(x) + p(x){{\tau}_1}'(x) + q(x){\tau _1}(x) = g(x), \\ {\tau _1}(0) = {\tau _1}(1) = 0, \end{gather*} where \[ p(x) = - \frac{\Gamma (\alpha )}{l_1(x)},\quad q(x) = \frac{m_1(x)}{l_1(x)},\quad g(x) = - \frac{1}{l_1(x)}\big[n_1(x) + 2\int_0^x {f_1(t,x)dt } \big]. \] The uniqueness of the solution of this problem follows from the uniqueness of the solution of problem DS. Note that this solution can be written by Green's function (see \cite{e2}). Since, function ${\tau _1}(x)$ is now known, from \eqref{e11} we find $\nu _1^-(y)$. Hence, the solution of the problem in ${\Omega_1}$ is known. The unknown function ${\tau _2}(y)$ can be found by the formula of the solution of the first boundary problem for the \eqref{e2} in ${\Omega _3}$ \cite{b4}: \begin{equation} \begin{aligned} u(x,y) &= \int_0^y {{G_\xi }(x,y,0,\eta ){\tau _2}(\eta )d\eta } +\int_0^1 {\tilde G(x - \xi ,y){\tau _1}(\xi )d\xi }\\ &\quad - \iint_{{\Omega _3}} {G(x,y,\xi ,\eta )f(\xi ,\eta )d\xi d\eta }, \end{aligned}\label{e19} \end{equation} where $G(x,y,\xi ,\eta )$ is the Green's function of the first boundary problem for the diffusion equation with the Riemann-Liouville fractional differential operator (see \cite[p. 108]{p1}) \begin{gather*} G(x,y,\xi ,\eta ) = \frac{{{{(y - \eta )}^{\beta - 1}}}}{2}\sum_{n = - \infty }^\infty \Big[ {e_{1,\beta }^{1,\beta } \Big( { - \frac{{| {x - \xi + 2n}|}}{{{{(y - \eta )}^\beta }}}} \Big) - e_{1,\beta }^{1,\beta }\Big( { - \frac{{| {x + \xi + 2n} |}}{{{{(y - \eta )}^\beta }}}} \Big)} \Big], \\ \tilde G(x - \xi ,y) = \frac{1}{{\Gamma (1 - \alpha )}} \int_0^y {{\eta ^{ - \alpha }}G(x,y,\xi ,\eta )d\eta }, \quad \beta = \frac{\alpha }{2}, \end{gather*} where $e_{1,\beta }^{1,\beta }( z)$ is the Wright's function, which has the form \[ e_{1,\beta }^{1,\delta }(z) = \sum_{n = 0}^\infty {\frac{{{z^n}}}{{n!\,\Gamma (\delta - \beta n)}}} . \] Calculating ${u_x}(x,y)$ from \eqref{e19} and letting $x$ go to zero, bearing in mind \cite[Lemma 2.2.2]{p1}, we get a relation between functions ${\tau _2}(y)$ and $\nu _2^ + (y)$ on $AA_0$: \[ \nu _2^ + (y) = - \int_0^y {K(y - t){{\tau}_2}'(t)dt} + {\Phi _0}(y),\label{e20} \] where \begin{gather*} K(y - t) = \frac{1}{{{{(y - t)}^\beta }}} \Big[ {\frac{1}{{\Gamma (1 - \beta )}} + 2\sum_{n = 1}^\infty {e_{1,\beta }^{1,1 - \beta } \Big( { - \frac{{2n}}{{{{(y - t)}^\beta }}}} \Big)} } \Big], \\ {\Phi _0}(y) = \lim _{x \to + 0} \Big[ {\int_0^1 {{{\tilde G}_x}(x - \xi ,y){\tau _1}(\xi )d\xi } - \iint_{{\Omega _3}} {{G_x}(x,y,\xi ,\eta )f(\xi ,\eta )d\xi d\eta }} \Big]. \end{gather*} Considering $\nu _2^ - (y) = \nu _2^ + (x) = {\nu _2}(x)$, excluding ${\nu _2}(x)$ from \eqref{e13} and (20), we get the Volterra integral equation of second kind regarding the unknown function ${\tau_2}'(y)$: \begin{equation} {\tau_2}'(y) + \int_0^y {K(y - t){{\tau}_2}'(t)dt} = \Phi (y), \label{e21} \end{equation} where \[ \Phi (y) = {\Phi _0}(y) + 2\int_0^y {{f_2}(t,y)dt}. \] Since a solution of the integral equation depends on the kernel $K(y - t)$, we shall study it in detail. The function $K(y - t)$ we represent as a sum of two kernels \[ K(y - t) = {K_1}(y - t) + {K_2}(y - t), \] where \[ {K_1}(y - t) = - \frac{{{{(y - t)}^{ - \beta }}}}{{\Gamma (1 - \beta )}},\quad {K_2}(y - t) = - \frac{2}{{{{(y - t)}^\beta }}} \sum_{n = 1}^\infty {e_{1,\beta }^{1,1 - \beta } \big( { - \frac{{2n}}{{{{(y - t)}^\beta }}}} \big)}. \] Note that ${K_1}(y - t)$ is a kernel with weak singularity. The kernel ${K_2}(y - t)$ represented as a series of Wright's type functions. From \cite[(2.2.5), (2.2.24)]{p1} it follows that the kernel ${K_2}(y - t)$ has also weak singularity. Therefore, equation \eqref{e21} is a Volterra equation of second kind with weak singularity. Using formulas \cite[2.2.19, 2.2.25, 2.3.8, 3.3.3]{p1}, it is not difficult to show that the right hand side of \eqref{e21} is a continuous function. Then, from the theory of integral equations follows that \eqref{e21} has a unique continuous solution (see, for example \cite{m2}). After finding the functions $\tau _1(x)$, $\tau _2(x)$, $\nu _1(x)$ and $\nu _2(y)$, the solution of problem DS in the domain $\Omega_3$ will be found by formula \eqref{e19}, and in the domains $\Omega_1$, $\Omega_2$ as a solution of the Cauchy problem by the formulas \eqref{e10} and \eqref{e12}, respectively. The proof of Theorem \ref{thm1} is complete. \subsection*{Acknowledgements} Author is grateful to Professor M. Kirane for his useful remarks and suggestions. \begin{thebibliography}{00} \bibitem{a1} O. P .Agrawal; \emph{Solution for Fractional Diffusion-Wave equation defined in a bounded domain.} Nonlnear Dynamics 29 (2002), pp. 145-155. \bibitem{a2} D. Amanov, B. J. Kadirkulov; \emph{A boundary problem for mixed parabolic equation of the fourth order with fractional derivative} (Russian). Uzbek Mathematical Journal. 3 (2009), pp. 8-17. \bibitem{b1} A. S. Berdyshev , A. Cabada, B. J. Kadirkulov; \emph{The Samarskii-Ionkin type problem for the fourth order parabolic equation with fractional differential operator.} Computers and Mathematics with Applications. 10 (62) (2011), pp. 3884-3893. \bibitem{b2} A. S. Berdyshev, A. Cabada, E. T. Karimov; \emph{On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator}. Nonlinear Anal., Theory Methods Appl. 75(6) (2012), pp. 3268-3273. \bibitem{b3} A. S. Berdyshev, B. J. Kadirkulov and J.J.Nieto; \emph{Solvability of an elliptic partial differential equation with boundary condition involving fractional derivatives}. Complex Variables and Elliptic Equations: An International Journal, DOI:10.1080/17476933.2013.777711. \bibitem{b4} A. S. Berdyshev, B. J. Kadirkulov; \emph{On boundary problems for fractional order diffusion equation.} Proceedings of the republican scientific conference ``Actual problems of mathematical analysis". Urgench, Uzbekistan, 2012, Vol. 1, pp. 70-71. \bibitem{b5} A. S. Berdyshev, E. T. Karimov, N. S. Akhtaeva; \emph{Boundary value problems with integral gluing conditions for fractional-order mixed-type equation}. Int. J. Differ. Equ. 2011, Article ID 268465 (2011). doi:10.1155/2011/268465 \bibitem{b6} A. S. Berdyshev, B. Kh. Turmetov, B. J. Kadirkulov; \emph{Some properties and applications of the integrodifferential operators of Hadamard-Marchaud type in the class of harmonic functions.} Siberian Mathematical Journal. 