Fundamental properties of solutions to fractional-order Maxwell's equations

Abstract

In this paper, fundamental properties of solutions to fractional-order (FO) Maxwell's equations are analysed. As a starting point, FO Maxwell's equations are introduced in both time and frequency domains. Then, we introduce and prove the fundamental properties of electromagnetic field in FO electromagnetics, i.e. energy conservation, uniqueness of solutions, and reciprocity. Furthermore, the algorithm of the plane wave simulation is developed and implemented in software, which allows us to demonstrate the general properties of electromagnetic field in the media described by FO models (FOMs). The differences in interpretation of the fundamental theorems of electromagnetics (i.e. Poynting's theorem, the uniqueness theorem and the Lorentz reciprocity theorem) in comparison to integer-order electromagnetics are analysed. It is demonstrated that all the properties of electromagnetic field, related to these fundamental theorems are preserved when time derivatives are generalized towards FO in Maxwell's equations.