Retarded Potentials in Fractional Electrodynamics
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RETICAL AND MATHEMATICAL PHYSICS
Retarded Potentials in Fractional Electrodynamics A. V. Pskhu1 and S. Sh. Rekhviashvili1* 1
Institute of Applied Mathematics and Automation, Kabardin-Balkar Scientific Center, Russian Academy of Sciences, Nalchik, 360000 Russia Received February 25, 2020; revised March 23, 2020; accepted April 10, 2020
Abstract—A general expression is obtained for retarded potentials for a system of equations of macroscopic electrodynamics with fractional Caputo derivatives with respect to time. An analog of the Lienard– Wiechert potentials is obtained. All expressions contain a nonlocal (time-distributed) delay, which takes the temporal dispersion in the system into account. Keywords: classical electrodynamics, fractional integro-differentiation, memory effect, time dispersion, retarded potentials, Lienard–Wiechert potentials. DOI: 10.3103/S0027134920040098
INTRODUCTION A variant of fractional electrodynamics of material media based on the application of the Caputo time derivative was proposed in [1, 2]. The results of these works imply that the introduction of a fractional derivative in the Maxwell equations allows taking the dissipative processes into account a priori. In the subsequent works [3, 4] based on this approach, the generalized case of material medium with spatialtemporal dispersion which is a power law function of time was analyzed. It was shown that the media of this type possess fractal properties in the time variable. We also note that in several later works on this topic (for instance, in [5–7]) the results from [1, 2] were in fact reproduced. Among other approaches to electrodynamics with fractional derivatives, we may note the approaches from [8–10]. In [8, 9] it was demonstrated that the electromagnetic fields in dielectric media are described by differential equations with fractional time derivatives if the susceptibility of these media is characterized by a fractional power dependence. In [10] the fractional analogs of the Green, Stokes, and Gauss theorems were formulated. Following from this, new wave equations were derived with the fractional derivative with respect to the spatial variable. In total, however, we have to state that the application of fractional derivatives in electrodynamics is not widely used thus far. The main difficulty is in the absence of a clear physical interpretation of the fractional integro-differential operators and of *
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the fractional-order diffusion-wave partial differential equations. The recent achievements in developing fractional and fractal models in electrodynamics were presented in [11]. In the current work we develop the ideas in [1, 2]. We obtain the Kirchhoff formula for the generalized diffusion-wave equation and determine the general form for retarded scalar and vector potentials with it. We find the fractional analog of the Lienard–Wiechert potentials. Based on the results, we discuss the physical meaning of fractional integro-differentiation in the electrodynamics of material media. THEORY In fractional electrodynamics
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