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\title{On the convergence of finite difference schemes for the heat equation with
concentrated capacity}
\author{Lubin G. Vulkov \\
%EndAName
Center of Applied Mathematics and Informatics\\
University of Rousse, 7017 Rousse, Bulgaria\\
e-mail:vulkov@ami.ru.acad.bg}
\date{}
\maketitle

\begin{abstract}
\end{abstract}

As a model problem we consider the initial boundary-value problem for the
heat equation

\[
\left( 1+C\delta \left( x-\xi \right) \right) \frac{\partial u}{\partial t}-%
\frac{\partial ^2u}{\partial x^2}=f\left( x,t\right) ,
\]
in the domain $Q_T=\Omega \times \left( 0,T\right) ,\quad \Omega =\left(
0,1\right) ,\quad 0<\xi <1;$

\[
u\left( 0,t\right) =u\left( 1,t\right) =0,\quad t\in \left[ 0,T\right]
;\quad u\left( x,0\right) =u_0\left( x\right) ,\quad x\in \Omega .
\]

Here $C=$constant$>0$ is the magnitude of the concentrated capacity, $\delta
\left( x-\xi \right) $ is the Dirac-delta (generalized) function. Stability
of weighted difference schemes in integral norms is studied. Convergence
rate estimates compatible with the smoothness of the generalized solution $%
u\in H^{2,1}\left( Q_T^{*}\right) \cap C\left( Q_T^{*}\right) ,\quad \Omega
^{*}=\Omega \backslash \xi ,\quad  Q_T^{*} =\Omega ^{*}\times
\left( 0,T\right) $ are obtained.

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