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\begin{center}
STATIONARY HEAT DISTRIBUTION IN THE NONHOMOGENEOUS
TRANSTROPIC PARABOLICAL CYLINDER

\vspace*{0.2cm}
{\it I. Khomasuridze}

\vspace*{0.2cm}
{\it I. Vekua Institute of Applied Mathematics}
\end{center}

\vspace*{0.2cm}
\par In the cylinder-parabolical coordinate system $\ro,$ $\al,$ 
$z,$ where $H_\ro=H_\al=$ $=\sqrt{\ro^2+\al^2},$ $H_z=1$ are Lam$\'{e}$ coeficients,
the following differential equation
\begin{equation}
\la_1\fr{\de^2 T}{\de\ro^2}+\fr{\de}{\de\al}\oc \la_2\fr{\de T}{\de\al}\cc+\oc \ro^2+\al^2\cc\la_1\fr{\de^2 T}{\de z^2}=0
\end{equation}
is considered in the domain $\Omega=\of -\infty<\ro<\infty; 0<\al_0<\al<\al_1; z_0<z<z_1\cf.$
\par Equation $(1)$ discribes the stationary heat distribution in transtropic body,
nonhomogeneous along $\al.$
In $(1),$ $\la_2=\la_2\oc\al\cc$ is heat conductivity along tangential to the coordinate line $\al,$ 
while $\la_1=\la_1\oc\al\cc$ is heat conductivity along tangential to the coordinate 
line $\ro$ and coordinate line $z.$
\par Boundary conditions of the problem have the form
\begin{equation}
\begin{array}{lll}
\text{For}  & z=z_0 & T=0,\\
            & z=z_1 & T=0.
\end{array}
\end{equation}
\begin{equation}
\begin{array}{lll}
\text{For}  & \al=\al_0 & T=F_0\oc\ro,z\cc,\\
            & \al=\al_1 & T=F_1\oc\ro,z\cc.
\end{array}
\end{equation}
\begin{equation}
\begin{array}{llll}

\text{For}  & \ro\rightarrow\infty  & T\rightarrow 0 &\text{exponential convergence,}\\
            & \ro\rightarrow-\infty & T\rightarrow 0 &\text{exponential convergence.}

\end{array}
\end{equation}
\par We assume that agreement
conditions on the edges of the curvilinear coordinate parallelepiped hold.
\par Using the method of separation of variables we can write the solution 
of problem $(1),$ $(2),$ $(3),$ $(4)$ in the form: 
\begin{equation}
T=\sum\limits^\infty\limits_{n=0}\sum\limits^\infty\limits_{k=1}
\oc e^{-\fr{\pi k\ro^2}{2z_1}} H_n\oc \sqrt{\fr{\pi k}{z_1}}\ro\cc\cc
sin\oc\fr{\pi k}{z_1}z\cc A
\end{equation}
$H_n$ Heremit's number $n$ polynomial, and $A$ is the general solution
of the following differential equation:
\begin{equation}
A''+\fr{\la_2'}{\la_2}A'-\oc \mu+\theta^2\al^2\cc\fr{\la_1}{\la_2}A=0
\end{equation}
where $\theta=\fr{\pi k}{z_1},$ $\mu=\theta\oc 2n+1\cc=\fr{\pi k}{z_1}\oc 2n+1\cc.$
\par General solution of $(6)$ in the case under
consideration 
($\la_2=e^{a\al},$ $\la_1=g^2e^{a\al};$ $a$ and $g$ are constants) 
can be presented by means
of elementary functions. Thus, the general solution of $(5)$ is presented as a
product of trigonometric functions, Hermit polinomials and general solution
of $(6).$
\par The investigated problem presents a mathematical model of a real object.
\par Convergence of the series and uniqueness of the solution are proved.
\end{document}