\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 221, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/221\hfil Boundary-value problems] {Boundary-value problems for a third-order loaded parabolic-hyperbolic equation with variable coefficients} \author[B. Islomov, U. I. Baltaeva \hfil EJDE-2015/221\hfilneg] {Bozor Islomov, Umida I. Baltaeva} \address{Bozor Islomov \newline National University of Uzbekistan, 100174, Tashkent, Uzbekistan} \email{islomovbozor@yandex.ru \newline http://math.nuu.uz/en/islomov-bozor-islomovich} \address{Umida Ismoilovna Baltaeva \newline Mathematical Institute University of G\"ottingen, 37073 G\"ottingen, Germany} \email{umida\_baltayeva@mail.ru} \thanks{Submitted February 25, 2015. Published August 25, 2015.} \subjclass[2010]{35M10} \keywords{Equations of mixed type; loaded equation; gluing condition; \hfill\break\indent boundary-value problem; integral equation} \begin{abstract} We prove the unique solvability of a boundary-value problems for a third-order loaded integro-differential equation with variable coefficients, by reducing the equation to a Volterra integral equation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{problem}[theorem]{Problem} \allowdisplaybreaks \section{Introduction} The theory of mixed type equations is one of the principal parts of the general theory of partial differential equations. The interest for these kinds of equations arises intensively due to both theoretical and practical uses of their applications. Many mathematical models of applied problems require investigations of this type of equations. The first fundamental results in this direction were obtained in 1923 by Tricomi. The works of Gellerstedt, Lavrent'ev, Bitsadze, Frankl, Protter and Morawetz, Salakhitdinov, Djuraev, Rassias have had a great impact in this theory, where outstanding theoretical results were obtained and pointed out important practical values. Currently, the concept of mixed-type equations has expanded to include all possible combinations of two or three classic types of equations. The necessity of the consideration of the parabolic-hyperbolic type equation was specified for the first time in 1956 by Gel'fand \cite{g1}. He gave an example connected to the movement of the gas in a channel surrounded by a porous environment. The movement of the gas inside the channel was described by the equation, outside by the diffusion equation \cite{b2,c1,r1,s3}. A systematic study of the third and higher order mixed and mixed-composite type PDEs, containing in the main part parabolic-hyperbolic, hyperbolic-elliptic and elliptic-parabolic operators began in the early seventies and intensively developed by many mathematicians \cite{d1,d2,s1,s2}. In the recent years, in connection with intensive research on problems of optimal control of the agro economical system, long-term forecasting and regulating the level of ground waters and soil moisture, it has become necessary to investigate a new class of equations called as ``loaded equations''. Such equations were investigated for the first time by Knezer \cite{k1}, Lichtenstein \cite{l1}. However, they did not use the term ``loaded equation''. This terminology has been introduced