Formal expressions for the solution of Dirichlet and Neumann problems

Abstract

Based on the relationship between the potential and its normal derivative on the boundary of the region of interest, the solution of the Dirichlet and the Neumann boundary value problems relative to the Laplacian is written explicitly in an operator form. The numerical computation of the potential for regions of arbitrary shape is performed by converting the two boundary operators involved into square matrices, which only depend on the boundary shape. The formulas presented are of interest since they allow a direct calculation of the potential in a given region under any Dirichlet or Neumann boundary conditions. These solution formulas represent for regions of arbitrary geometry what the direct integration Green formulas represent for some canonical regions for which the problem-specific Green functions can be constructed analytically.