A new integral identity for potential polygonal domain problems described by parametric linear functions

Abstract

The paper presents a modification of the classical integral identity for two-dimensional potential boundary value problems with a linear segments boundary. The modification consists of describing integral contours in the integral identity by means of parametric linear functions. As a result of the modification, it was possible to obtain a new integral identity, which included the geometry of the boundary in its subinterval functions. The identity can be used for finding solutions in polygonal domains under the condition that the solutions previously obtained on the boundary are arrived at by the new parametric integral equation system. The proposed method makes it possible to obtain solutions of domain problems with no need for the discretization of the boundary geometry. The effectiveness of the method is illustrated by three testing examples.