by Nakhushev \cite{n1}, where the most general definition of a loaded equation is given and various loaded equations are classified in detail, e.g., loaded differential, integral, integro-differential, functional equations etc., and numerous applications are described \cite{w1,n2}. \begin{definition} \label{def1.1} \rm An equation \begin{equation}\label{0} Au(x)=f(x) \end{equation} is called loaded in an $n$-dimensional Euclidean domain $\Omega$ if (part of) the operator~A depends on the restriction of the unknown function $u(x)$ to a closed subset of $\overline{\Omega}$, of measure strictly less than $n$. \end{definition} \begin{definition} \label{def1.2} \rm A loaded equation is called a loaded differential equation in the domain $\Omega \subseteq \mathbb{R}^n$ if it contains at least one derivative of the unknown solution in a subset of $\overline\Omega$ of nonzero measure. \end{definition} Basic questions of the theory of boundary value problems for PDEs are the same for the boundary value problems for the loaded differential equations. However, the existence of the loaded part operator $A$ does not always make it possible to apply directly the known theory of boundary value problems for equations \[ L(x)=f(x). \] On the other hand, searching for solutions of loaded differential equation pre-assigned classes it might reduce to new problems for non-loaded equations. Works of Nakhushev, Shkhankov, Borodin, Borok, Kaziev, Pomraning, Larsen, Pul`kina, Eleev, Dzhenaliev, Attaev, Wiener, Islomov, Khubiev et al. are devoted to loaded second-order partial differential equations. However, we would like to note that boundary-value problems for third-order loaded equations of a hyperbolic, parabolic-hyperbolic, elliptic-hyperbolic types are not well studied. We indicate only the works \cite{e1,e2,v1} in which study-case, when loaded part contain only track or derivative track from unknown solutions. It can be explained with the absence of the representation of the general solution for such equations; on the other hand, these problems will be reduced to integral equations with stir \cite{b1}, which are not investigated in detail. \section{Formulating of the problem} Let $\Omega $ be a simple connected domain located in the plane of independent variables $x$ and $y$, in the case $y > 0$, is bounded by the segments $AA_0$, $BB_0$, and $ A_0 B_0(A(0,0)$, $B(1,0)$, $ A_0 (0,h)$, $B_0(1,h))$, of the straight lines $x = 0$, $x = 1$, and $ y = h$, respectively, and in the case $y < 0$, with segments $AC:x + y = 0$, $BC:\eta=x - y = 1$ originating at the point $C(1/2,-1/2)$. We use the following designation: $$ I = \big\{ (x,y):0 < x < 1,\; y = 0 \big\},\quad \Omega _1 = \Omega \cap \{{y>0}\}, \Omega _2 =\Omega \cup \{y<0\}. $$ We consider a linear loaded integro-differential equation \begin{equation}\label{eq1} \big( {a{\frac{{\partial }}{{\partial x}}} + c}\big) Lu = 0, \end{equation} where \[ Lu \equiv \begin{cases} L_1 u \equiv u_{xx} + a_1 (x,y)u_{x} + b_1 (x,y)u_{y} + c_1 (x,y)u\\ - \sum_{i = 1}^{n} {d_i }D_{0x}^{\alpha _i } u(x,0), &\text{if } y \geqslant 0, \\[4pt] L_2 u \equiv u_{xx} - u_{yy} + a_2 (x,y)u_{x} + b_2 (x,y)u_{y} + c_2 (x,y)u \\ - \sum_{i = 