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

Tomasz P Stefański1, Jacek Gulgowski2

  • 1The Faculty of Electronics, Telecommunications and Informatics, Gdansk University of Technology, 80-233 Gdansk, Poland.

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Summary

This study introduces time-fractional (TF) electrodynamics using the Riemann-Silberstein vector, enabling modeling of electromagnetic systems with memory and energy dissipation. The research analyzes TF Maxwell

Keywords:
Maxwell’s equationsRiemann-Silberstein vectorfractional derivatives

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Area of Science:

  • Electromagnetism
  • Fractional Calculus
  • Mathematical Physics

Background:

  • Classical electrodynamics describes electromagnetic phenomena but often neglects memory effects and energy dissipation.
  • Fractional calculus offers a powerful framework for modeling systems with non-local and memory-dependent behaviors.
  • The Riemann-Silberstein (RS) vector provides a compact and elegant representation of electromagnetic fields.

Purpose of the Study:

  • To formulate time-fractional (TF) electrodynamics using the Riemann-Silberstein (RS) vector.
  • To analyze the properties of TF Maxwell's equations concerning classical electrodynamics principles.
  • To derive and analyze classical solutions for wave propagation problems within the TF electrodynamics framework.

Main Methods:

  • Derivation of TF Maxwell's equations utilizing the RS vector and fractional-order derivatives.
  • Analysis of TF Maxwell's equations for conservation laws (energy, momentum), reciprocity, and causality.
  • Derivation of classical wave propagation solutions with helical, spherical, and cylindrical symmetries.
  • Numerical simulations to validate theoretical results.

Main Results:

  • A compact formulation of TF Maxwell's equations is presented, incorporating energy dissipation and memory effects.
  • The analysis confirms that TF Maxwell's equations maintain key classical electrodynamic properties.
  • Classical solutions for wave propagation problems are derived and numerically verified.
  • Connections between TF Schrödinger equation and TF electrodynamics are explored.

Conclusions:

  • The RS vector provides an effective tool for formulating TF electrodynamics, extending classical electromagnetism to systems with memory.
  • TF electrodynamics offers a robust framework for modeling complex electromagnetic phenomena with dissipation.
  • The derived solutions and analyses contribute to a deeper understanding of fractional electromagnetism and its applications.