On volume integral equations

The focus of this paper is on the volume integral representations to be used in constructing integral equations for composite volume media. The major thrust of the paper is to identify where derivatives of a discontinuous function arise in the derivation of the volume representation. Three different derivation methods are presented, resulting in identical representation independent of the derivation method. These representations agree with some in the existing literature and disagree with others. When an electric field formulation is considered, the source of disagreement manifests itself only when magnetic materials are present. Likewise, for the dual situation, the inconsistency appears for a magnetic field formulation of dielectric materials. This paper identifies the sources of error in the incorrect representations and its major contribution is the rigorously correct derivation of the representations to be used in volume integral equations. We also present numerical results for an integral equation derived from our representation. The numerical results employ only the E-field as the unknown and the singularity is handled in a manner analogous to a standard numerical treatment of the electric field integral equation.