\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 09, 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-2015/09\hfil Generalized Euler-Poisson-Darboux equation]
{Particular solutions of generalized Euler-Poisson-Darboux equation}

\author[R. B. Seilkhanova, A. H. Hasanov \hfil EJDE-2015/09\hfilneg]
{Rakhila B. Seilkhanova, Anvar H. Hasanov}  % in alphabetical order

\address{Rakhila B.  Seilkhanova \newline
S. Baishev University Aktobe, Department for Information Systems and Applied Mathematics,
030000 Aktobe, Zhubanov str. 302, Kazakhstan}
\email{srahila@inbox.ru}

\address{Anvar H. Hasanov \newline
S. Baishev University Aktobe, Department for Information Systems and Applied Mathematics,
030000 Aktobe, Zhubanov str. 302, Kazakhstan}
\email{anvarhasanov@yahoo.com}

\thanks{Submitted October 21, 2014. Published January 5, 2015.}
\subjclass[2000]{35Q05, 35L80, 35C65}
\keywords{Generalized Euler-Poisson-Darboux equation;
 hyperbolic equation; \hfill\break\indent
 Lauricelli hypergeometric functions}

\begin{abstract}
 In this article we consider the generalized Euler-Poisson-Darboux equation
 $$
 {u}_{tt}+\frac{2\gamma }{t}{{u}_{t}}={u}_{xx}+{u}_{yy}
 +\frac{2\alpha }{x}{{u}_{x}}+\frac{2\beta }{y}{{u}_y},\quad x>0,\;y>0,\;t>0.
 $$
 We construct particular solutions in an explicit form expressed by
 the Lauricella hypergeometric function of three variables.
 Properties of each constructed solutions have been investigated in sections
 of surfaces of the characteristic cone. Precisely, we prove that found
 solutions have singularity $1/r$ at  $r\to 0$, where
 ${{r}^2}={{( x-{{x}_0})}^2}+{{( y-{{y}_0})}^2}-{{( t-{{t}_0})}^2}$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks

\section{Introduction}\label{intr}

Many problems of modern mathematics and theoretical physics lead to the
investigation of hypergeometric functions of many variables.
In particular, problems of superstring theory \cite{can-os-gr-pa},
analytical continuations of Mellin-Barnes integrals \cite{pas-ts-ch} and
algebraic geometry \cite{hor-de}. Systems of hypergeometric type differential
equations have numerous applications as nontrivial model examples in
realization of algorithms for symbolic calculations, which are used in modern
systems of computer algebra \cite{saito-st-tak}. Hypergeometric functions of
many variables appear in quantum field theory as solutions of Knizhnik-Zamolodchikov
equation \cite{var-de}. These equations can be considered as generalized
hypergeometric type equations and their solutions have integral representations,
which generalize classic Euler integrals for hypergeometric functions of one variable.
 This approach allows us to link the special functions of hypergeometric type and
challenges the theory of representations of Lie algebras and quantum groups
\cite{var-de}.

Initially hypergeometric functions introduced by many authors with different
 methods, which are not related with each other. Their occurrence is determined,
as a rule, by the need to solve problems, that led to a differential equation
(or system of equations), insoluble in the class of elementary functions.
Thus arose  Bessel functions, Hermite functions, Gauss hypergeometric function.
Hypergeometric functions occupy an important place among the special functions
of mathematical physics. Many problems of gas dynamics are reduced to boundary
value problems for degenerate equations of mixed type.

It is known that the degenerate equation of mixed type in the hyperbolic part
of the domain is reduced to the generalized equation of Euler-Poisson-Darboux
\begin{equation}\label{e1.1}
{u}_{tt}+\frac{2\gamma }{t}{{u}_{t}}={u}_{xx}+{u}_{yy}
+\frac{2\alpha }{x}{{u}_{x}}+\frac{2\beta }{y}{{u}_y},\quad x>0,\; y>0,\; t>0,
\end{equation}
where $\alpha >0$, $\beta >0$ and $\gamma >0$ are constants.

We note that Euler-Poisson-Darboux equation
\begin{equation}\label{e1.2}
{{u}_{\xi \eta }}-\frac{\beta }{\xi -\eta }{{u}_{\xi }}
+\frac{\alpha }{\xi -\eta }{{u}_{\eta }}=0,\quad \alpha >0,\;\beta >0,\;
\alpha +\beta <1,
\end{equation}
was considered in  \cite{smirnov-de}, where the Cauchy problem for \eqref{e1.2}
was solved. In \cite{sai-1, sai-2, sai-3}, non-local boundary problems
for \eqref{e1.2} were investigated in characteristic triangles.
In \cite{has-1}, two confluent hypergeometric functions of three variables
were introduced. Further, for introduced hypergeometric functions
authors prove formulas of analytical continuation. Using the introduced
confluent hypergeometric functions, they constructed the Riemann
function for the generalized Euler -Poisson-Darboux equation
\begin{equation}\label{e1.3}
{{u}_{\xi \eta }}+[ \frac{\alpha }{\eta +\xi }
+\frac{\beta }{\eta -\xi }]{{u}_{\xi }}
+[ \frac{\alpha }{\eta +\xi }-\frac{\beta }{\eta -\xi }]{{u}_{\eta }}+\gamma u=0.
\end{equation}
Further, by the Riemann-function method the Cauchy problem for \eqref{e1.3}
 was solved in characteristic triangle. Solution is written in an explicit form.
Note that Euler -Poisson-Darboux equations \eqref{e1.2} and \eqref{e1.3} are
written in characteristic coordinates.

In \cite{seil}, the unique solvability of the Darboux problem with deviation
for the Euler -Poisson-Darboux equation was proved outside of characteristic cone.
Other type of the Euler -Poisson-Darboux equations were investigated in
 works \cite{bentrad, bres, bers, yag, fran, rassias, barros-1, barros-2, barros-3}.