1}^{n} e_i D_{0\xi}^{\beta _i }u(\xi,0), &\text{if }y \leqslant 0, \ \end{cases} \] where $a,c$ are given real parameters, $a_i$, $b_i$, $c_i$, $d_i$, $e_i$ are given functions on $\Omega _i$ $(i = 1,2)$, and $b_1(x,y) < 0$, $c_1 (x,y) \leqslant 0$ on $\bar {\Omega }_1$; moreover the functions $ a_1$, $b_1$, $c_1$, $d_i$, $a_{1x}$, $ a_{1y}$, $b_{1x}$, $b_{1y}$, $d_{ix}$, $d_{iy}$ on $ \Omega _1 $ satisfy a H\"{o}lder condition, and $ a_2,b_2 \in C^{2}(\bar{\Omega }_2)$, $c_2 \in C^{1}(\Omega_2)$, $ e_i \in C^{1}(\bar{\Omega}_i)$. $D_{0x}^{\alpha _i }$ is integro-differential operator (in the sense of Riemann-Liouville), $\alpha_i, \beta_i<1$, $i=1,\dots,n$. For equation \eqref{eq1} we investigate the following problems $(a\ne 0)$. \begin{problem} \label{pr1} \rm Find a function $u(x,y)$ possessing the following properties: \begin{enumerate} \item $ u(x,y) \in C(\bar {\Omega })\cap C^{1}(\Omega ) $; \item $ u_{x} (u_{y} ) $ is continuous up to $ AA_0 \cup AC$, $( AC)$; \item $u(x,y)$ is a regular solution of equation \eqref{eq1} in the domains $\Omega _1 $ and $\Omega _2 $; \item $u(x,y)$ satisfies the boundary value conditions \begin{gather} \begin{gathered} u(x,y)\big|_{AA_0} = \varphi _1(y),\quad u(x,y) \big|_{BB_0}= \varphi _2(y), \\ u_{x}(x,y) \big|_{AA_0} = \varphi _3(y),\quad 0 \leqslant y \leqslant h, \end{gathered} \label{eq2} \\ u(x,y) \bigr|_{AC} = \psi _1 (x),\quad 0 \leqslant x \leqslant \frac12, \label{eq3} \\ \frac{{\partial u(x,y)}}{{\partial n}} \bigr|_{AC} = \psi _2 (x),\quad 0 \leqslant x \leqslant \frac{1}{2}, \label{eq4} \end{gather} where $n$ is an inner normal, $\varphi _1 (y)$, $\varphi _2 (y)$, $\varphi _3 (y)$, $\psi _1 (x)$ and $\psi _2 (x)$ are given real-valued functions, moreover $\varphi _1 (0) = \psi _1(0)$, ${\psi }'_1 (0) = \sqrt {2} \psi _2 (0) - 2{\varphi }'_1 (0)$. \end{enumerate} \end{problem} \begin{problem}\label{pr2} \rm Find a function $u(x,y)$, satisfying the following conditions: \begin{enumerate} \item $ u(x,y) \in C(\bar {\Omega })\cap C^{1}(\Omega )$; \item $ u_{x} (u_{y} ) $ is continuous up to $AA_0 \cup BC$, $(BC)$; \item $u(x,y)$ is a regular solution of equation \eqref{eq1} in the domains $\Omega _1 $ and $\Omega _2$; \item $u(x,y)$ satisfies the boundary value conditions \eqref{eq2} and \begin{gather} \label{eq5} u(x,y) \big|_{BC} = \psi _3(x),\quad \frac{1}{2} \leqslant x \leqslant 1, \\ \label{eq6} \frac{{\partial u(x,y)}}{\partial n} \big|_{BC} = \psi _4 (x),\quad \frac{1}{2} \leqslant x \leqslant 1, \end{gather} where $n$ is an inner normal, $\varphi _1(y)$, $\varphi _2 (y)$, $\varphi _3 (y)$, $\psi _3 (x)$ and $\psi _4 (x)$ are given real-valued functions, moreover $\varphi _2(0) = \psi _3 (0)$. \end{enumerate} \end{problem} \section{Main results} From condition (1) problems \ref{pr1} and \ref{pr2} it follows that \begin{gather} \label{eq7} u(x,+ 0) = u(x,-0) =\tau (x), \\ \label{eq8} u_{y} (x,+ 0) = u_{y} (x,-0) = \nu(x), \\ \label{eq9} u_{x} (x,+ 0) = u_{x} (x,-0) = {\tau }'(x), \end{gather} where $\tau (x)$ and $\nu(x)$, are still unknown functions. Assuming that \[ u(x,y) = \begin{cases} u_1 (x,y), &(x,y) \in \bar {\Omega }_1 , \\ u_2 (x,y), &(x,y) \in \bar {\Omega }_2 , \end{cases} \] equation \eqref{eq1} can be represented by two systems: \begin{gather} \label{eq10} \begin{gathered} L_1 u_1+ \sum_{i = 1}^{n} d_i D_{0x}^{\alpha _i } u_1(x,0) = \upsilon _1 (x,y), \quad (x,y) \in \bar {\Omega }_1 ,\\ a\upsilon _{1x} + c\upsilon _1 = 0, \end{gathered} \\ \label{eq11} \begin{gathered} L_2 u_2+ \sum_{i = 1}^{n} e_i D_{0\xi}^{\beta _i } u_2(\xi,0) = \upsilon _2 (x,y), \quad (x,y) \in \bar {\Omega }_2 ,\\ a\upsilon _{2x} + c\upsilon _2 = 0, \end{gathered} \end{gather} where $\upsilon _1 (x,y)$, $\upsilon _2 (x,y)$ are continuous differentiable functions. \begin{theorem}\label{thm1} If $b_1 (x,y) < 0$, $ c _1 (x,y) \leq 0$ and $ a_i (x,y) \geq 0$ for all $(x,y)\in \Omega_i$, \begin{gather}\label{eq12} \varphi _i (y) \in C^{1}[0,h],\;(i = 1,2),\quad \varphi _3 (y) \in C[0,h] \cap C^{1}(0,h), \\ \label{eq13} \psi _1 (x) \in C^{1}[0,1/2] \cap C^{3}(0,1/2),\quad \psi _2(x) \in C[0,1/2] \cap C^{2}(0,1/2), \end{gather} then there exists a unique solution to the problem \ref{pr1} in the domain $\Omega $. \end{theorem} \begin{theorem}\label{thm2} If $b_1 (x,y) < 0$, $ c _1 (x,y) \leq 0$ and $ a_i (x,y) \geq 0$ for all $(x,y)\in \Omega_i$, condition \eqref{eq12} is satisfied and \begin{equation}\label{eq14} \psi _3 (x) \in C^{1}{[1/2,1]} \cap C^{3}(1/2,1), \quad \psi _4(x) \in C{[1/2,1]} \cap C^{2}(1/2,1), \end{equation} then there exists a unique solution to the problem \ref{pr2} in the domain $\Omega$. \end{theorem} \begin{proof}[Proof of Theorem \ref{thm1}] Bearing in mind \cite{d2} that system \eqref{eq11} is reduced to the form \begin{equation}\label{eq15} L_2 u_2 + {\sum_{i = 1}^{n} {e_i } }D_{0\xi}^{\beta _i } u_2(\xi,0) = w_2 (y)\exp \big( { - {\frac{{c}}{{a}}}x} \big). \end{equation} Hence going over to the characteristic coordinates $\xi=x+y$, $\eta=x-y$, we obtain \begin{equation} \label{eq16} \begin{aligned} &u_{2\xi \eta } + a_3 (\xi ,\eta )u_{2\xi } + b_3 (\xi ,\eta )u_{2\eta } + c_3 (\xi ,\eta )u_2 \\ &= E_i (\xi ,\eta )D_{0\xi }^{\beta _i } \tau + {\frac{{1}}{{4}}}\omega _2 \big( {{\frac{{\xi - \eta }}{{2}}}} \big)\exp \big( { - {\frac{{c}}{{2a}}}(\xi+\eta)} \big), \end{aligned} \end{equation} where $a_3 (\xi ,\eta )$, $b_3 (\xi ,\eta )$, $c_3 (\xi ,\eta )$ depend on the coefficients of equation \eqref{eq15}, \[ E_i (\xi ,\eta ) = {\frac{{1}}{{4}}}e_i \big( {{\frac{{\xi + \eta }}{{2}}},{\frac{{\eta - \xi }}{{2}}}} \big), \] with recurring index $i=1,2,\dots ,n$ implied summation. The boundary value conditions \eqref{eq3} and \eqref{eq4} is reduced to the form \begin{equation} \label{eq17} u_2 (\xi,\eta) \big|_{\xi = 0} = \psi _1 \big( {{\frac{{\eta }}{{2}}}} \big),\quad 0 \leqslant \eta \leqslant 1, \end{equation} and \begin{equation}\label{eq18} \frac{{\partial u_2 (\xi,\eta)}}{\partial \xi } \big|_{\xi = 0} = {\frac{{1}}{{\sqrt {2} }}}\psi _2 \big( {{\frac{{\eta }}{{2}}}} \big),\quad 0 <\eta < 1. \end{equation} The solution of the equation \eqref{eq16}, with boundary conditions \eqref{eq17} and \begin{equation}\label{eq19b} ( u_{2\xi } - u_{2\eta } )\big|_{\eta = \xi } = \nu(\xi ),\quad 0 < \xi < 1, \end{equation} (problem Cauchy-Goursat), is represented analogously as \cite{p1} \begin{equation} \label{eq14b} \begin{aligned} u_2 (\xi,\eta) & = F(\xi ,\eta ) + {\frac{{1}}{{4}}}{\int_0^{\xi } {dt} }\int_{t}^{\eta } T(t,\tau ;\xi ,\eta ) \exp \big( { - {\frac{{c}}{{2a}}(t + \tau )}} \big)\omega _2 \big( {{\frac{{t- \tau }}{{2}}}} \big)d\tau \\ &\quad + \int_0^{\xi } T_0 ( {\xi ,\eta ;t})\nu(t)dt\\ &\quad +\frac{{1}}{{4}}{\int_0^{\xi } {dt} }{\int_{t}^{\xi } {S(t,\tau ;\xi ,\eta )\exp \big( { - {\frac{{c}}{{2a}}(t + \tau )}} \big)\omega _2 \big( {{\frac{{t - \tau }}{{2}}}} \big)d\tau } } \\ &\quad + \int_0^{\xi }{dt} \int_{t}^{\xi } E_i (t,\tau )D_{0t}^{\beta _i } \tau (t)S(t,\tau ;\xi ,\eta )d\tau \\ &\quad + {\int_0^{\xi } {dt} }\int_{t}^{\eta } E_i (t,\tau )D_{0t}^{\beta _i } \tau (t)T(t,\tau ;\xi,\eta )d\tau , \end{aligned} \end{equation} where \begin{align*} F(\xi ,\eta ) &= \psi _1 (\frac{\eta }{2}) +\psi _1(\frac{\xi}{2}) - \psi _1 (0) \\ &\quad + {\int_0^{\xi } {dt} }\int_{t}^{\xi } K(t,\tau )S(t,\tau;\xi ,\eta )d\tau + {\int_0^{\xi } {dt} }{\int_{t}^{\eta } {K(t,\tau )T(t,\tau ;\xi ,\eta )d\tau ,} } \end{align*} \[ K(\xi ,\eta ) = - {\frac{1}{2}}a_3 (\xi ,\eta ){\psi }'_1 (\frac{\xi}{2}) - {\frac{1}{2}}b_3 (\xi ,\eta ){\psi }'_1 (\frac{\eta }{2}) - c_3 (\xi ,\eta )\Big( {\psi _1 (\frac{\xi}{2}) + \psi _1 (\frac{\eta }{2}) - \psi _1 (0)} \Big), \] \begin{align*} T_0 (\xi,\eta;t) &= 1 - {\int_{t}^{\xi } {a_3 \left( {t,\tau } \right)S(t,\tau ;\xi ,\eta )d\tau - {\int_{t}^{\xi } {a_3 \left( {t,\tau } \right)T(t,\tau ;\xi ,\eta )d\tau } }} } \\ &\quad - {\int_{t}^{\xi } {ds} }\int_{s}^{\xi } c_3 (s,\tau )S(s,\tau ;\xi ,\eta )d\tau \\ &\quad - {\int_0^{\xi } {ds} }{\int_{t}^{\eta } {c_3 (s,\tau ) T(s,\tau ;\xi ,\eta )d\tau ,} } \end{align*} where $S(t,\tau ;\xi ,\eta )$ and $T(t,\tau ;\xi ,\eta )$ are expressed via coefficients $a_3$, $b_3$, $c_3 $ and continuous in $\bar {\Omega }_2 \times \bar {\Omega }_2 $ functions $S_{\xi }$, $S_{\eta }$, $T_{\eta } $ are continuous in $\bar {\Omega }_2 \times \bar {\Omega }_2 $, and function $T_{\xi } $ it can have discontinuities of the first kind on compact subsets of this domains. More properties of these functions are established in \cite{p1}. Substituting \eqref{eq14b} in \eqref{eq18}, taking into account that $\nu(0) = u_{2\eta } (0,0) = u_{1\eta } (0,0) = \varphi' _1 (0)$ and $\varphi _1' (0) = {\frac{1}{2}}\big( {\sqrt {2} \psi_2 (0) - \psi _1' (0)} \big)$, we obtain \begin{equation} \label{eq15b} \begin{aligned} &\int_0^{\eta } T(0,\tau;0,\eta)\exp \left( { - {\frac{{c}}{{2a}}\tau}} \right)\omega _2 \left( { - {\frac{{\tau }}{{2}}}} \right)d\tau \\ &= 2\sqrt {2} \psi _2 (\frac{\eta }{2}) - 2{\psi }'_1 (0) - 4\int_0^{\eta } K(0,\tau)T(0,\tau;0,\eta)d\tau \\ &\quad - 4\varphi' _1 (0)\Big( {1 - {\int_0^{\eta } {a_3 (0,\tau)T(0,\tau;0,\eta)d\tau } }} \Big). \end{aligned} \end{equation} From here with regard \eqref{eq13}, differentiating \eqref{eq15b} with respect to $\eta $, reduction in this integral equation of the second kind \begin{equation} \label{eq16b} \omega _2 \big(-\frac{\eta}{2}\big) - {\int_0^{\eta } {T_{\eta } (0,\tau ;0,\eta )\exp \left( {{\frac{{c}}{{2a}}}(\eta-\tau)} \right)\omega _2 \left( { - {\frac{{\tau }}{{2}}}} \right)d\tau = g\left( {\eta } \right)} }, \end{equation} \begin{align*} g\left( {\eta } \right) & = \Big(\sqrt {2} \,{\psi }'_2 (\frac{\eta }{2}) - 4K\left( {0,\eta } \right) - 4\int_0^{\eta } K(0,\tau )T_{\eta } (0,\tau;0,\eta)d\tau \\ &\quad + 4\,{\varphi }'_1 (0)\Big( {a_3 \left( {0,\eta } \right) + {\int_0^{\eta } {a_3 (0,\tau )T_{\eta } \left({0,\tau ;0,\eta } \right)d\tau } }} \Big)\Big) \exp \left( {{\frac{{c }}{{2a}}\eta}} \right). \end{align*} From \eqref{eq13}, we conclude that the kernel and $g(\eta)$ are continuous. Then it leads to a unique solutions in the class of continuous functions. Solving this, we obtain $\omega _2 \big(-\frac{\eta}{2}\big)$ in $ - {\frac{1}{2}} \leqslant - {\frac{{\eta }}{{2}}} \leqslant \,0$. Therefore in instead of $\omega _2 \big(-\frac{\eta}{2}\big)$ we can take $\omega _2 \left( {{\frac{{\xi - \eta }}{{2}}}}\right)$. Substituting in \eqref{eq14b} the expression $\omega _2\left( {{\frac{{\xi - \eta }}{{2}}}} \right)$ we find the solution $u_2 (\xi,\eta)$ in the form \begin{equation} \label{eq17b} \begin{aligned} u_2 (\xi,\eta) &= M(\xi ,\eta ) +\int_0^{\xi } T_0 (\xi,\eta;t)\nu(t)dt\\ &\quad + {\int_0^{\xi } {dt} }{\int_{t}^{\xi } {E_i (t,\tau )D_{0t}^{\beta _i } \tau (t)S(t,\tau ;\xi ,\eta)d\tau } }\\ &\quad + {\int_0^{\xi } {dt} }{\int_{t}^{\eta } {E_i (t,\tau )D_{0t}^{\beta _i } \tau (t)T(t,\tau ;\xi ,\eta )d\tau,} } \end{aligned} \end{equation} where \begin{align*} M (\xi,\eta) &= F(\xi ,\eta ) +{\frac{{1}}{{4}}}{\int_0^{\xi } {dt} }{\int_{t}^{\xi } {S(t,\tau ;\xi ,\eta ) \exp \left( { - {\frac{{c}}{{2a}}(t + \tau )}} \right) \omega _2 \big(\frac{t-\tau}{2})d\tau } } \\ &\quad + {\frac{{1}}{{4}}}{\int_0^{\xi } {dt} }{\int_{t}^{\eta } {T(t,\tau ;\xi ,\eta )\exp \left( { - {\frac{{c}}{{2a}}(t + \tau )}} \right)\omega _2 \big(\frac{t-\tau}{2})d\tau }}, \end{align*} depend on a given function. In $\eta = \xi = x$, setting $M(x) = M(x,x)$, $T_0 (x,t) = T_0 (x,x;t),\tau (x) = u_2(x,x)$, from \eqref{eq14b} we obtain \begin{align*} \tau (x) & = M(x) + {\int_0^{x} {T_0 (x,t)\nu(t)dt } } + {\int_0^{x} {D_{0t}^{\beta _i } \tau (t)dt} }{\int_{t}^{x} {E_i (t,\tau )S(t,\tau ;x,x)d\tau }}\\ &\quad + {\int_0^{x} {D_{0t}^{\beta _i } \tau (t)dt} }\int_{t}^{x} E_i (t,\tau )T(t,\tau ;x,x)d\tau . \end{align*} Differentiating the last relation, obtain integral equation second kind relative to $\nu (x)$: \begin{equation}\label{eq18b} \nu (x) + {\int_0^{x} {T'_{ox} (x,t)\nu (t)ds = {\tau }'(x) - } }{\int_0^{x} {L(x,t)D_{_{0t} }^{\beta _i } \tau (t)dt - {M}'(x),} } \end{equation} where \begin{align*} L(x,t)& = E_i (t,x)\left( {S(t,x;x,x) - T(t,x;x,x)}\right) \\ &\quad + \int_{t}^{x} E_i \left( {t,\tau } \right)\left( {{S}'(t,\tau ;x,x) + {T}'(t,\tau ;x,x)} \right)d\tau. \end{align*} The right-hand side equation \eqref{eq18b} is continuous and kernel can be discontinuous of the first kind. Therefore $\nu (x)$: \begin{equation} \label{eq19} \begin{aligned} \nu (x) &= {\tau }'(x) - \int_0^{x} L(x,t)D_{_{0t} }^{\beta _i } \tau (t)dt - {M}'(x) \\ &\quad + \int_0^{x} \Gamma _0 (x,t)\Big( {M}'(t) - {\tau }'(t) + \int_0^{t} L(t,s)D_{0s}^{\beta _i } \tau (s)ds \Big)dt, \end{aligned} \end{equation} where $\Gamma _0 (x,t)$ is the resolvent of the kernel ${T}'_{0x} (x,t)$. This is the first functional relation between the function $\tau (x)$ and $\nu (x)$ transferred from