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NON-LOCAL PROBLEMS FOR A VIBRATING STRING

\vspace*{0.2cm}
{\it G. Avalishvili, D. Gordeziani}

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

\vspace*{0.2cm}
\par
Non-local problems naturally arise while modelling various phenomena and 
processes of physics, ecology, chemistry and other fields. The investigation 
of such a problems is caused not only by theoretical interests, but also by
practical necessity. These problems are of practical interest in solving  
mathematical problems of the mechanics of a solid body. They allow us to 
control the stress-strain state of the body.

First a certain class of non-local problems was formulated and studied in [1].
Further, in the works [2],[3], the problem stated in [1] was called  
Bitsadze-Samarskii problem and solution methods for such 
problems were suggested in the case of rather general elliptic equations. Later, in the
[4]-[8] interesting generalizations of Bitsadze-Samarskii
conditions were suggested.

We consider the non-local problems for the following 1-dimensional hyperbolic
equation
\begin{equation}
\displaystyle\frac{\partial^2 u}{\partial t^2}=
\displaystyle\frac{\partial^2 u}{\partial x^2}+c^2 u,
\;\;\;\;
0<x<l,
\;\;
0<t<T,
\end{equation}
with classical initial conditions 
\begin{equation}
\begin{array}{l}
u(x,0)=\varphi(x),
\\
u_t(x,0)=\psi(x),
\end{array}
\;\;\;\;
0\le x\le l,
\end{equation}
and non-local boundary conditions of the following type 
\begin{equation}
\begin{array}{l}
\alpha u(0, t)+\beta\displaystyle\frac{\partial u}{\partial x}(0, t)=
\sum\limits_{i=1}^{m}\alpha_i u(\xi_i, t)+f(t),
\\
\hspace*{85mm} 0\le t\le T,
\\
\gamma u(l, t)+\delta\displaystyle\frac{\partial u}{\partial x}(l, t)=
\sum\limits_{j=1}^{n}\beta_j u(\eta_j, t)+g(t),
\end{array}
\end{equation}
where $\alpha,\beta,\gamma,\delta,\alpha_i,\beta_j,$ are real numbers, 
such that $\alpha\beta\ne 0$, $\gamma\delta\ne 0$, $\xi_i,\eta_j(i=1,\ldots,m,\;\; j=1,\ldots,n)$
are prescribed points of the interval $(0,l)$. Note, that if $c\ne 0$, then we obtain
the non-local problem for telegraph equation. The following theorem is true.

{\bf Theorem 1.} {\it If $\varphi\in C ^2[0,l],\;\;\psi\in C^1[0,l],\;\;f,g\in C^2[0,T],$
$c$ is a real or imaginary number, then there exists unique solution $u(x,t)$
of the problem {\rm(1)-(3)}, which is twice continuously differentiable on
$[0,l]\times[0,T],$ satisfies equation {\rm(1)} and conditions {\rm(2), (3)}.}

As we see, non-local conditions (3) contain the linear combination of the 
values of function on the discrete layers. In contrast to this case, we can 
consider the problem with integral non-local conditions of the following type 
\begin{equation}
\begin{array}{l}
\alpha u(0, t)+\beta\displaystyle\frac{\partial u}{\partial x}(0, t)=
p\int\limits_{0}^{\xi}u(x,t)dx+f(t),
\\
\hspace*{85mm} 0\le t\le T,
\\
\gamma u(l, t)+\delta\displaystyle\frac{\partial u}{\partial x}(l, t)=
q\int\limits_{\eta}^{l}u(x,t)dx+g(t),
\end{array}
\end{equation}
where $\xi,\eta$ are given points of the interval $(0,l),\;\;f,g\in C^2[0,T].$
Here we have

{\bf Theorem 2.} {\it There exists a unique regular solution $u(x,t)$ of the problem 
{\rm(1), (2), (4)}.}

The proofs of the formulated theorems are based on the formula of general 
solution of classical initial-boundary value problem, in the case of string 
oscillation equation, and on the potential of special type, in the case of 
telegraph equation.

It must be pointed out, that increasing the number of space variables the 
direct definition of the solution becomes very difficult, but nevertheless under
the rather general conditions the uniqueness theorem is true.

