Content-Type: multipart/mixed; boundary="-------------0511100932730"
This is a multi-part message in MIME format.
---------------0511100932730
Content-Type: text/plain; name="05-385.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="05-385.comments"
8 pages
---------------0511100932730
Content-Type: text/plain; name="05-385.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="05-385.keywords"
boundary value problem ,odd order partial equation,Fredholm type.
---------------0511100932730
Content-Type: application/x-tex; name="ijmest01.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="ijmest01.tex"
\documentclass[11pt,a4paper,reqno]{amsart}
\usepackage{amsfonts}
\usepackage{graphicx} % graphic package for postscript figures
\usepackage{graphics}
\usepackage{epsfig}
\input{psfig.sty}
\usepackage{anysize}
\marginsize{2.5cm}{2.5cm}{2.5cm}{2.5cm}
%========================================================%
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\numberwithin{equation}{section}
\newcommand{\abs}[1]{\lvert#1\rvert}
\newcommand{\norm}[1]{\|#1\|}
\newcommand{\Norm}[1]{\Big\|#1\Big\|}
\newcommand{\bta}[1][]{\beta_{#1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\norm#1{{\Vert #1 \Vert}}
\def\norat#1#2{|\hskip -0.1em |\hskip -0.1em | #1
|\hskip -0.1em |\hskip -0.1em |_{#2}}
\def\absol#1{{\vert #1 \vert}}
\def\abs#1{{\vert #1 \vert}}
\def\inner#1#2{{(#1,#2)}}
\def\IR {\text{\bf R}}
\def\Th {{\Cal T_h}}
\def\Ltwoomega{{L_2(\Omega)}}
\def\Ltwo{{L_2}}
\def\Honeomega{{H^1(\Omega)}}
\def\Hone{{H^1}}
\def\Honeoomega{{H^1_0(\Omega)}}
\def\Honeo{{H^1_0}}
\def\Htwoomega{{H^2(\Omega)}}
\def\Htwo{{H^2}}
\def\({\left(}
\def\){\right)}
\def\vh{{v_h}}
\def\uh{{u_h}}
\def\uht{{u_{h,t}}}
\def\uoh{{u_{0h}}}
\def\Sh{{S_h}}
\def\Linftyomega{{L_\infty(\Omega)}}
\def\Linfty{{L_\infty}}
\def\Lpomega{{L_p(\Omega)}}
\def\Lp{{L_p}}
\def\Lfouromega{{L_4(\Omega)}}
\def\Lfour{{L_4}}
\def\Wspomega{{W^s_p(\Omega)}}
\def\Wsp{{W^s_p}}
\def\qtext#1{\quad\text{#1}}
\def\D{{\text {\bf D}}}
\def\x{\bold x}
\def\n{\mathbf n}
\def\tripnorm#1{|\hskip -0.1em |\hskip -0.1em | #1
|\hskip -0.1em |\hskip -0.1em |}
\def\Ch {{\Cal C_h}}
\def\Vh {{\Cal V_h}}
\def\Wh {{\Cal W_h}}
\def\xx{{x_{\perp}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\bigskip
\textbf{On a boundary value problem for composite type third order}
\begin{center}
\textbf{equations with non-local boundary conditions}
\end{center}
\bigskip
\begin{center}
\textbf{Nehan Aliyev and Akhmedali Aliyev}
\end{center}
\bigskip
\begin{center}
\textbf{Institute of Applied Mathematics, Baku State University,}
\end{center}
\begin{center}
\textbf{Z.Khalilov-23, Az1148, Baku, Azerbaijan}
\end{center}
\bigskip
\subsubsection*{N.Aliyev}
\textit{104 S.Rahimov, Baku AZ1009, Azerbaijan}
\textit{Tel: (994 12) 955228. Email: .}\textit{\underline {jeff@azdata.net}}
\bigskip
\subsubsection*{A.Aliyev}
\textit{47Ataturk, apt.6, Baku AZ1069, Azerbaijan}
\textit{Tel: (994 12) 625142. Email: .}\textit{\underline
{ahmadm19}}\underline {@}\textit{\underline {yahoo.com}}
\bigskip
The paper is devoted to the investigation of solution of a boundary value
problem of the odd order partial equation where the boundary of the
considered domain holds boundary conditions of this problem.