the $\Omega _2 $. Present we need obtain second functional relation between this functions. To this end equation \eqref{eq1} at $y > 0$ rewrite in the form \[ L_1 u_1 \equiv u_{1xx} + a_1 u_{1x} + b_1 u_{1y} + c_1 u_1 + {\sum_{i = 1}^{n} {d_i } }D_{0x}^{\alpha _i } u_1 (x,0)= w_1 (y)\exp \left( { - {\frac{{c}}{{a}}}x} \right), \] where $w_1 (y)$ is arbitrary continuous functions. Hence, considering property of the problem \ref{pr1}, in $b_1 = - 1$, passage to the limit, we obtain second functional relation between the function $\tau (x)$ and $\nu (x)$ transferred from the $\Omega _1 $: \begin{equation}\label{eq20} \tau ''(x) + a_1 (x,0)\tau '(x) + c_1 (x,0)\tau (x) - {\sum_{j = 1}^{n} {d_{j} D_{0x}^{\alpha _{j} } } }\tau (x) - \nu (x) = \omega _1 (0)\exp \left( { - {\frac{{c}}{{a}}}x} \right). \end{equation} Substituting \eqref{eq19} in \eqref{eq20}, results \begin{equation} \label{eq21} \begin{aligned} &\tau ''(x) + p(x)\tau '(x) + q(x)\tau (x) - {\sum_{j = 1}^{n} {d_{j} D_{0x}^{\alpha _{j} } } }\tau (x) \\ &+ {\int_0^{x} {\Gamma _0 } }(x,t){\tau }'(t)dt + {\int_0^{x} {\Gamma _1 } }(x,t)D_{0t}^{\beta _i } \tau (t)dt\\ & = \omega _1 (0)\exp \left( { - {\frac{{c}}{{a}}}x} \right) + m(x), \end{aligned} \end{equation} where \begin{gather*} p(x) = a_1 (x,0) - 1,\quad q(x) = c_1 (x,0),\\ \Gamma _1 (x,t) = L(x,t) - {\int_{t}^{x} {\Gamma _0 } }\left( {x,s} \right)L\left( {s,t} \right)ds, \\ m(x) = {\int_0^{x} {\Gamma _0 } }(x,t){M}'(t)dt - {M}'(x). \end{gather*} Solve \eqref{eq21} under the initial condition \[ \tau (0) = \varphi _1 (0) = \psi _1 (0),\quad {\tau}'(0) = \sqrt {2} \psi _2 (0) - {\varphi }'_1 (0). \] Introduce new unknown function ${\tau }''(x) = z(x)$. Then with regards the next conditions we have \begin{gather*} {\tau }'(x) = {\int_0^{x} {z(t)dt} } + \sqrt {2} \psi _2 (0) - {\varphi }'_1 (0), \\ \tau (x) = {\int_0^{x} {\left( {x - t} \right)z(t)dt} } + \left( {\sqrt {2} \psi _2 (0) - {\varphi }'_1 (0)} \right)x + \psi _1 (0). \end{gather*} Bearing mind this, we rewrite equation \eqref{eq21} in form \begin{equation}\label{eq22} z(x) + {\int_0^{x} {Q} }(x,t;\alpha_j,\beta_i)z(t)dt = \omega _1 (0)\exp \left( { - {\frac{{c}}{{a}}}x} \right) + M(x), \end{equation} where \begin{gather*} Q(x,t;\alpha_j,\beta_i) = p(x) + q(x)\left( {x - t} \right) - Q_1\left( {x,t;\alpha _{j} } \right) + Q_2 \left( {x,t;\beta _i } \right) + {\int_{t}^{x} {\Gamma _0 \left( {x,s} \right)} }ds, \\ Q_1 (x,t;\alpha _{j} ) = \begin{cases} {\sum_{j = 1}^{n} }{\frac{{d_{j} }{B\left( {2; - \alpha _{j} } \right)}} {{\Gamma\left( { - \alpha _{j} } \right)}}} \left({x - t} \right)^{1 - \alpha _{j} }, &\alpha _{j} < 0, \\[4pt] {\sum_{j = 1}^{n} }{\frac{{d_{j}(2-\alpha_{j})}{B\left( {2;1 - \alpha _{j} } \right)}} {{\Gamma \left( {1 - \alpha _{j} } \right)}}}\left( {x - t} \right)^{1 - \alpha _{j} }, & 0 < \alpha _{j} < 1, \end{cases} \\ Q_2 (x,t;\beta _i ) = \begin{cases} {\frac{{B\left( {2; - \beta _i } \right)}}{{\Gamma \left( { - \beta _i } \right)}}}{\int_{t}^{x} {\Gamma _1 \left( {x,s} \right)} }\left({s - t} \right)^{1 - \beta _i }ds, &\beta _i < 0, \\[4pt] {\frac{{B\left( {2;1 - \beta _i } \right)}}{{\Gamma \left( {1 - \beta _i } \right)}}}{\int_{t}^{x} {\Gamma _1 \left( {x,s} \right)}}\left( {s - t} \right)^{2 - \beta _i }ds, & 0 < \beta _i < 1. \end{cases} \end{gather*} where $B$ is the Beta function and $\Gamma(z)$ is the Gamma function. The Kernel and the right-hand side of \eqref{eq22} are continuous. Therefore, $z(x) \in C[0,1]$. Solving we find $z(x)$: \begin{align*} z(x) &= M(x) + {\int_0^{x} {R} }(x,t;\alpha_j,\beta_i)M(t)dt \\ &\quad + \omega _1 (0)\Big( {\exp \left( { - {\frac{{c}}{{a}}}x} \right) + {\int_0^{x} {R} }(x,t;\alpha_j,\beta_i)\exp \left( { - {\frac{{c}}{{a}}}t} \right)dt} \Big), \end{align*} where $R(x,t;\alpha_j,\beta_i) $ is a resolvent of the kernel $Q(x,t;\alpha_j,\beta_i)$. Taking into account the last equality, we obtain \begin{equation} \label{eq23} \begin{aligned} \tau (x) &= \omega _1(0)\int_0^{x} (x - t) \Big( \exp \left( { - {\frac{{c}}{{a}}}t} \right) \\ &\quad + {\int_0^{t} {R} }\left( {t,s;\alpha _{j},\beta _i } \right)\exp \left( { - {\frac{{c}}{{a}}}s} \right)ds \Big)dt + M_1 (x), \end{aligned} \end{equation} where \begin{align*} M_1 (x) & = {\int_0^{x} {(x - t)} }\Big( {M(t) + {\int_0^{t} {R} }\left( {t,s;\alpha _{j} ,\beta _i } \right)M\left( {s} \right)ds} \Big)dt \\ &\quad + \left( {\sqrt {2} \psi _2 (0) - {\varphi }'_1 (0)} \right)x + \psi _1 (x). \end{align*} Hence, by the condition $\tau (1) = \varphi _2 (0)$, $w_1 (0)$ are determined uniquely. Thus, from function $\tau (x)$ using relation \eqref{eq20} we uniquely define function $\nu(x)$. Set value function $\tau (x)$ and $\nu(x)$ in \eqref{eq14b}, we obtain function $u_2 (\xi,\eta)$ in domain $\Omega_2$. For determination function $u_1 (x,y)$ in domain $\Omega_1 $ reduce to problem \ref{pr2} and \[ u_1 (x,0) = \tau (x), \] for the equation \begin{equation} \label{eq24} \big( {a{\frac{{\partial }}{{\partial x}}} + c}\big) \left( {u_{1xx} + a_1 (x,y)u_{1x} + b_1 (x,y) u_{1y} + c_1 (x,y)u_1 } \right) = F(x,y), \end{equation} where $F(x,y) = \left( {a{\frac{{\partial }}{{\partial x}}} + c} \right){\sum_{i = 1}^{n} {d_i D_{0x}^{\alpha _i } \tau (x)} }$ is a well-known function. Unique solvability this problem was proved in \cite[\S 2, chapter 4]{d2}. We can conclude from these that, there exists a regular solution of problem in $\Omega _1 $. Therefore, we can conclude from these that, there exists a regular solution of problem \ref{pr1}. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm2}] The proof for Problem \ref{pr2} is analogous to the proof for Problem \ref{pr1}. We omit it. \end{proof} \begin{remark} \rm For problems \ref{pr1} and \ref{pr2} it is possible examine with general discontinuous gluing conditions. In this case 1, problems \ref{pr1} and \ref{pr2}, change in the following way: function $u(x,y)$ is continuous in each closed domains $\bar {\Omega }_1 $ and $\bar {\Omega }_2 $, conditions (2), (3) and (4) it remains invariant. Indeed, the following conditions are fulfilled: \begin{gather*} u(x, + 0) = \alpha _1 (x)\,u(x, - 0) + \gamma _1^{} (x), \quad 0 < x <1, \\ u_{y} (x, + 0) = \beta _1 (x)\,u_{y} (x, - 0) + \alpha _2 (x)\,u_{y} (x, - 0) + \gamma _2^{} (x), \quad 0 < x < 1, \end{gather*} where $\alpha _1,\gamma _1\in C^{3}$, $\alpha_2,\beta_1,\gamma _2\in C^{2}$ are given functions, and $\alpha _1\beta _1\ne 0$ for $0