\section{Reduction of the Euler-Poisson-Darboux equation to a
 system of Lauricella hypergeometric functions}\label{FP}

Solution of the Euler-Poisson-Darboux equation \eqref{e1.1} is searched in the form
\begin{equation}\label{e2.1}
u=P\omega ( \xi ,\eta ,\zeta),
\end{equation}
where $\omega ( \xi ,\eta ,\zeta)$ is unknown function and
\begin{gather}\label{e2.2}
P={{( {{r}^2})}^{-\alpha -\beta -\gamma -\frac{1}{2}}} ,\\
\xi =\frac{{{r}^2}-r_1^2}{{{r}^2}},\quad
\eta =\frac{{{r}^2}-r_2^2}{{{r}^2}},\quad
\zeta =\frac{{{r}^2}-r_3^2}{{{r}^2}}, \nonumber \\
{{r}^2}={{( x-{{x}_0})}^2}+{{( y-{{y}_0})}^2}-{{( t-{{t}_0})}^2},\quad
r_1^2={{( x+{{x}_0})}^2}+{{( y-{{y}_0})}^2}-{{( t-{{t}_0})}^2}, \nonumber \\
r_2^2={{( x-{{x}_0})}^2}+{{( y+{{y}_0})}^2}-{{( t-{{t}_0})}^2},\quad
r_3^2={{( x-{{x}_0})}^2}+{{( y-{{y}_0})}^2}-{{( t+{{t}_0})}^2}. \nonumber
\end{gather}
Calculating necessary derivatives from \eqref{e2.1} and substituting them
into \eqref{e1.1}, we obtain
\begin{equation}\label{e2.3}
\begin{aligned}
&{{A}_1}{{\omega }_{\xi \xi }}+{{A}_2}{{\omega }_{\eta \eta }}
+{{A}_3}{{\omega }_{\zeta \zeta }}+{{B}_1}{{\omega }_{\xi \eta }}
+{{B}_2}{{\omega }_{\xi \zeta }}+{{B}_3}{{\omega }_{\eta \zeta }}\\
&+{{C}_1}{{\omega }_{\xi }}+{{C}_2}{{\omega }_{\eta }}
+{{C}_3}{{\omega }_{\zeta }}+D\omega =0,
\end{aligned}
\end{equation}
where
\begin{gather*}
{{A}_1}=P\xi _{x}^2+P\xi _y^2-P\xi _{t}^2,\quad
{{A}_2}=P\eta _{x}^2+P\eta _y^2-P\eta _{t}^2,\quad
{{A}_3}=P\zeta _{x}^2+P\zeta _y^2-P\zeta _{t}^2, \\
{{B}_1}=2P{{\xi }_{x}}{{\eta }_{x}}+2P{{\xi }_y}{{\eta }_y}
-2P{{\xi }_{t}}{{\eta }_{t}},\quad
{{B}_2}=2P{{\xi }_{x}}{{\zeta }_{x}}+2P{{\xi }_y}{{\zeta }_y}
 -2P{{\xi }_{t}}{{\zeta }_{t}}, \\
{{B}_3}=2P{{\eta }_{x}}{{\zeta }_{x}}+2P{{\eta }_y}{{\zeta }_y}
-2P{{\eta }_{t}}{{\zeta }_{t}}, \\
{{C}_1}=P( {{\xi }_{xx}}+{{\xi }_{yy}}-{{\xi }_{tt}})
+2( {{P}_{x}}{{\xi }_{x}}+{{P}_y}{{\xi }_y}-{{P}_{t}}{{\xi }_{t}})
+P\Big( \frac{2\alpha }{x}{{\xi }_{x}}+\frac{2\beta }{y}{{\xi }_y}
-\frac{2\gamma }{t}{{\xi }_{t}}\Big),
\\
{{C}_2}=P( {{\eta }_{xx}}+{{\eta }_{yy}}-{{\eta }_{tt}})
+2( {{P}_{x}}{{\eta }_{x}}+{{P}_y}{{\eta }_y}-{{P}_{t}}{{\eta }_{t}})
+P\Big( \frac{2\alpha }{x}{{\eta }_{x}}+\frac{2\beta }{y}{{\eta }_y}
-\frac{2\gamma }{t}{{\eta }_{t}}\Big),
\\
{{C}_3}=P( {{\zeta }_{xx}}+{{\zeta }_{yy}}-{{\zeta }_{tt}})
+2( {{P}_{x}}{{\zeta }_{x}}+{{P}_y}{{\zeta }_y} -{{P}_{t}}{{\zeta }_{t}})
+P\Big(\frac{2\alpha }{x}{{\zeta }_{x}}+\frac{2\beta }{y}{{\zeta }_y}
-\frac{2\gamma }{t}{{\zeta }_{t}}\Big),
\\
D={{P}_{xx}}+{{P}_{yy}}-{{P}_{tt}}+{{P}_{x}}\frac{2\alpha }{x}
+{{P}_y}\frac{2\beta }{y}-\frac{2\gamma }{t}{{P}_{t}.}
\end{gather*}
Now we consider ${{A}_1}$. Since
\begin{gather*}
\xi _{x}^2=\frac{4{{( x+{{x}_0})}^2}{{( {{r}^2})}^2}-8( x+{{x}_0})
( x-{{x}_0}){{r}^2}r_1^2+4{{( x-{{x}_0})}^2}{{( r_1^2)}^2}}{{{( {{r}^2})}^{4}}},
\\
\xi _y^2=\frac{4{{( y-{{y}_0})}^2}{{( {{r}^2})}^2}-8{{( y-{{y}_0})}^2}{{r}^2}r_1^2
+4{{( y-{{y}_0})}^2}{{( r_1^2)}^2}}{{{( {{r}^2})}^{4}}},
\\
\xi _{t}^2=\frac{4{{( t-{{t}_0})}^2}{{( {{r}^2})}^2}-8{{( t-{{t}_0})}^2}{{r}^2}r_1^2
+4{{( t-{{t}_0})}^2}{{( r_1^2)}^2}}{{{( {{r}^2})}^{4}}},
\end{gather*}
 we obtain
\[
{{A}_1}=4Pr_1^2{{( {{r}^2})}^{-3}}[ ( {{r}^2})