The objective is to clarify the linear non-local boundary conditions with
which the boundary value problem is of Fredholm type. The investigation of
the problem is based on fundamental solution [12] of the considered
equation. The necessary conditions similar to the ones in [5]-[10] are
obtained. After regularizing these necessary conditions, and joining them
with given boundary conditions, the condition is found for the given problem
to belong to Fredholm type. A similar problem for Schrodinger equation and
for parabolic equation of first order is considered in [5,7,8].
The stated problem is devoted to the investigation of solution for a
boundary value problem of the odd order partial equation when all the
boundary of the considered domain is a support of boundary conditions of
this problem.
It is known that boundary value problems may have no solutions owing to
equations, or the considered domain, or boundary conditions. The problem on
the existence of solutions for boundary value problems owing for to boundary
conditions were was considered in our investigations [5]-[10]. The study of
these problems led us to necessary conditions related to which are connected
only with the domain where the boundary value problem is studied, and with
to the considered equation.
Is should be noted that these necessary conditions don't depend on boundary
conditions. They dictate determine us the kind of boundary conditions of a
concrete specific problem. A.A. Dezin [1] obtained these conditions for an
ordinary linear differential equation. For Laplace equations in bounded
domain they were obtained by A.V. Bitzadze [2]. and they don't yield to
generalization.
A.V. BitzadzeHe named then the conditions that could not be generalized as
``necessary and sufficient'' conditions.
As it follows from A.N. Tikhonov [3] and M.M. Lavrentiev's [4]
inveatigations investigations, mathematically ill-ill-posedposed
problems may occur more and morefrequently. Then there arises a
question: how should be the statement of the is the boundary value
problem stated for at random chosenan arbitrary equation? The
scheme of derivation of the abovementioned necessary conditions
answers to the this question.
These necessary conditions lead us to non-local boundary condition
(sometimes boundary conditions may contain even global addendsaddends), in
which the number of boundary conditions coincide with the highest order of
derivative on spatial variable contained in the equations of the considered
problem (this regularity was observed for in ordinary linear differential
equations). Many boundary value problems, for instance [5]-[10], were
investigated by this method, e.g. [5]-[10].
These problems were stated for the first and second order equations. A
boundary value problem for two-dimensional first order mixed type equation
was considered in [5]. A problem for the Laplace equation with non-local and
global addends addends in boundary conditions was stated in [6]. In [7] a
mixed problem for a Schrodinger equation with non-local boundary conditions
was investigated. In [8] a mixed problem was considered for a parabolic
equation whose boundary conditions contain both non-local and global
addendsaddends. Here the results on the whole of the mixed problem is are
stated briefly. In [9] the Cauchy problem (on the whole of space) was
considered for Navier-Stokes equations system. Paper [10] is mainly devoted
to the Fredholm boundary value problem obtained by means of Laplace
transformation for a parabolic equation with non-local and global boundary
conditions. The Nnecessary conditions were investigated in detail.