4 (53) (2012), pp. 600-610. \bibitem{c1} A. Carpinteri, F. Mainardi; (Eds.) \emph{Fractals and Fractional Calculus in Continuum Mechanics}, Telos, Springer-Verlag, 1998. \bibitem{d1} T. D. Djuraev, A. Sopuev, M.Mamajonov; \emph{Krayevye zadachi dlya uravneniy parabolo-giperbolicheskogo tipa}. [Boundary-value Problems for the Parabolic-hyperbolic Type Equations]. Tashkent: FAN, 1986. \bibitem{d2} T. D. Djuraev; \emph{Krayevye zadachi dlya uravneniy smeshannogo I smeshanno-sostavnogo tipov.} [Boundary value problems for mixed and mixed-composite type equations] Tashkent, 1979. \bibitem{e1} A. V. Efimov; \emph{On boundary value problems with Saigo operators for the mixed type equation with a fractional derivative} (Russian). Vestnik Samarsk. Gos. Tekh. Univ. 26 (2004), pp. 16-20. \bibitem{e2} U. Egamberdiev; \emph{Boundary problems for mixed parabolic-hyperbolic equation with two lines of type changing.} PhD thesis, Tashkent, 1984. \bibitem{e3} V. A. Eleev, V. N. Lesev; \emph{On two boundary problems for mixed type equations with perpendicular lines of changing type}. Vladikavkaz math. journ., 3 (4) (2001), pp. 9-22. \bibitem{g1} I. M. Gelfand; \emph{Some questions of analysis and differential equations}. Uspehi Mat. Nauk, 3 (87) (1959), pp. 3-19. \bibitem{h1} R. Hilfer (Ed.); \emph{Applications of Fractional Calculus in Physics}, World Scientific, Singapore, 2000. \bibitem{k1} B. J. Kadirkulov, B. Kh. Turmetov; \emph{On a generalization of a heat equation}. Uzbek Mathematical Journal. 3 (2006), pp. 40-45. \bibitem{k2} E. T. Karimov, A. I. Sotvoldiev; \emph{Existence of solutions to non-local problems for parabolic-hyperbolic equations with three lines of type changing}. Electronic Journal of Differential Equations, Vol. (2013) 2013, No 138, pp. 1-5. \bibitem{k3} A. A. Kilbas, O. A. Repin; \emph{An analog of the Bitsadze-Samarskii problem for a mixed type equation with a fractional derivative}. Differ. Equ. 39(5) (2003), pp. 674-680; transl. from Differ. Uravn. 39, No 5 (2003), pp. 638-644. \bibitem{k4} A. A. Kilbas, O. A. Repin; \emph{An analog of the Tricomi problem for a mixed type equation with a fractional derivative}. Fractional Calculus and Applied Analysis. 13(1) (2010), pp. 69-84. \bibitem{k5} A. A. Kilbas, H. M. Srivastava, J. J. Trujillo; Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science B. V., Amsterdam, 2006. xvi+523 pp. \bibitem{m1} R. L. Magin; \emph{Fractional Calculus in Bioengineering}, Begell House Inc., Redding, CT, 2006. \bibitem{m2} S. G. Mikhlin; Leksii po lineynim integralnym uravneniyam. [Lectures on linear integral equations]. Moscow, 1959. 232 p. \bibitem{n1} A. M. Nakhushev; \emph{Drobniye ischisleniya i yego primenenie}. [Fractional Calculus and its Applications], Moscow: FIZMATLIT, 2003. \bibitem{n2} V. A. Nakusheva; \emph{First boundary problem for mixed type equation in a characteristic polygon}. Dokl. AMAN, 1(14) (2012), pp. 58-65. \bibitem{p1} A. V. Pskhu; \emph{Uravneniya v chastnykh proizvodnykh drobnogo poryadka}. [Partial differential equations of fractional order], Moscow: NAUKA, 2005. \bibitem{s1} A. Saadatmandi, M. Dehghan; \emph{A tau approach for solution of the space fractional diffusion equation}. Computers and Mathematics with Applications. 62 (2011), pp. 1135-1142. \end{thebibliography} \end{document}