+( r_1^2)-2( x+{{x}_0})( x-{{x}_0})-2{{( y-{{y}_0})}^2}+2{{( t-{{t}_0})}^2}],
\]
 or ${{A}_1}=4Pr_1^2{{( {{r}^2})}^{-3}}4x_0^2$.
By  the equality $4{{x}_0}=-{{x}^{-1}}{{r}^2}\xi $, we  obtain
\begin{equation}\label{e2.4}
{{A}_1}=-4P{{x}^{-1}}{{x}_0}{{( {{r}^2})}^{-1}}\xi ( 1-\xi).
\end{equation}
Similarly we have
\begin{gather}\label{e2.5}
{{A}_2}=-4P{{y}^{-1}}{{y}_0}{{( {{r}^2})}^{-1}}\eta ( 1-\eta), \\
\label{e2.6}
{{A}_3}=-4P{{t}^{-1}}{{t}_0}{{( {{r}^2})}^{-1}}\zeta ( 1-\zeta).
\end{gather}
Further, we calculate representation of ${{B}_1}$.
Finding necessary derivatives from the arguments and substituting them
 we obtain
\begin{align*}
{{B}_1}&=2P\Big\{ \frac{2( x+{{x}_0}){{r}^2}
-2( x-{{x}_0})r_1^2}{{{( {{r}^2})}^2}}\frac{2( x-{{x}_0}){{r}^2}
-2( x-{{x}_0})r_2^2}{{{( {{r}^2})}^2}}\\
&\quad +\frac{2( y-{{y}_0}){{r}^2}-2( y-{{y}_0})r_1^2}{{{( {{r}^2})}^2}}
\frac{2( y+{{y}_0}){{r}^2}-2( y-{{y}_0})r_2^2}{{{( {{r}^2})}^2}} \\
&\quad -\frac{2( t-{{t}_0}){{r}^2}-2( t-{{t}_0})r_1^2}{{{( {{r}^2})}^2}}
 \frac{2( t-{{t}_0}){{r}^2}-2( t-{{t}_0})r_2^2}{{{( {{r}^2})}^2}} \Big\},
\end{align*}
or
\begin{align*}
{{B}_1}&=8P{{( {{r}^2})}^{-4}}\Big\{ [ ( x+{{x}_0}){{r}^2}-( x-{{x}_0})r_1^2]
 [ ( x-{{x}_0}){{r}^2}-( x-{{x}_0})r_2^2]  \\
&\quad +[ ( y-{{y}_0}){{r}^2}-( y-{{y}_0})r_1^2]
 [ ( y+{{y}_0}){{r}^2}-( y-{{y}_0})r_2^2] \\
&\quad - [ ( t-{{t}_0}){{r}^2}-( t-{{t}_0})r_1^2]
 [ ( t-{{t}_0}){{r}^2}-( t-{{t}_0})r_2^2] \Big\}.
\end{align*}
After some evaluations, we deduce
\begin{equation}\label{e2.7}
{{B}_1}=4P{{y}^{-1}}{{y}_0}{{( {{r}^2})}^{-1}}\xi \eta
+4P{{x}^{-1}}{{x}_0}{{( {{r}^2})}^{-1}}\xi \eta.
\end{equation}
Similarly one obtains
\begin{gather}\label{e2.8}
{{B}_2}=4P{{t}^{-1}}{{t}_0}{{( {{r}^2})}^{-1}}\xi \zeta
 +4P{{x}^{-1}}{{x}_0}{{( {{r}^2})}^{-1}}\xi \zeta, \\
\label{e2.9}
{{B}_3}=4P{{t}^{-1}}{{t}_0}{{( {{r}^2})}^{-1}}\eta \zeta
 +4P{{y}^{-1}}{{y}_0}{{( {{r}^2})}^{-1}}\eta \zeta.
\end{gather}
Further, considering the following expressions
\begin{gather*}
{{\xi }_{xx}}+{{\xi }_{yy}}-{{\xi }_{tt}}=2{{( {{r}^2})}^{-1}}
( 2{{x}^{-1}}{{x}_0}\xi -\xi),
\\
{{P}_{x}}{{\xi }_{x}}+{{P}_y}{{\xi }_y}-{{P}_{t}}{{\xi }_{t}}
=2P{{( {{r}^2})}^{-1}}\big(\alpha +\beta +\gamma +\frac{1}{2}\big)
 ( {{x}^{-1}}{{x}_0}\xi +\xi),
\\
\begin{aligned}
\frac{2\alpha }{x}{{\xi }_{x}}+\frac{2\beta }{y}{{\xi }_y}
-\frac{2\gamma }{t}{{\xi }_{t}}
&=-4{{( {{r}^2})}^{-1}}\big[ \alpha {{x}^{-1}}{{x}_0}+\alpha \xi
 +\alpha {{x}^{-1}}{{x}_0}( 1-\xi)-\beta {{y}^{-1}}{{y}_0}  \\
&\quad +\beta \xi +\beta {{y}^{-1}}{{y}_0}( 1-\xi)
 -\gamma {{t}^{-1}}{{t}_0}+\gamma \xi +\gamma {{t}^{-1}}{{t}_0}( 1-\xi)\big],
\end{aligned}
\end{gather*}
we define
\begin{equation}\label{e2.10}
\begin{aligned}
{{C}_1}&=-4P{{( {{r}^2})}^{-1}}{{x}^{-1}}{{x}_0}
[ 2\alpha -( 2\alpha +\beta +\gamma +\frac{1}{2}+1)\xi] \\
&\quad +4P{{( {{r}^2})}^{-1}}{{y}^{-1}}{{y}_0}\beta \xi
 +4P{{( {{r}^2})}^{-1}}{{t}^{-1}}{{t}_0}\gamma \xi .
\end{aligned}
\end{equation}
Similarly, we define
\begin{gather}\label{e2.11}
\begin{aligned}
{{C}_2}&=-4P{{( {{r}^2})}^{-1}}{{y}^{-1}}{{y}_0}[ 2\beta
-( \alpha +2\beta +\gamma +\frac{1}{2}+1)\eta] \\
&\quad +4P{{( {{r}^2})}^{-1}}{{x}^{-1}}{{x}_0}\alpha \eta