Thus, consider the following problem:
\begin{equation}
\label{eq1}
\begin{array}{l}
\frac{{\partial ^{3}u\left( {x} \right)}}{{\partial x_{2}^{3}} } +
\frac{{\partial ^{3}u\left( {x} \right)}}{{\partial x_{1}^{2}
\partial x_{2} }} =
0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \in D
\subset R^{2}
\end{array}
\end{equation}
\begin{equation}
\label{eq2}
\begin{array}{l}
\left. {\frac{{\partial ^k u(x)}}{{\partial x_1^k }}} \right|_{x_2
= \gamma _1 (x_1 )} = \left. {\frac{{\partial ^k u(x)}}{{\partial
x_2^k }}} \right|_{x_2 = \gamma _2 (x_1 )} + \varphi _k (x_1
){\rm }k = 0,1,2{\rm }x_1 \in [a,b]
\end{array}
\end{equation}
\noindent where $\gamma _{k} \left( {x_{1}} \right)\,\,\left( {k
= 1,2} \right) - $ are the equations of open lines $\Gamma _{k}
\left( {\Gamma _{1} \cup \Gamma _{2} = \Gamma} \right)$ obtained
from the boundary $\Gamma$ of the domain $D,$ by means of
orthogonal transformation of the domain on the axis $x_{1} , \quad
\left[ {a_{1} ,b_{1}} \right] = οπ_{x_{1}} \Gamma _{1} =
οπ_{x_{1}} \Gamma _{2} $ while $\gamma _{1} \left( {x_{1}}
\right) < \gamma _{2} \left( {x_{1}} \right),\,\,\,x_{1} \in
\left( {a_{1} ,b_{1}} \right).$
Proceeding from Fourier transformation [12], [13] we obtain for equation (\ref{eq1})
a fundamental solution in the form of integral on the plane, which contains
supersingularity. Regularizing it by means of bilateral Hermander stairs
[11] and performing some calculations, we getthe following is obtained:
\begin{equation}
\label{eq3} U(x - \xi ) = \frac{{x_2 - \xi _2 }}{{2\pi }}\left[
{\ln \sqrt {\left| {x_1 - \xi _1 } \right|^2 + (x_2 - \xi _2
)^2 } - 1} \right] + \frac{{\left| {x_1 - \xi _1 }
\right|}}{{2\pi }}arctg\frac{{x_2 - \xi _2 }}{{\left| {x_1 - \xi
_1 } \right|}}
\end{equation}
Then, by applying the method of papers [5]-[10] to solve equation
(\ref{eq1}) and its derivative up to the second order inclusively
we get necessary conditions both for $x_{2} = \gamma _{1} \left(
{x_{1}} \right)$ and $x_{2} = \gamma _{2} \left( {x_{1}}
\right)$. Note that only six from of these twelve expressions
(i.e. necessary conditions connected with related to the second
order of derivative) contain singular integrals.
To reduce these singular addends addends, from fundamental
solution (\ref{eq3}) and allowing for
\begin{equation}
\left. {\frac{{\partial ^2 U}}{{\partial x_2^2 }}}
\right|_{\scriptstyle x_2 = \gamma _p (x_1 ) \hfill \atop
\scriptstyle \xi _2 = \gamma _p (\xi _1 ) \hfill} = \frac{1}{{2\pi }}
\cdot \frac{{\gamma '_p (\sigma _p (x_1 ,\xi _1 ))}}{{(x_1 - \xi _1 )
\left( {1 + \gamma '_p^2 (\tau _p )} \right)}},
\end{equation}
\begin{equation}
\label{eq5}
\begin{array}{l}
\left. {\frac{{\partial ^2 U}}{{\partial x_1^{} \partial x_2 }}}
\right|_{\scriptstyle x_2 = \gamma _p (x_1 ) \hfill \atop
\scriptstyle \xi _2 = \gamma _p (\xi _1 ) \hfill} = \frac{1}{{2\pi }}\frac{1}
{{(x_1 - \xi _1 )\left( {1 + \gamma '_p^2 (\tau _p )} \right)}},
\end{array}
\end{equation}
\begin{equation}
\label{eq6}
\begin{array}{l}
\left. {\frac{{\partial ^2 U}}{{\partial x_1^2 }}}
\right|_{\scriptstyle x_2 = \gamma _p (x_1 ) \hfill \atop
\scriptstyle \xi _2 = \gamma _P (\xi _1 ) \hfill} = - \frac{1}{{2\pi }}
\cdot \frac{{\gamma '_p (\tau _p )}}{{(x_1 - \xi _1 )\left( {1 + \gamma '_p^2 (\tau _p )} \right)}},
\end{array}
\end{equation}
\begin{equation}
\label{eq7}
\begin{array}{l}
\left. {\frac{{\partial ^2 U}}{{\partial x_1^2 }}}
\right|_{\scriptstyle x_2 = \gamma _p (x_1 ) \hfill \atop
\scriptstyle \xi _2 = \gamma _q (\xi _1 ) \hfill} =
\delta (x_1 - \xi _1 )e(\gamma _p (x_1 ) - \gamma _q (\xi _1 )) - \frac{1}{{2\pi }}
\cdot \frac{{\gamma '_p (x_1 ) - \gamma _q (\xi _1 )}}{{(x_1 - \xi _1 )^2 +
\left( {\gamma _p^{} (x_1 ) - \gamma _q (\xi _1 )} \right^2 }},
\end{array}
\end{equation}
\noindent
havethe following is derived:
\begin{equation}
\label{eq8}
\begin{array}{l}
\left. {\frac{{\partial ^{2}u\left( {\xi} \right)}}{{\partial \xi _{1}^{2}
}}} \right|_{\xi _{2} = \gamma _{1} \left( {\xi _{1}} \right)} - \left.
