The Integral Equation Method

Abstract

The (boundary) integral equation method denotes the transformation of partial differential equations with d spatial variables into an integral equation over a (d-1)-dimensional surface. In §7.4, the method has already been introduced for the two-dimensional Laplace equation. There, results concerning the singular Cauchy kernel were used to find integral equation formulations for the Laplace equation (7.4.1a). According to this approach, the Laplace equation with its close connection to holomorphic functions seems to represent a very particular case (cf. Remark 1.1) and the question remains whether other equations can be treated. The integral equation method or boundary integral method starts from a differential equation Lu = 0 with suitable boundary conditions and looks for an equivalent formulation as integral equation. The numerical treatment of the integral equation, which thus arises, is the subject of the next chapter (§9) entitled «boundary element method».