+4P{{( {{r}^2})}^{-1}}{{t}^{-1}}{{t}_0}\gamma \eta ,
\end{aligned} \\
\label{e2.12}
\begin{aligned}
{{C}_3}&=-4P{{( {{r}^2})}^{-1}}{{t}^{-1}}{{t}_0}
[ 2\gamma -( \alpha +\beta +2\gamma +\frac{1}{2}+1)\zeta] \\
&\quad +4P{{( {{r}^2})}^{-1}}{{x}^{-1}}{{x}_0}\gamma \zeta
 +4P{{( {{r}^2})}^{-1}}{{y}^{-1}}{{y}_0}\beta \zeta .
\end{aligned}
\end{gather}
After simple calculations we obtain
\begin{equation}\label{e2.13}
\begin{aligned}
 D&=4( \alpha +\beta +\gamma +\frac{1}{2})P{{( {{r}^2})}^{-1}}
 \alpha {{x}^{-1}}{{x}_0}
 +4( \alpha +\beta +\gamma +\frac{1}{2})P{{( {{r}^2})}^{-1}}\beta {{y}^{-1}}{{y}_0}\\
&\quad  +4( \alpha +\beta +\gamma +\frac{1}{2})
 P{{( {{r}^2})}^{-1}}\gamma {{t}^{-1}}{{t}_0}.
\end{aligned}
\end{equation}
Substituting \eqref{e2.4}-\eqref{e2.13} in \eqref{e2.3}, we obtain
\begin{equation}\label{e2.14}
\begin{aligned}
& -\frac{4P{{x}_0}}{x{{r}^2}}\Big\{ \xi ( 1-\xi){{\omega }_{\xi \xi }}
-\xi \eta {{\omega }_{\xi \eta }}-\xi \zeta {{\omega }_{\xi \zeta }}
-\alpha \eta {{\omega }_{\eta }}-\gamma \zeta {{\omega }_{\zeta }}  \\
&  +[ 2\alpha -( \alpha +\beta +\gamma +\frac{1}{2}+\alpha +1)\xi]{{\omega }_{\xi }}
-( \alpha +\beta +\gamma +\frac{1}{2})\alpha \omega  \Big\} \\
& -\frac{4P{{y}_0}}{y{{r}^2}}\Big\{ \eta ( 1-\eta){{\omega }_{\eta \eta }}
 -\xi \eta {{\omega }_{\xi \eta }}-\eta \zeta {{\omega }_{\eta \zeta }}
 -\beta \xi {{\omega }_{\xi }}-\beta \zeta {{\omega }_{\zeta }}  \\
& \quad +[ 2\beta -( \alpha +\beta +\gamma
 +\frac{1}{2}+\beta +1)\eta]{{\omega }_{\eta }}
 -( \alpha +\beta +\gamma +\frac{1}{2})\beta \omega  \Big\} \\
& -\frac{4P{{t}_0}}{t{{r}^2}}\Big\{ \zeta ( 1-\zeta){{\omega }_{\zeta \zeta }}
 -\xi \zeta {{\omega }_{\xi \zeta }}-\eta \zeta {{\omega }_{\eta \zeta }}
 -\gamma \xi {{\omega }_{\xi }}-\gamma \eta {{\omega }_{\eta }}  \\
&  +[ 2\gamma -( \alpha +\beta +\gamma +\frac{1}{2}+\gamma +1)
 \zeta]{{\omega }_{\zeta }}-( \alpha +\beta +\gamma +\frac{1}{2})\gamma \omega
 \Big\}=0.
\end{aligned}
\end{equation}
Hence, the following equality is valid
\begin{equation}\label{e2.15}
\begin{aligned}
& \xi ( 1-\xi){{\omega }_{\xi \xi }}-\xi \eta {{\omega }_{\xi \eta }}
 -\xi \zeta {{\omega }_{\xi \zeta }}
 +[ 2\alpha -( \alpha +\beta +\gamma +\frac{1}{2}+\alpha +1)\xi]{{\omega }_{\xi }} \\
& -\alpha \eta {{\omega }_{\eta }}-\gamma \zeta {{\omega }_{\zeta }}
 -( \alpha +\beta +\gamma +\frac{1}{2})\alpha \omega =0,
  \\
& \eta ( 1-\eta){{\omega }_{\eta \eta }}-\xi \eta {{\omega }_{\xi \eta }}
 -\eta \zeta {{\omega }_{\eta \zeta }}+[ 2\beta
 -( \alpha +\beta +\gamma +\frac{1}{2}+\beta +1)\eta]{{\omega }_{\eta }} \\
& -\beta \xi {{\omega }_{\xi }}-\beta \zeta {{\omega }_{\zeta }}
 -( \alpha +\beta +\gamma +\frac{1}{2})\beta \omega =0,
\\
& \zeta ( 1-\zeta){{\omega }_{\zeta \zeta }}-\xi \zeta {{\omega }_{\xi \zeta }}
 -\eta \zeta {{\omega }_{\eta \zeta }}+[ 2\gamma -( \alpha +\beta +\gamma
 +\frac{1}{2}+\gamma +1)\zeta]{{\omega }_{\zeta }} \\
& -\gamma \xi {{\omega }_{\xi }}-\gamma \eta {{\omega }_{\eta }}
 -( \alpha +\beta +\gamma +\frac{1}{2})\gamma \omega =0.
\end{aligned}
\end{equation}
Thus, the Euler-Poisson-Darboux equation \eqref{e1.1} equivalently reduced
to the system \eqref{e2.15}.

