Dissipativity Constraints in Zener-Type Time Dispersive Electromagnetic Materials of the Fractional Type

Abstract

Thermodynamic constraints must be satisfied for the parameters of a constitutive relation, particularly for a model describing an electromagnetic (or any other) material with the intention of giving that model a physical meaning. We present sufficient conditions for the parameters of the constitutive relation of an electromagnetic Zener-type fractional 2D and 3D anisotropic model so that a weak form of the thermodynamic (entropy) inequality is satisfied. Moreover, for such models, we analyze the corresponding thermodynamic constraints for field reconstruction and regularity in the 2D anisotropic case. This is carried out by the use of the matrix version of the Bochner theorem in the most general form, including generalized functions as elements of a matrix, which appear in that theorem. The given numerical results confirm the calculus presented in the paper.