IJPAM: Volume 27, No. 1 (2006)

ON A NUMERICAL SOLUTION TO
THE DIRICHLET PROBLEM

Alexander Kozhevnikov$^1$, Olga Lepsky$^2$
$^1$Department of Mathematics
University of Haifa
Haifa, 31905, ISRAEL
e-mail: kogevn@math.haifa.ac.il
$^2$Natural Sciences Program
Lesley University
34 Wendell Str., Cambridge, MA 02138-2790, USA
e-mail: olga@lesley.edu


Abstract.It is proved in this paper that any solution to the Dirichlet boundary value problem for the homogeneous equation $\left(
I-\Delta \right) u=0$ in a bounded domain $\Omega \subset $ ${\Bbb R}^{n}$ $%
\;$($I$ and $\Delta $ being the identity operator and the Laplacian, respectively) is represented in the form of the volume potential with a density supported in an arbitrarily thin boundary layer exterior to $%
\partial \Omega $. As a result, the Dirichlet problem is reduced to an integral equation with an unknown density defined in the thin boundary layer. An approximate solution to the latter integral equation generates a rather simple new numerical algorithm solving the $3D$ Dirichlet problems. This algorithm is different from the finite difference, finite element, or boundary element methods. It can be called a boundary layer element method. Examples of its accuracy are presented. All the results are obtained not just for the operator $I-\Delta $ but also for an arbitrary elliptic differential operator in ${\Bbb R}^{n}$ of an even order with constant coefficients, as well as for boundary value problems in interior and exterior domains of ${\Bbb R}^{n}$.

Received: March 3, 2006

AMS Subject Classification: 35J40, 35J05, 65N99

Key Words and Phrases: Dirichlet boundary-value problem, numerical solution

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 27
Issue: 1