\section{Particular solutions of the Euler-Poisson-Darboux equation}\label{AP}

In \cite{Appell}, system  \eqref{e2.15}  was
considered for the $n$-dimensional case
\begin{align*}
& {{x}_{j}}( 1-{{x}_{j}})\frac{{{\partial }^2}F_{A}^{( n)}}{\partial x_{j}^2}
 -{{x}_{j}}\sum_{ k=1,\, k\ne j} ^{n}
{{x}_{k}}\frac{{{\partial }^2}F_{A}^{( n)}}{\partial {{x}_{k}}\partial {{x}_{j}}}
+[ {{c}_{j}}-( a+{{b}_{j}}+1){{x}_{j}}]
\frac{\partial F_{A}^{( n)}}{\partial {{x}_{j}}} \\
 & -{{b}_{j}}\sum_{ k=1, \,  k\ne j}^{n} {{x}_{k}}
\frac{\partial F_{A}^{( n)}}{\partial {{x}_{k}}}
-a{{b}_{j}}F_{A}^{( n)}=0,\quad j=1,2,\dots,n.
\end{align*}
There were found $2^n$ particular solutions of this system.
 All of them are expressed by Lauricella hypergeometric functions
$ F_{A}^{( n)} $. In particular case, system of hypergeometric
functions \eqref{e2.15} has the following solutions \cite{Appell}
\begin{gather}\label{e3.1}
{{\omega }_1}=F_{A}^{( 3)}\Big( \alpha +\beta +\gamma +\frac{1}{2};\alpha ,\beta ,
\gamma ;2\alpha ,2\beta ,2\gamma ;\xi ,\eta ,\zeta\Big), \\
\label{e3.2}
{{\omega }_2}={{\xi }^{1-2\alpha }}F_{A}^{( 3)}\Big( -\alpha +\beta +\gamma +\frac{3}{2};
1-\alpha ,\beta ,\gamma ;2-2\alpha ,2\beta ,2\gamma ;\xi ,\eta ,\zeta\Big), \\
\label{e3.3}
{{\omega }_3}={{\eta }^{1-2\beta }}F_{A}^{( 3)}\Big( \alpha -\beta +\gamma
+\frac{3}{2};\alpha ,1-\beta ,\gamma ;2\alpha ,2-2\beta ,2\gamma ;\xi ,\eta ,
 \zeta\Big), \\
\label{e3.4}
\begin{aligned}
{{\omega }_4}={{\zeta }^{1-2\gamma }}F_{A}^{( 3)}\Big( \alpha +\beta -\gamma
+\frac{3}{2};\alpha ,\beta ,1-\gamma ;2\alpha ,2\beta ,2-2\gamma ;
 \xi ,\eta ,\zeta\Big),
\end{aligned} \\
\label{e3.5}
\begin{aligned}
{{\omega }_5}&={{\xi }^{1-2\alpha }}{{\eta }^{1-2\beta }}F_{A}^{( 3)}
\Big( -\alpha -\beta +\gamma +\frac{5}{2};1-\alpha ,1-\beta ,\gamma ;\\
&\quad 2-2\alpha ,2-2\beta ,2\gamma ;\xi ,\eta ,\zeta\Big),
\end{aligned} \\
\label{e3.6}
\begin{aligned}
{{\omega }_6}&={{\xi }^{1-2\alpha }}{{\zeta }^{1-2\gamma }}F_{A}^{( 3)}
\Big( -\alpha +\beta -\gamma +\frac{5}{2};1-\alpha ,\beta ,1-\gamma ;2-2\alpha ,
 2\beta , \\
&\quad 2-2\gamma ;\xi ,\eta ,\zeta\Big),
\end{aligned} \\
\label{e3.7}
\begin{aligned}
{{\omega }_7}
&={{\eta }^{1-2\beta }}{{\zeta }^{1-2\gamma }}F_{A}^{( 3)}
\Big( \alpha -\beta -\gamma +\frac{5}{2};\alpha ,1-\beta ,1-\gamma ;2\alpha , 2-2\beta ,\\
&\quad 2-2\gamma ;\xi ,\eta ,\zeta\Big),
\end{aligned} \\
\label{e3.8}
\begin{aligned}
{{\omega }_8}&={{\xi }^{1-2\alpha }}{{\eta }^{1-2\beta }}{{\zeta }^{1-2\gamma }}
 F_{A}^{( 3)}\Big( -\alpha -\beta -\gamma +\frac{7}{2};1-\alpha ,1-\beta ,1-\gamma ;\\
&\quad 2-2\alpha ,2-2\beta ,2-2\gamma ;\xi ,\eta ,\zeta\Big),
 \end{aligned}
\end{gather}
where the Lauricella hypergeometric function is defined as
$$
F_{A}^{( 3)}( a;{{b}_1},{{b}_2},{{b}_3};{{c}_1},{{c}_2},{{c}_3};x,y,z)
=\sum_{m,n,p=0}^{\infty } \frac{{{( a)}_{m+n+p}}
{{( {{b}_1})}_{m}}{{( {{b}_2})}_{n}}{{( {{b}_3})}_{p}}}