{\frac{{\partial ^{2}u\left( {\xi} \right)}}{{\partial \xi _{1}^{2}} }}
\right|_{\xi _{2} = \gamma _{2} \left( {\xi _{1}} \right)} - \left.
{\frac{{\partial ^{2}u\left( {\xi} \right)}}{{\partial \xi _{2}^{2}} }}
\right|_{\xi _{2} = \gamma _{2} \left( {\xi _{1}} \right)} = \\
= - \left. {\frac{{1}}{{\pi} }\int\limits_{a_{1}} ^{b_{1}}
{\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{1} \partial x_{2}
}}}} \right|_{x_{2} = \gamma _{1} \left( {x_{1}} \right)} \frac{{dx_{1}
}}{{x_{1} - \xi _{1}} } + ... \\
\end{array}
\end{equation}
\begin{equation}
\label{eq9}
\begin{array}{l}
\left. {\frac{{\partial ^{2}u\left( {\xi} \right)}}{{\partial \xi _{1}^{2}
}}} \right|_{\xi _{2} = \gamma _{2} \left( {\xi _{1}} \right)} - \left.
{\frac{{\partial ^{2}u\left( {\xi} \right)}}{{\partial \xi _{1}^{2}} }}
\right|_{\xi _{2} = \gamma _{1} \left( {\xi _{1}} \right)} - \left.
{\frac{{\partial ^{2}u\left( {\xi} \right)}}{{\partial \xi _{2}^{2}} }}
\right|_{\xi _{2} = \gamma _{1} \left( {\xi _{1}} \right)} = \\
= - \left. {\frac{{1}}{{\pi} }\int\limits_{a_{1}} ^{b_{1}}
{\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{1} \partial x_{2}
}}}} \right|_{x_{2} = \gamma _{2} \left( {x_{1}} \right)} \frac{{dx_{1}
}}{{x_{1} - \xi _{1}} } + ... \\
\end{array}
\end{equation}
\begin{equation}
\label{eq10}
\left. {\frac{{\partial ^{2}u\left( {\xi} \right)}}{{\partial \xi _{2}^{2}
}}} \right|_{\xi _{2} = \gamma _{k} \left( {\xi _{1}} \right)} = \left.
{\frac{{\left( { - 1} \right)^{k - 1}}}{{\pi} }\int\limits_{a_{1}} ^{b_{1}}
{\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{1}^{} \partial x_{2}
}}}} \right|_{x_{2} = \gamma _{k} \left( {x_{1}} \right)} \frac{{dx_{1}
}}{{x_{1} - \xi _{1}} } + ...\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k = 1,2
\end{equation}
\begin{equation}
\left. {\frac{{\partial ^2 u(\xi )}}{{\partial \xi _1 \partial \xi
_2 }}} \right|_{\xi _2 = \gamma _k (\xi _1 )} = \left. {\frac{{(
- 1)^{k - 1} }}{\pi }\int\limits_{a_1 }^{b_1 } {\frac{{\partial ^2
u(x)}}{{\partial x_1 \partial x_2 }}} } \right|_{x_2 = \gamma _k
(x_1 )} \frac{{dx_1 }}{{x_1 - \xi _1 }} + ...k = 1,2
\end{equation}
\section*{Thus the following statement is established.}
Theorem 1. Let D be a bounded, convex in the direction $x_{2} ,$
plane domain with the Liapunov $\Gamma $-line boundary. Then
boundary values of each solution of equations (\ref{eq1})
determined on domain D satisfy necessary conditions, a part of
which has no singularities, and the remaining part are connected
withrelated to the boundary value of the second derivative and
contain singularity represented in the form (\ref{eq8})-(11).
