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.

中文翻译：

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分数阶麦克斯韦方程组解的基本性质

在本文中，分析了分数阶 (FO) 麦克斯韦方程组解的基本性质。作为起点，在时域和频域中都引入了 FO 麦克斯韦方程。然后，我们介绍并证明了FO电磁学中电磁场的基本性质，即能量守恒、解的唯一性和互易性。此外，平面波模拟的算法是在软件中开发和实现的，这使我们能够展示由 FO 模型 (FOM) 描述的介质中电磁场的一般特性。分析了电磁学基本定理（即坡印廷定理、唯一性定理和洛伦兹互易定理）与整数阶电磁学在解释上的差异。