Formulation of Time-Fractional Electrodynamics Based on Riemann-Silberstein Vector

Tomasz P Stefański; Jacek Gulgowski

doi:10.3390/e23080987

Formulation of Time-Fractional Electrodynamics Based on Riemann-Silberstein Vector

Entropy (Basel). 2021 Jul 30;23(8):987. doi: 10.3390/e23080987.

Authors

Tomasz P Stefański¹, Jacek Gulgowski²

Affiliations

¹ The Faculty of Electronics, Telecommunications and Informatics, Gdansk University of Technology, 80-233 Gdansk, Poland.
² The Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-308 Gdansk, Poland.

Abstract

In this paper, the formulation of time-fractional (TF) electrodynamics is derived based on the Riemann-Silberstein (RS) vector. With the use of this vector and fractional-order derivatives, one can write TF Maxwell's equations in a compact form, which allows for modelling of energy dissipation and dynamics of electromagnetic systems with memory. Therefore, we formulate TF Maxwell's equations using the RS vector and analyse their properties from the point of view of classical electrodynamics, i.e., energy and momentum conservation, reciprocity, causality. Afterwards, we derive classical solutions for wave-propagation problems, assuming helical, spherical, and cylindrical symmetries of solutions. The results are supported by numerical simulations and their analysis. Discussion of relations between the TF Schrödinger equation and TF electrodynamics is included as well.

Keywords: Maxwell’s equations; Riemann-Silberstein vector; fractional derivatives.