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

Stefański, Tomasz P.; Gulgowski, Jacek

doi:10.3390/e23080987

Cited by 8 publications

(4 citation statements)

References 59 publications

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“…In this case, the conduction current keeps its usual definition. The suggested expression for the modified displacement current, Equation (104), aligns with the form recently utilized in certain generalizations of Maxwell's equations [83,84] for material media that incorporate fractional-order operators. To avoid this work going down the road that involves analyzing problems related to the covariance of Maxwell's equations, from now on, we restrict ourselves to analyzing the extensions of the PNP model to the fractional field that implies a change in the conduction current.…”

Section: The Displacement Currentmentioning

confidence: 76%

A Brief Review of Fractional Calculus as a Tool for Applications in Physics: Adsorption Phenomena and Electrical Impedance in Complex Fluids

Barbero,

Evangelista,

Zola

et al. 2024

Fractal Fract

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Many fundamental physical problems are modeled using differential equations, describing time- and space-dependent variables from conservation laws. Practical problems, such as surface morphology, particle interactions, and memory effects, reveal the limitations of traditional tools. Fractional calculus is a valuable tool for these issues, with applications ranging from membrane diffusion to electrical response of complex fluids, particularly electrolytic cells like liquid crystal cells. This paper presents the main fractional tools to formulate a diffusive model regarding time-fractional derivatives and modify the continuity equations stating the conservation laws. We explore two possible ways to introduce time-fractional derivatives to extend the continuity equations to the field of arbitrary-order derivatives. This investigation is essential, because while the mathematical description of neutral particle diffusion has been extensively covered by various authors, a comprehensive treatment of the problem for electrically charged particles remains in its early stages. For this reason, after presenting the appropriate mathematical tools based on fractional calculus, we demonstrate that generalizing the diffusion equation leads to a generalized definition of the displacement current. This modification has strong implications in defining the electrical impedance of electrolytic cells but, more importantly, in the formulation of the Maxwell equations in material systems.

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Section: The Displacement Currentmentioning

confidence: 76%

A Brief Review of Fractional Calculus as a Tool for Applications in Physics: Adsorption Phenomena and Electrical Impedance in Complex Fluids

Barbero,

Evangelista,

Zola

et al. 2024

Fractal Fract

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“…In particular, a generalizations of Debye's permittivity and a weak spatial dispersion of power-law type in plasma-like media were suggested. Using the formulation of TF electrodynamics with the Riemann-Silberstein vector, a compact form of FO Maxwell equations was presented in [28]. The proposed formulation allows for inclusion of energy dissipation and establishes a relation between the TF Schrödinger equation and TF electrodynamics.…”

Section: Fractional Vector Calculusmentioning

confidence: 99%

“…where υ l and υ k are the integers obtained by rounding up the real numbers β l and 1 + α k to their next highest integer, respectively. Taking the inverse Fourier transform of (28) gives…”

Section: Fdtd Schemementioning

confidence: 99%

FDTD-Based Electromagnetic Modeling of Dielectric Materials with Fractional Dispersive Response

Mescia

Bia²,

Caratelli

2022

Electronics

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The use of fractional derivatives and integrals has been steadily increasing thanks to their ability to capture effects and describe several natural phenomena in a better and systematic manner. Considering that the study of fractional calculus theory opens the mind to new branches of thought, in this paper, we illustrate that such concepts can be successfully implemented in electromagnetic theory, leading to the generalizations of the Maxwell’s equations. We give a brief review of the fractional vector calculus including the generalization of fractional gradient, divergence, curl, and Laplacian operators, as well as the Green, Stokes, Gauss, and Helmholtz theorems. Then, we review the physical and mathematical aspects of dielectric relaxation processes exhibiting non-exponential decay in time, focusing the attention on the time-harmonic relative permittivity function based on a general fractional polynomial series approximation. The different topics pertaining to the incorporation of the power-law dielectric response in the FDTD algorithm are explained, too. In particular, we discuss in detail a home-made fractional calculus-based FDTD scheme, also considering key issues concerning the bounding of the computational domain and the numerical stability. Finally, some examples involving different dispersive dielectrics are presented with the aim to demonstrate the usefulness and reliability of the developed FDTD scheme.

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“…Assuming the plane-wave, spherical and cylindrical symmetries of solutions to (77), one obtains, respectively, the following transfer functions describing the 1-D wave propagation [64]:…”

Section: Frequency Responsementioning

confidence: 99%

Analytical Methods for Causality Evaluation of Photonic Materials

2022

Self Cite

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We comprehensively review several general methods and analytical tools used for causality evaluation of photonic materials. Our objective is to call to mind and then formulate, on a mathematically rigorous basis, a set of theorems which can answer the question whether a considered material model is causal or not. For this purpose, a set of various distributional theorems presented in literature is collected as the distributional version of the Titchmarsh theorem, allowing for evaluation of causality in complicated electromagnetic systems. Furthermore, we correct the existing material models with the use of distribution theory in order to obtain their causal formulations. In addition to the well-known Kramers–Krönig (K–K) relations, we overview four further methods which can be used to assess causality of given dispersion relations, when calculations of integrals involved in the K–K relations are challenging or even impossible. Depending on the given problem, optimal approaches allowing us to prove either the causality or lack thereof are pointed out. These methodologies should be useful for scientists and engineers analyzing causality problems in electrodynamics and optics, particularly with regard to photonic materials, when the involved mathematical distributions have to be invoked.

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

Cited by 8 publications

References 59 publications

A Brief Review of Fractional Calculus as a Tool for Applications in Physics: Adsorption Phenomena and Electrical Impedance in Complex Fluids

A Brief Review of Fractional Calculus as a Tool for Applications in Physics: Adsorption Phenomena and Electrical Impedance in Complex Fluids

FDTD-Based Electromagnetic Modeling of Dielectric Materials with Fractional Dispersive Response

Analytical Methods for Causality Evaluation of Photonic Materials

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