There $\gamma _{k} \left( {x_{1}} \right)\,\,\,k =
1,2;\,\,\,\,x_{1} \in \left( {a_{1} ,b_{1} } \right)$ are the
equations of the part $\Gamma _{k} \,\,\,\left( {k = 1,2} \right)$
of the boundary $\Gamma$ of the domain D which are obtained by
orthogonaly projecting this domain on the axis $x_{1} $
\[
\gamma _{1} \left( {x_{1}} \right) < \gamma _{2} \left( {x{}_{1}}
\right)\,\,\,\,\,\,\,\,\,\,\,\,x_{1} \left( {a_{1} ,b_{1}} \right)
\]
\underline {Remark Comment 1}. Integrating (\ref{eq1}) with
respect to the variable $x_{2} $ from $\gamma _{k} \left( {x_{1}}
\right)$ to $x_{2}$ we have the following necessary conditions
\begin{equation}
\label{eq12}
\left. {\Delta u\left( {x} \right)} \right|_{x_{2} = \gamma _{1} \left(
{x_{1}} \right)} = \left. {\Delta u\left( {x} \right)} \right|_{x_{2} =
\gamma _{2} \left( {x_{1}} \right),\,\,\,\,\,\,\,\,\,\,} x_{1} \in \left[
{a_{1} ,b_{1}} \right]
\end{equation}
\noindent which are also the corollaries result of the
abovementioned conditions.
\underline {Remark Comment 2.} Taking into account the remark
comment made above it is easy to see that (\ref{eq8}) and
(\ref{eq9}) are reduced to necessary condition (\ref{eq10}).
Differentiating the given boundary conditions (\ref{eq2}) 2-k times with respect to
the variable $x_{1} $ and allowing for (\ref{eq12}), we havethe following is
derived:
\begin{equation}
\label{eq13}
\begin{array}{l}
\left. {\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{1}^{2}} }}
\right|_{x_{2} = \gamma _{1} \left( {x_{1}} \right)} = \left.
{\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{2}^{2}} }}
\right|_{x_{2} = \gamma _{2} \left( {x_{1}} \right)} + \varphi _{2} \left(
{x_{1}} \right) \\
\left. {\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{1}^{2}} }}
\right|_{x_{2} = \gamma _{2} \left( {x_{1}} \right)} = \left.
{\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{2}^{2}} }}
\right|_{x_{2} = \gamma _{1} \left( {x_{1}} \right)} + \varphi _{2} \left(
{x_{1}} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
\quad x_{1} \in \left[ {a_{1} ,b_{1}} \right] \\
\end{array}
\end{equation}
\noindent
and
\begin{equation}
\label{eq14}
\left\{ {\begin{array}{l}
{\left. {\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{2}^{2}} }}
\right|_{x_{2} = \gamma _{1} \left( {x_{1}} \right)} \left[ {1 - {\gamma
}'_{1}^{2} \left( {x_{1}} \right)} \right] - 2\left. {\frac{{\partial
^{2}u\left( {x} \right)}}{{\partial x_{1} \partial x_{2}} }} \right|_{x_{2}
= \gamma _{1} \left( {x_{1}} \right)} {\gamma} '_{1} \left( {x_{1}}
\right) -} \\
{ - \left. {\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{2}^{2}
}}} \right|_{x_{2} = \gamma _{2} \left( {x_{1}} \right)} \left[ {1 -
{\gamma} '_{2}^{2} \left( {x_{1}} \right)} \right] + 2\left.
{\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{1} \partial x_{2}
}}} \right|_{x_{2} = \gamma _{2} \left( {x_{1}} \right)} {\gamma} '_{2}
\left( {x_{1}} \right) = - {\varphi} ''_{0} \left( {x_{1}} \right) +} \\
{ + \left. {\frac{{\partial u\left( {x} \right)}}{{\partial x_{2}^{}} }}
\right|_{x_{2} = \gamma _{1} \left( {x_{1}} \right)} \cdot {\gamma} ''_{1}
\left( {x_{1}} \right) - \left. {\frac{{\partial u\left( {x}
\right)}}{{\partial x_{2}^{}} }} \right|_{x_{2} = \gamma _{2} \left( {x_{1}
} \right)} \cdot {\gamma} ''_{2} \left( {x_{1}} \right),\,\,\,} \\
{\left. {\frac{{\partial ^{2}u}}{{\partial x_{2}^{2}} }} \right|_{x_{2} =
\gamma _{2} \left( {x_{1}} \right)} \left[ {1 - {\gamma} '_{2}^{2} \left(
{x_{1}} \right)} \right] + \left. {\frac{{\partial ^{2}u\left( {x}
\right)}}{{\partial x_{1} \partial x_{2}} }} \right|_{x_{2} = \gamma _{1}
\left( {x_{1}} \right)} {\gamma} '_{1} \left( {x_{1}} \right) -} \\
{ - \left. {\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{1}
\partial x_{2}} }} \right|_{x_{2} = \gamma _{2} \left( {x_{1}} \right)} =
{\varphi} '_{1} \left( {x_{1}} \right) - \varphi _{2} \left( {x_{1}}
\right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x_{1}
\in \left[ {a_{1} ,b_{1}} \right]} \\
\end{array}} \right.