{{{( {{c}_1})}_{m}}{{( {{c}_2})}_{n}}{{( {{c}_3})}_{p}}m!n!p!}
{{x}^{m}}{{y}^{n}}{{z}^{p}}.
$$
Further, substituting \eqref{e3.1} - \eqref{e3.8} in \eqref{e2.1}, we obtain
\begin{gather}\label{e3.9}
\begin{aligned}
& {{q}_1}( x,y,t;{{x}_0},{{y}_0},{{t}_0})\\
&={{k}_1}{{( {{r}^2})}^{-\alpha -\beta -\gamma -\frac{1}{2}}}
 {{F}_{A}}\Big( \alpha +\beta +\gamma +\frac{1}{2};\alpha ,\beta ,\gamma ;
2\alpha ,2\beta ,2\gamma ;\xi ,\eta ,\zeta\Big),
\end{aligned} \\
\label{e3.10}
\begin{aligned}
&{{q}_2}( x,y,t;{{x}_0},{{y}_0},{{t}_0})\\
&={{k}_2}{{( {{r}^2})}^{\alpha -\beta -\gamma -\frac{3}{2}}}
  {{( x{{x}_0})}^{1-2\alpha }}F_{A}^{( 3)}\Big( -\alpha +\beta +\gamma
 +\frac{3}{2};1-\alpha ,\beta ,\gamma ;\\
&\quad 2-2\alpha ,2\beta ,2\gamma ;\xi ,\eta ,\zeta\Big),
\end{aligned}\\
\label{e3.11}
\begin{aligned}
 & {{q}_3}( x,y,t;{{x}_0},{{y}_0},{{t}_0})\\
&={{k}_3}{{( {{r}^2})}^{-\alpha +\beta -\gamma -\frac{3}{2}}}
  {{( y{{y}_0})}^{1-2\beta }}F_{A}^{( 3)}\Big( \alpha -\beta +\gamma
+\frac{3}{2};\alpha ,1-\beta ,\gamma ;2\alpha , \\
&\quad 2-2\beta , 2\gamma ;\xi ,\eta ,\zeta\Big),
\end{aligned}\\
\label{e3.12}
\begin{aligned}
& {{q}_4}( x,y,t;{{x}_0},{{y}_0},{{t}_0})\\
&={{k}_4}{{( {{r}^2})}^{-\alpha -\beta +\gamma -\frac{3}{2}}}
  {{( t{{t}_0})}^{1-2\gamma }}F_{A}^{( 3)}\Big( \alpha +\beta -\gamma
+\frac{3}{2};\alpha ,\beta ,1-\gamma ;2\alpha ,2\beta ,\\
&\quad 2-2\gamma ;\xi ,\eta ,\zeta\Big),
\end{aligned} \\
\label{e3.13}
\begin{aligned}
& {{q}_5}( x,y,t;{{x}_0},{{y}_0},{{t}_0})\\
&={{k}_5}{{( {{r}^2})}^{\alpha +\beta -\gamma
-\frac{5}{2}}}{{( x{{x}_0})}^{1-2\alpha }}{{( y{{y}_0})}^{1-2\beta }}
 F_{A}^{( 3)}\Big( -\alpha -\beta +\gamma +\frac{5}{2};  1-\alpha , \\
&\quad 1-\beta ,\gamma ;2-2\alpha ,2-2\beta ,2\gamma ;\xi ,\eta ,\zeta\Big),
\end{aligned} \\
\label{e3.14}
\begin{aligned}
 & {{q}_6}( x,y,t;{{x}_0},{{y}_0},{{t}_0})\\
&={{k}_6}{{( {{r}^2})}^{\alpha -\beta
+\gamma -\frac{5}{2}}}{{( x{{x}_0})}^{1-2\alpha }}{{( t{{t}_0})}^{1-2\gamma }}
 F_{A}^{( 3)}\Big( -\alpha +\beta -\gamma +\frac{5}{2};1-\alpha , \beta ,\\
&\quad 1-\gamma ; 2-2\alpha ,2\beta ,2-2\gamma ;\xi ,\eta ,\zeta\Big),
\end{aligned} \\
\label{e3.15}
\begin{aligned}
 & {{q}_7}( x,y,t;{{x}_0},{{y}_0},{{t}_0})\\
&={{k}_7}{{( {{r}^2})}^{-\alpha +\beta +\gamma
 -\frac{5}{2}}}{{( y{{y}_0})}^{1-2\beta }}{{( t{{t}_0})}^{1-2\gamma }}
 F_{A}^{( 3)}\Big( \alpha -\beta -\gamma +\frac{5}{2};\alpha ,1-\beta ,\\
&\quad 1-\gamma ;2\alpha ,2-2\beta ,2-2\gamma ;\xi ,\eta ,\zeta\Big),
\end{aligned} \\
\label{e3.16}
\begin{aligned}
 & {{q}_8}( x,y,t;{{x}_0},{{y}_0},{{t}_0})\\
&={{k}_8}{{( x{{x}_0})}^{1-2\alpha }}{{( y{{y}_0})}^{1-2\beta }}
{{( t{{t}_0})}^{1-2\gamma }}{{( {{r}^2})}^{\alpha +\beta +\gamma -\frac{7}{2}}}
  F_{A}^{( 3)}\Big( -\alpha -\beta -\gamma +\frac{7}{2};\\
&\quad 1-\alpha , 1-\beta ,1-\gamma ;2-2\alpha ,2-2\beta ,2-2\gamma ;\xi ,\eta ,\zeta\Big) \\
\end{aligned}
\end{gather}
where $k_i$, $i=\overline{1,\,8}$ are constants.