Let us consider the bounded region $\Omega\subset R^n,n\ge 2$, where 
$\Gamma$ is the boundary of $\Omega.$ Let $\Omega_i\;(i=1,\ldots,m)$ be regions with 
boundaries $\Gamma_i$, such that each of the regions $\bar{\Omega}_i$ is placed
strictly inside $\bar{\Omega}$. Additionaly, $\Gamma_i$ represents a 
diffeomorphical image of $\Gamma$, i.e. $x^{(i)}=I_i(x),x^{(i)}\in\Gamma_i,$
$x\in\Gamma,I_i(.)-$ diffeomorphism, $\Gamma,\;\Gamma_i$ are Lyapunov surfaces
$(i=1,\ldots,m).$

Let $L$ be a uniformly elliptic operator of the following type
$$
L\equiv\sum\limits_{i=1}^{n}a_{ik}(x)\frac{\partial^2}{\partial x_i \partial x_k}+
\sum\limits_{i=1}^{n}b_i(x)\frac{\partial}{\partial x_i}+c(x),
$$
where $a_{ik},b_i,c\;(c\le 0)$ are prescribed functions.

We consider a non-local problem for the hyperbolic equation
\begin{equation}
\frac{\partial^2 u}{\partial t^2}-Lu=f(x,t),\;\;\;\;\;\;\;\;\;
(x,t)\in Q_T=\Omega\times(0,T),
\end{equation}
with initial conditions

\begin{equation}
\begin{array}{l}
u(x,0)=u_0(x),
\\
u_t(x,0)=u_1(x),
\end{array}
\;\;\;\;x\in\Omega,
\end{equation}
and non-local boundary condition  
\begin{equation}
u(x,t)=\sum\limits_{i=1}^{m}q_i(x,t)u(x^{(i)},t)+\varphi(x,t),
\;\;\;\;(x,t)\in S_T=\Gamma\times[0,T],
\end{equation}
where $f,\varphi,u_0,u_1$ are sufficiently smooth functions, such that 
classical Cauchy-Dirichlet problem for equation (5) with initial conditions 
(6) and boundary condition $u(x,t)\bigg|_{S_T}=\varphi(x,t)$ has a unique solution. Using
the properties of solution of hyperbolic equation on characteristic cones we 
can prove 
 
{\bf Theorem 3.} {\it Non-local problem {\rm(5)-(7)} can not possess more than one regular
solution, which is twice continuously differentiable in $\displaystyle\bar{Q_T}.$}

\vspace*{0.5cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.3cm}
\end{center}
[1] Bitsadze A.V., Samarskii A.A., On the Simplest Generalizations of Linear 
Elliptic Problems. (Russian), Dokl. Akad. Nauk SSSR, 185, 1969, 
No. 4, 739-740.

[2] Gordeziani D.G., On One Method of Resolution of Bitsadze-Samarskii 
Boundary Value Problem. (Russian), Abstracts of Reports of Inst. Appl. 
Math. Tbilis. State Univ., 2, 1970, 38-40.

[3] Gordeziani D.G., Djioev T.Z., On Solvability of One Boundary Value 
Problem for Non-Linear Elliptic Equation. (Russian), Bull. Akad. Sci.
Georgian SSR, 68, 1972, No. 2, 289-292. 

[4] Gordeziani D., Gordeziani N., On the Investigation of One Problem in the 
Linear Theory of the Elastic Mixtures., Seminar of I. Vekua Inst. 
of Appl. Math. Reports, v. 22, 1993, 115-122.

[5] Gordeziani D., Gordeziani N., Avalishvili G., On One Class of
Non-Local Problems for Partial Differential Equations.
Reports of Enlarged Session of I.Vekua Inst. of Appl. Math., Tbilisi State Univ., v.10,
1995, No. 3, 14-16. 

[6] Gordeziani D., Gordeziani N., Avalishvili G., On the Investigation and
Resolution of Non-Local Boundary and Initial-Boundary Value Problems.
Reports of Enlarged Session of I.Vekua Inst. of Appl. Math., Tbilisi State Univ., v.12,
1997, No. 3. 

[7] Gordeziani D., Gordeziani N., Avalishvili G., Non-Local Boundary 
Value Problems in Mathematical Physics.
Abstracts of Intern. Symposium Dedicatied to the $90^{th}$ Birthday 
Anniversary of Academician I. Vekua DEMPh, 1997, 127-128. 

[8] Gordeziani D., Gordeziani N., Avalishvili G., Non-Local Boundary 
Value Problems for Some Partial Differential Equations.
Bull. Georgian Acad. Sci., (to appear).
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