\end{equation}
\noindent in which (\ref{eq13}) determines $ \left.
{\frac{{\partial ^2 u(x)}}{{\partial x_1^2 }}} \right|_{x_2 =
\gamma _k (x_1 )} $ if $\left. {\frac{{\partial ^{2}u\left( {x}
\right)}}{{\partial x_{2}^{2}} }} \right|_{x_{2} = \gamma _{k}
\left( {x_{1}} \right)} $ are known, and (\ref{eq14}) gives
usproduces two relations between four unknowns
$\left. {\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{2}^{}} }}
\right|_{x_{2} = \gamma _{k} \left( {x_{1}} \right)} $ and $\left.
{\frac{{\partial ^{2}u\left( {x} \right)}}{{\partial x_{1}^{} \partial x_{2}
}}} \right|_{x_{2} = \gamma _{k} \left( {x_{1}} \right)}
\,\,\,\,\,\,\,\,\,\,\,\,\left( {k = 1,2} \right)$
One more pair of regular relations (regular) with respect to these
unknowns are obtained by the scheme method of papers [5]-[10]
allowing for (\ref{eq14}) proceeding from necessary conditions
(\ref{eq10}) and (11).
\begin{equation}
\label{eq15}
\begin{array}{l}
- \sum\limits_{k = 1}^{2} {2{\gamma} '_{k} \left( {\xi _{1}} \right)}
\left. {\frac{{\partial ^{2}u\left( {\xi} \right)}}{{\partial \xi _{2}^{2}
}}} \right|_{\xi _{2} = \gamma _{k} \left( {\xi _{1}} \right)} -
\sum\limits_{k = 1}^{2} {\left[ {1 - {\gamma} '_{k}^{2} \left( {\xi _{1}}
\right)} \right]} \left. {\frac{{\partial ^{2}u\left( {\xi}
\right)}}{{\partial \xi _{1}^{} \partial \xi _{2}} }} \right|_{\xi _{2} =
\gamma _{k} \left( {\xi _{1}} \right)} = \\
= \frac{{1}}{{\pi} }\int\limits_{a_{1}} ^{b_{1}} {\left[ { - {\varphi
}''_{0} \left( {x_{1}} \right) + \left. {\frac{{\partial u\left( {x}
\right)}}{{\partial x_{2}} }} \right|_{x_{2} = \gamma _{1} \left( {x_{1}}
\right)} {\gamma} ''_{1} \left( {x_{1}} \right) - \left. {\frac{{\partial
u\left( {x} \right)}}{{\partial x_{2}} }} \right|_{x_{2} = \gamma _{2}
\left( {x_{1}} \right)} {\gamma} ''_{2} \left( {x_{1}} \right)} \right]}
\frac{{dx_{1}} }{{x_{1} - \xi _{1}} } + ... \\
\end{array}
\end{equation}
\begin{equation}
\begin{array}{l}
\gamma '_1 (x_1 )\left. {\frac{{\partial ^2 u(\xi )}}{{\partial \xi _2^2 }}} \right|_{\xi _2 = \gamma _1 (\xi _1 )} + \left. {\frac{{\partial ^2 u(\xi )}}{{\partial \xi _2^2 }}} \right|_{\xi _2 = \gamma _2 (\xi _1 )} + [1 - \gamma '_2 (\xi _1 )]\left. {\frac{{\partial ^2 u(\xi )}}{{\partial \xi _1 \partial \xi _2^{} }}} \right|_{\xi _2 = \gamma _2 (\xi _1 )} = \\
= \frac{1}{\pi }\int\limits_{a_1 }^{b_1 } {\left[ {\varphi '_1 (x_1 ) - \varphi _2 (x_1 )} \right]} \frac{{dx_1 }}{{x_1 - \xi _1 }} + ... \\
\end{array}
\end{equation}
Taking into account that the necessary conditions obtained for
$\left. {\frac{{\partial u\left( {x} \right)}}{{\partial x_{2}} }}
\right|_{x_{2} = \gamma _{k} \left( {x_{1}} \right)} $ contain
under the integral sign the normal derivatives from $ \left.