\section{Some properties of particular solutions}\label{PP}

\begin{theorem}\label{thm1}
If $\alpha ,\beta ,\gamma >0$, then particular solutions
\eqref{e3.9}--\eqref{e3.16} tends to infinity of the order $1/r$ at $r\to 0$.
\end{theorem}

\begin{proof}
By the expansion for the Lauricella hypergeometric function \cite{has-sri}
\begin{equation}\label{e4.1}
\begin{aligned}
& F_{A}^{( 3)}( a;{{b}_1},{{b}_2},{{b}_3};{{c}_1},{{c}_2},{{c}_3};x,y,z) \\
 & =\sum_{i,j,k=0}^{\infty } \frac{{{( a)}_{i+j+k}}{{( {{b}_1})}_{i+j}}{{( {{b}_2})}_{i+k}}{{( {{b}_3})}_{j+k}}}{{{( {{c}_1})}_{i+j}}{{( {{c}_2})}_{i+k}}{{( {{c}_3})}_{j+k}}i!j!k!}{{x}^{i+j}}{{y}^{i+k}}{{z}^{j+k}} \\
 & \times F( a+i+j,{{b}_1}+i+j;{{c}_1}+i+j;x)F( a+i+j+k,{{b}_2}+i+k;{{c}_2}+i+k;y) \\
 & \times F( a+i+j+k,{{b}_3}+j+k;{{c}_3}+j+k;z),
\end{aligned}
\end{equation}
the particular solution \eqref{e3.9} is rewritten as follows
\begin{equation}\label{e4.2}
\begin{aligned}
 & {{q}_1}( x,y,t;{{x}_0},{{y}_0},{{t}_0})={{k}_1}{{( {{r}^2})}^{-\alpha -\beta -\gamma -\frac{1}{2}}} \\
 & \times \sum_{i,j,k=0}^{\infty } \frac{{{( \alpha +\beta +\gamma +\frac{1}{2})}_{i+j+k}}{{( \alpha)}_{i+j}}{{( \beta)}_{i+k}}{{( \gamma)}_{j+k}}}{{{( 2\alpha)}_{i+j}}{{( 2\beta)}_{i+k}}{{( 2\gamma)}_{j+k}}i!j!k!} \\
 & \times {{\big( \frac{{{r}^2}-r_1^2}{{{r}^2}}\big)}^{i+j}}
{{\big( \frac{{{r}^2}-r_2^2}{{{r}^2}}\big)}^{i+k}}
{{\big( \frac{{{r}^2}-r_3^2}{{{r}^2}}\big)}^{j+k}} \\
 & \times F\Big( \alpha +\beta +\gamma +\frac{1}{2}+i+j,\alpha +i+j;2\alpha +i+j;
 \frac{{{r}^2}-r_1^2}{{{r}^2}}\Big) \\
 & \times F\Big( \alpha +\beta +\gamma +\frac{1}{2}+i+j+k,\beta +i+k;2\beta +i+k;
 \frac{{{r}^2}-r_2^2}{{{r}^2}}\Big) \\
 & \times F\Big( \alpha +\beta +\gamma +\frac{1}{2}+i+j+k,\gamma +j+k;2\gamma +j+k;
 \frac{{{r}^2}-r_3^2}{{{r}^2}}\Big). \\
\end{aligned}
\end{equation}
Using the formula $ F( a,b;c;x)={{( 1-x)}^{-b}}F( c-a,b;c;x/( x-1)) $ \cite{erdel},
from \eqref{e4.2} we obtain
\begin{equation}\label{e4.3}
\begin{aligned}
{{q}_1}( x,y,t;{{x}_0},{{y}_0},{{t}_0})={{k}_1}{{( {{r}^2})}^{-\frac{1}{2}}}{{( r_1^2)}^{-\alpha }}{{( r_2^2)}^{-\beta }}{{( r_3^2)}^{-\gamma }}q_1^{*}( x,y,t;{{x}_0},{{y}_0},{{t}_0}),
\end{aligned}
\end{equation}
where
\begin{equation}\label{e4.4}
\begin{aligned}
 q_1^{*}( x,y,t;{{x}_0},{{y}_0},{{t}_0})
& =\sum_{i,j,k=0}^{\infty } \frac{{{( \alpha +\beta +\gamma +\frac{1}{2})}_{i+j+k}}{{( \alpha)}_{i+j}}{{( \beta)}_{i+k}}{{( \gamma)}_{j+k}}}{{{( 2\alpha)}_{i+j}}{{( 2\beta)}_{i+k}}{{( 2\gamma)}_{j+k}}i!j!k!} \\
 & \times {{\big( \frac{r_1^2-{{r}^2}}{r_1^2}\big)}^{i+j}}
{{\big( \frac{r_2^2-{{r}^2}}{r_2^2}\big)}^{i+k}}
{{\big( \frac{r_3^2-{{r}^2}}{r_3^2}\big)}^{j+k}} \\
 & \times F\Big( \alpha -\beta -\gamma -\frac{1}{2},\alpha +i+j;
 2\alpha +i+j;\frac{r_1^2-{{r}^2}}{r_1^2}\Big) \\
 & \times F\Big( \beta -\alpha -\gamma -\frac{1}{2}-j,\beta +i+k;2\beta +i+k;
\frac{r_2^2-{{r}^2}}{r_2^2}\Big) \\
 & \times F\Big( \gamma -\alpha -\beta -\frac{1}{2}-i,\gamma +j+k;2\gamma +j+k;
\frac{r_3^2-{{r}^2}}{r_3^2}\Big).
\end{aligned}
\end{equation}
Now we show that $ q_1^{*}( x,y,t;{{x}_0},{{y}_0},{{t}_0}) $ is bounded
aa $ r\to 0 $. Since
$$
\lim_{r\to 0} \frac{r_1^2-{{r}^2}}{r_1^2}
=\lim_{r\to 0} \frac{r_2^2-{{r}^2}}{r_2^2}
=\lim_{r\to 0} \frac{r_3^2-{{r}^2}}{r_3^2}=1,
$$
hence, one can find
\begin{gather}\label{e4.5}
\begin{aligned}
 & F\Big( \alpha -\beta -\gamma -\frac{1}{2},\alpha +i+j;2\alpha +i+j;1\Big) \\
 & =\frac{\Gamma ( 2\alpha)\Gamma ( \beta +\gamma +\frac{1}{2})
{{( 2\alpha)}_{i+j}}}{\Gamma ( \alpha)\Gamma ( \alpha +\beta +\gamma
+\frac{1}{2}){{( \alpha +\beta +\gamma +\frac{1}{2})}_{i+j}}},
\end{aligned} \\
\label{e4.6}
\begin{aligned}
 & F\Big( \beta -\alpha -\gamma -\frac{1}{2}-j,\beta +i+k;2\beta +i+k;1\Big) \\
 & =\frac{\Gamma ( 2\beta)\Gamma ( \alpha +\gamma
 +\frac{1}{2}){{( 2\beta)}_{i+k}}{{( \alpha +\gamma
 +\frac{1}{2})}2_{j}}}{\Gamma ( \beta)\Gamma ( \alpha +\beta +\gamma
 +\frac{1}{2}){{( \alpha +\beta +\gamma +\frac{1}{2})}_{i+j+k}}},
\end{aligned} \\
\label{e4.7}
\begin{aligned}
 & F\Big( \gamma -\alpha -\beta -\frac{1}{2}-i,\gamma +j+k;2\gamma +j+k;1\Big) \\
 & =\frac{\Gamma ( 2\gamma){{( 2\gamma)}_{j+k}}\Gamma ( \alpha +\beta
+\frac{1}{2}){{( \alpha +\beta +\frac{1}{2})}_i}}{\Gamma ( \gamma)\Gamma ( \alpha
+\beta +\gamma +\frac{1}{2}){{( \alpha +\beta +\gamma +\frac{1}{2})}_{i+j+k}}}.
