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.

Keywords:

• Computational electromagnetics
• electromagnetic propagation
• Maxwell's equations
• fractional calculus

Acknowledgments

Computations were carried out at the Academic Computer Centre in Gdańsk, Poland.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Tomasz P. Stefański

Tomasz P. Stefański received the M.Sc. degree in telecommunications and the Ph.D. degree in electronics engineering from Gdansk University of Technology (GUT), Gdansk, Poland, in 2002 and 2007, respectively. He is currently an Associate Professor at the Faculty of Electronics, Telecommunications and Informatics at GUT. Before joining GUT in 2011, he was with the Swiss Federal Institute of Technology (ETH Zurich) conducting research on parallelization of electromagnetic solvers on modern computing architectures using OpenCL programming language. Between 2006 and 2009 he worked at the University of Glasgow developing parallel alternating direction implicit finite-difference time-domain (ADI-FDTD) full-wave solvers for general purpose high-performance computers and graphics processing units. His current research interests include parallel processing, computational electromagnetics and scientific computing.

Jacek Gulgowski

Jacek Gulgowski received the M. Sc. degree in computer science from Gdansk University of Technology, Gdansk, Poland, in 1996 and the M.Sc. degree in mathematics from University of Gdansk in 1997. He received the Ph.D. degree in mathematics from University of Gdansk in 2002. He is currently an Associate Professor at the Faculty of Mathematics, Physics and Informatics, University of Gdansk. His scientific interests were always related both to the theoretical and to applied mathematics. Currently he is focused on the research related to fractional order models of electromagnetism.