Electromagnetic fields in fractal continua

Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity

“…[43,[47][48][49][50][51][52]). Furthermore, the mapping F ⊂ E 3 → F ν ⊆ E 3 implies the use of metric partial derivatives ∂/∂ i instead of conventional partial derivatives ∂/∂x i [43].…”

“…The development of vector differential calculus in the fractional dimensional space with ν < 3 is not so straightforward (see Refs. [7,43,[50][51][52] and references therein). Fortunately, in the special case of one-dimensional flow through a fractally permeable medium, as studied in our experiments (see Fig.…”

“…(4a). Consequently, the gradient of function f (x j ) in the flow direction i = 1 reads as ∂ f /∂ 1 = (∂ 1 /∂x 1 ) −1 ∂ f /∂x 1 [43,[47][48][49][50][51][52]. Accordingly, for the one-dimensional laminar flow through a fractal the Darcy-like equation can be obtained by changing the spatial variable (x 1 → 1 ) and gradient operator (∂/∂ 1 → ∂/∂x 1 ) in the equation q = (k/μ)∂ P /∂x 1 derived in Ref.…”

Fractal electrodynamics via non-integer dimensional space approach

“…Then these continuum models of fractal electrodynamics have been applied and developed in two directions: (a) fractional integral models by Baskin and Iomin [6,7], by Ostoja-Starzewski [8] to describe anisotropic fractal cases; (b) fractional (non-integer) dimensional models by Muslih, Baleanu and coauthors [9][10][11], by Zubair, Mughal, Naqvi [12][13][14][15][16], by Balankin with coauthors [17], to describe an anisotropic case, multipoles, and electromagnetic waves in fractional space. Effective continuum models of fractal electrodynamics, which is considered in papers [9][10][11][12][13][14][15][16][17], are based on Stillinger and Palmer-Stavrinou generalizations of the scalar Laplacian that are suggested in [18] and [19], respectively. In these papers [18,19], the authors have proposed only the second order differential operators for scalar fields in the form of the scalar Laplacian in the non-integer dimensional space.…”

Cauchy representation formulas for Maxwell equations in 3-dimensional domains with fractal boundaries

“…In particular, in [2,41,42,44] the authors have presented a formulation of electromagnetic fields theory on fractals through the fractional differential topological formulation of Maxwell equations. The interest in fractional integrals and derivatives has been growing continuously during the last few years because of numerous applications in fractal media [34,35].…”