{\frac{{\partial U(x - \xi )}}{{\partial x_2 }}}
\right|_{\scriptstyle x_2 = \gamma _k (x_1 ) \hfill \atop
\scriptstyle \xi _2 = \gamma _2 (\xi _1 ) \hfill}
$ which are regular for Lyapunov line [1], we havethe following
is stated:
Theorem 2. By the conditions of Theorem 1, if $\varphi _{k}
\left( {x_{1}} \right) \in C^{3 - k}\left( {a_{1} ,b_{1}}
\right)$ and $\varphi _{k}^{\left( {2 - k} \right)} \left( {a_{1}}
\right) = \varphi _{k}^{\left( {2 - k} \right)} \left( {b_{1}}
\right) = 0,\,\,\,\,k = \overline {0,2,} $ the relations
(\ref{eq15}) and (16) are regular.
Thus, we get Hence the following statement:
Theorem 3. Onder the conditions of the theorem 2 the condition of
Fredholm property of the boundary value problem (1)-(2) is
equvivalent to the fact the certain matrix has non-zero
determinant.
\bigskip
\subsection*{REFERENCES}
\bigskip
1. Dezin A.A. General problems of the theory of boundary value problems.
Moskow, Nauka, 19806, (Russian).
2. Bitzadze A.V. Boundary value problems for second order elliptic
equations. Moscow, Nauka, 1966 (russianRussian).
3. Tikhonov A.N. On the solution of ill-posed problems and regularization
method. DAN SSSR, 1963, 151, 3, 501-504 (russianRussian).
4. Lavrentiev M.M. On the some ill-inadequately posed problems of
mathematical physics. Novosibirsk, 1962. (russianRussian).
5. Nehan Aliyev, and Mohammad Jahanshahi. Sufficient conditions for
reduction of the BVP including a mixed PDE with non- local boundary
conditions to Fredholm integral equations. Int. J. Math. Educ. Sci.
Technol.IJMEST, vol 28, N3, 419-425.
6. Nihan Nehan Aliyev, and Mohammad Jahanshani. Solution of Poisson's
equation with global, local and non-local boundary conditions. Int. J. Math.
Educ. Sci. Technol.IJMEST, 2002, vol. 33, N2, 241-247.
7. G.Kavei and N.Aliyev An analytical method to the solution of the
time-dependent Schrodinger equation using half cylinder space system -I.
Bulletin of Pure and Applied Sciences. Vol. 16E (N2). 1997, p.253-263.
8. S.M. Hosseini and N.Aliyev. A mixed Parabolic with a non-local and global
linear conditions. J. Sci. I.R. Iran vol 11, N3, Summer 2000.
9. N.Aliyev, S.M. Hosseini Cauchy problem for the Navier-Stokes equation and
its reductions to a non-linear system of second kind Fredholm integral
equations. International Journal of Pure and Applied Mathematics, vol 3,
N.3, 2002, 317-324.
10. N.Aliyev S.M. Hosseini An analysis of a parabolic problem with a general
(non-local and global) supplementary linear conditions. Italian Journal of
pure and Applied Mathematics-N. 12-2002 (143-154).
11. Hormander L. Linear Partial Differential Operators, Sprenger-Verlag,
Berlin, Gettingen, Heldelberg,1963.
12. Vladimirov V.S. Equations of Mathematical Physics ``Nauka'', Moskow,
1971.
13. Petrovski I.G. Lectures on Partial Differential Equations, Inter.
Science POublisher, New York, 1954.
\end{document}
---------------0511100932730--