\end{aligned}
\end{gather}
Due to \eqref{e4.5}--\eqref{e4.7} at $ r\to 0 $ from \eqref{e4.4} we obtain
\begin{equation}\label{e4.8}
\begin{aligned}
&\lim_{r\to 0} q_1^{*}( x,y,t;{{x}_0},{{y}_0},{{t}_0}) \\
 & =\frac{\Gamma ( 2\alpha)\Gamma ( 2\beta)\Gamma ( 2\gamma)\Gamma ( \alpha +\beta +\frac{1}{2})\Gamma ( \alpha +\gamma +\frac{1}{2})\Gamma ( \beta +\gamma +\frac{1}{2})}{\Gamma ( \alpha)\Gamma ( \beta)\Gamma ( \gamma){{\Gamma }^{3}}( \alpha +\beta +\gamma +\frac{1}{2})} \\
 & \quad\times \sum_{i,j,k=0}^{\infty } \frac{{{( \alpha +\beta +\frac{1}{2})}_i}{{( \alpha +\gamma +\frac{1}{2})}_{j}}{{( \alpha)}_{i+j}}{{( \beta)}_{i+k}}{{( \gamma)}_{j+k}}}{{{( \alpha +\beta +\gamma +\frac{1}{2})}_{i+j}}{{( \alpha +\beta +\gamma +\frac{1}{2})}_{i+j+k}}i!j!k!}. \\
\end{aligned}
\end{equation}
It is easy to show that
\begin{equation}\label{e4.9}
\begin{aligned}
  & \sum_{i,j,k=0}^{\infty } \frac{{{( \alpha +\beta
+\frac{1}{2})}_i}{{( \alpha +\gamma +\frac{1}{2})}_{j}}{{( \alpha)}_{i+j}}
{{( \beta)}_{i+k}}{{( \gamma)}_{j+k}}}{{{( \alpha +\beta +\gamma
+\frac{1}{2})}_{i+j}}{{( \alpha +\beta +\gamma +\frac{1}{2})}_{i+j+k}}i!j!k!} \\
 & =\frac{\Gamma ( \frac{1}{2}){{\Gamma }^2}( \alpha +\beta +\gamma
+\frac{1}{2})}{\Gamma ( \alpha +\beta +\frac{1}{2})\Gamma ( \alpha +\gamma
+\frac{1}{2})\Gamma ( \beta +\gamma +\frac{1}{2})}.
\end{aligned}
\end{equation}
Thus, from \eqref{e4.4} we deduce
\begin{equation}\label{e4.10}
\begin{aligned}
\lim_{r\to 0} q_1^{*}( x,y,t;{{x}_0},{{y}_0},{{t}_0})
=\frac{\sqrt{\pi }\Gamma ( 2\alpha)\Gamma ( 2\beta)\Gamma ( 2\gamma)}
{\Gamma ( \alpha)\Gamma ( \beta)\Gamma ( \gamma)\Gamma ( \alpha +\beta
+\gamma +\frac{1}{2})}.
\end{aligned}
\end{equation}
From here considering \eqref{e4.10}, from \eqref{e4.3} at $ r\to 0 $
we have the estimate
\begin{equation}\label{e4.11}
| {{q}_1}( x,y,t;{{x}_0},{{y}_0},{{t}_0}) |\le \frac{{{c}_0}}{r},
\end{equation}
where
$$
{{c}_0}=\frac{{{k}_1}{{2}^{2\alpha +2\beta +2\gamma -1}}
\Gamma ( \alpha +\beta +\gamma)\Gamma ( 2\alpha)\Gamma
( 2\beta)\Gamma ( 2\gamma)}{\Gamma ( \alpha)\Gamma
( \beta)\Gamma ( \gamma)\Gamma ( 2\alpha +2\beta
+2\gamma){{( r_1^2)}^{\alpha }}{{( r_2^2)}^{\beta }}{{( r_3^2)}^{\gamma }}}.
$$
Estimate \eqref{e4.11} states that function
$ {{q}_1}( x,y,t;{{x}_0},{{y}_0},{{t}_0}) $ at  $ r\to 0 $ tends to
infinity of the order $ 1/r$. Similarly one can prove that every function
$ {{q}_i}( x,y,t;{{x}_0},{{y}_0},{{t}_0})$, $i=2,3,\dots,8 $ has singularity
$ 1/r $ as $ r\to 0$.
\end{proof}
\begin{theorem}\label{thm2}
If $\alpha ,\beta ,\gamma >0$, then the found particular solutions have the
following properties
\begin{gather*}
{{{x}^{2\alpha }}\frac{\partial }{\partial x}{{q}_1} \big|}_{x=0}=0,\quad
{{{y}^{2\beta }}\frac{\partial }{\partial y}{{q}_1} \big|}_{y=0}=0,\quad
{{{t}^{2\gamma }}\frac{\partial }{\partial t}{{q}_1}\big|}_{t=0}=0,
\\
{{{{q}_2} \big|}_{x=0}}=0,\quad
{{{{y}^{2\beta }}\frac{\partial }{\partial y}{{q}_2} \big|}_{y=0}}=0,\quad
{{{{t}^{2\gamma }}\frac{\partial }{\partial t}{{q}_2} \big|}_{t=0}}=0,
\\
{{{{x}^{2\alpha }}\frac{\partial }{\partial x}{{q}_3} \big|}_{x=0}}=0,\quad
{{{{q}_3} \big|}_{y=0}}=0,\quad
{{{{t}^{2\gamma }}\frac{\partial }{\partial t}{{q}_3} \big|}_{t=0}}=0,
\\
{{{{x}^{2\alpha }}\frac{\partial }{\partial x}{{q}_4} \big|}_{x=0}}=0,\quad
{{{{y}^{2\beta }}\frac{\partial }{\partial y}{{q}_4} \big|}_{y=0}}=0,\quad
{{{{q}_4} \big|}_{t=0}}=0,
\\
{{{{q}_5} \big|}_{x=0}}=0,\quad
{{{{q}_5} \big|}_{y=0}}=0,\quad
{{{{t}^{2\gamma }}\frac{\partial }{\partial t}{{q}_5} \big|}_{t=0}}=0,
\\
{{{{q}_6} \big|}_{x=0}}=0,\quad
{{{{y}^{2\beta }}\frac{\partial }{\partial y}{{q}_6} \big|}_{y=0}}=0,\quad
{{{{q}_6} \big|}_{t=0}}=0,
\\
{{{{x}^{2\alpha }}\frac{\partial }{\partial x}{{q}_7} \big|}_{x=0}}=0,\quad
{{{{q}_7} \big|}_{y=0}}=0,\quad
{{{{q}_7} \big|}_{t=0}}=0,
\\
{{{{q}_8} \big|}_{x=0}}=0,\quad
{{{{q}_8} \big|}_{y=0}}=0,\quad
{{{{q}_8} \big|}_{t=0}}=0.
\end{gather*}
\end{theorem}

Proofs of the above equalities are based on elementary calculations.
These properties could be used in studying various boundary problems
for the equation \eqref{e1.1}.

\subsection*{Acknowledgements}
We are grateful to the Professor H. M. Srivastava for his suggestion
to consider this problem.

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\end{document}