# IJPAM: Volume 27, No. 1 (2006)

**ON A NUMERICAL SOLUTION TO**

THE DIRICHLET PROBLEM

THE DIRICHLET PROBLEM

Department of Mathematics

University of Haifa

Haifa, 31905, ISRAEL

e-mail: kogevn@math.haifa.ac.il

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
in a bounded domain
( and 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
. 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 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 but also for an arbitrary elliptic
differential operator in of an even order with constant
coefficients, as well as for boundary value problems in interior and
exterior domains of .

**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