Elliptic problems

Abstract

An effective method for solving boundary value problems for the Laplace and Helmoltz equations (in domains possessing a definite symmetry) is the method of separation of variables. The general idea of this method is to find a set of solutions of the homogeneous partial differential equation in question that satisfy certain boundary conditions. These solutions then serve as “atoms”. from which, based on the linear superposition principle, one constructs the “general” solution, Since each of these “atoms” is a solution of the corresponding homogeneous equation, their linear combination is also a solution of the same equation. The solution of our problem is given by a series $\sum\nolimits_{n = 1}^\infty {{c_n}{u_n}\left( x \right)}$ (where u _n (x) are the atom solutions, x = (x ₁,…,x _N ) is the current point of the domain of space under consideration, and c _n are arbitrary constants). It remains to find constants c _n such that the boundary conditions are satisfied.