Short Communications
On an application of the connection between the Brownian motion and the Dirichlet problem
R. V. Ambartzumian Armenian Academy of Sciences, Calculating Center
Abstract:
It is known that in the domain
$G$ with a piecewise smooth boundary
$\Gamma$ the solution
$f(P)$ of the Dirichlet problem with continuous boundary values
$f(S)$,
$S\in\Gamma$ , can be represented in the form
$$
f(P)=\int_\Gamma u(P,S)f(S)\,dS
$$
where
$u(P,S)$ is the probability density for a brownian particle to be absorbed at a point
$S\in\Gamma$ starting from a point
$P$ of the domain
$G$ with the absorbing boundary
$\Gamma$.
It is shown that the construction of the function
$u_0(P,S)$ for a domain
$G_0$ which splits into two non-intersecting domains
$G_1$ and
$G_2$ with common boundary points and with known functions
$u_1(P,S)$ and
$u_2(P,S)$ is reduced to solving some Fredholm integral equation of the second kind.
The uniqueness of the solution of this integral equation is proved.
Received: 02.11.1963
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