Maxwell ’ s Equations on Cantor Sets : A Local Fractional Approach

1 College of Instrumentation & Electrical Engineering, Jilin University, Changchun 130061, China 2 Electronic and Information Technology Department, Jiangmen Polytechnic, Jiangmen 529090, China 3Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia 4Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey 5 Institute of Space Sciences, Magurele, 077125 Bucharest, Romania 6Department of Mathematics, University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy 7 Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu 221008, China


Introduction
Nondifferentiability, complexity, and similarity represent the basic properties of the nature.Fractals [1] are the basic characteristics of nature, which are that fractal geometry of substances generalizes to noninteger dimensions.Microphysics reveals the fractal behaviors of matter distribution in the universe [2] and soft materials [3].
Based on the fractal distribution of charged particles, the electric and magnetic fields in time-space Ω 3+1 ⊂ Ω 4 were developed in [14] and fractional Maxwell's equations were proposed in [15].In [16,17], the concept of static fractional electric potential was developed.Recently, based on the Hausdorff derivative, fractal continuum electrodynamics in time-space Ω 4 ⊂ Ω 4 was proposed [18].The fractional differential form of Maxwell's equations on fractal sets was suggested in [19].In [20], the Maxwell equations of fractional electrodynamics in time-space Ω 3+ ⊂ Ω 4 were considered.The Maxwell equations on anisotropic fractal media in timespace Ω 3+1 ⊂ Ω 4 were developed in [21].
The local fractional calculus theory [22,23] was applied to model some dynamics systems with nondifferentiable characteristics.In [22][23][24][25][26], the heat-conduction equation on Cantor sets was considered.In [27], the Navier-Stokes equations on Cantor sets based on local fractional vector calculus were proposed.Helmholtz and diffusion equations via local fractional vector calculus were reported in [28].The Fokker-Planck equation with local fractional space derivative was suggested in [29].In [30], Lagrangian and Hamiltonian mechanics with local fractional space derivative was presented.The measuring structures of time in fractal, fractional, classical, and discrete electrodynamics are shown in Figure 1.
The aim of this paper is to structure Maxwell's equations on Cantor sets from the local fractional calculus theory [23,27,28] point of view.This paper is structured as follows.In Section 2, we introduce the basic definitions and theorems for local fractional vector calculus.In Section 3, Maxwell's equations on Cantor sets in the local fractional vector integral form are presented.Maxwell's equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates are given in Section 4. Finally, Section 5 is devoted to conclusions.

Fundaments
In this section, we recall the basic definitions and theorems for local fractional vector calculus, which are used throughout the paper.
Local fractional gradient of the scale function  is defined as [23,27] ∇   = lim where S () is its bounding fractal surface,  is a small fractal volume enclosing , and the local fractional surface integral is given by [23,27,28] with  elements of area with a unit normal local fractional vector n  , Δ → 0 as  → ∞ for  = (3/2) = 3, and ∇  is denoted as the local fractional Laplace operator [22,23].
The local fractional divergence of the vector function u is defined through [23] ∇  ⋅ u = lim where the local fractional surface integral is suggested by [23,27,28] with  elements of area with a unit normal local fractional vector n  , Δ The local fractional curl of the vector function u [23] is defined as follows: where the local fractional line integral of the function u along a fractal line   is given by [23] ∫ with the elements of line Δl ()   requiring that all |Δ ()  | → 0 as  → ∞ and  = 2.
The local fractional Gauss theorem of the fractal vector field states that [23,27] ∭ where the local fractional volume integral of the function u is written as [23] ∭ u with the elements of volume Δ The local fractional Stokes theorem of the fractal field states that [23,27] ∮ The Reynolds transport theorem of the local fractional vector field states that [27] where  is the fractal fluid velocity.

Local Fractional Integral Forms of Maxwell's Equations on Cantor Sets
According to fractional complex transform method [31], the fact that the classical differential equations always transform into the local fractional differential equations leads to the idea of yielding Maxwell's equations on Cantor sets using the local fractional vector calculus.

Charge Conservations in Local Fractional
Field.Let us consider the total charge, which is described as follows: and the total electrical current is as follows where (, ) is the fractal electric charge density and (, ) is the fractal electric current density.
The Reynolds transport theorem in the fractal field gives (, )    () + ∯ which leads to From ( 14), we have (, )    () + ∯ where  =  represents the current density in the fractal field.
Hence, from (15), we get By analogy with electric charge density in the fractal field, we obtain the conservation of fractal magnetic charge, namely, where   (, ) is the fractal magnetic charge density and   (, ) is the fractal magnetic current density in the fractal field.

Formulation of Maxwell's Equations on Cantor
Sets.We now derive Maxwell's equations on Cantor set based on the local fractional vector calculus.

Gauss's
Law for the Fractal Electric Field.From (3), the electric charge density can be written as where  is electric displacement in the fractal electric field.From ( 7), (18) becomes () .
Hence, we obtain Gauss's law for the fractal electric field in the form () . (20)

3.2.2.
Ampere's Law in the Fractal Magnetic Field.Mathematically, Ampere's law in the fractal magnetic field can be suggested as [23] ∮ where  is the magnetic field strength in the fractal field.The current density in the fractal field can be written as which leads to (∇  × ) ⋅ S () . ( The total current in the fractal fried reads where   is the conductive current and which satisfies the following condition: ⋅ S () . ( Hence, Ampere's law in the fractal field is expressed as follows: Advances in High Energy Physics

Faraday's Law in the Fractal Electric
Field.Mathematically, Faraday's law in the local fractional field is expressed as where  is the magnetic potential in the fractal field and  is the electrical field strength in the fractal field.From ( 28), we have where  is the magnetic induction in the fractal field.
In view of ( 9), we rewrite (29) as So, from (30), Faraday's law in the fractal field reads as

Magnetic Gauss's Law for the Fractal Magnetic Field.
From (3), we derive the local fractional divergence of the magnetic induction in the fractal field, namely, Furthermore, the magnetic Gauss' law for the fractal magnetic field reads as ⋅ S () = 0. (33)

The Constitutive Equations in the Fractal
Field.Similar to the constitutive relations in fractal continuous medium mechanics [23], the constitutive relationships in fractal electromagnetic can be written as where   is the fractal dielectric permittivity and   is the fractal magnetic permeability.

Local Fractional Differential Forms of Maxwell's Equations on Cantor Sets
In this section, we investigate the local fractional differential forms of Maxwell's equations on Cantor sets.
The Cantor-type cylindrical coordinates can be written as follows [26]: with  ∈ (0, +∞),  ∈ (−∞, +∞),  ∈ (0, ], and  2 +  2 =  2 . Making use of (35), we have From ( 7) and (20), the local fractional differential form of Gauss's law for the fractal electric field is expressed by which leads to where  =   e   +   e   +   e   .In view of ( 9) and ( 27), we can present the local fractional differential forms of Ampere's law in the fractal magnetic field where  =   e   +   e   +   e   .From ( 7) and (33), the local fractional differential forms of the magnetic Gauss law for the fractal magnetic field read as follows: where  =   e   +   e   +   e   .

Conclusions
In this work, we proposed the local fractional approach for Maxwell's equations on Cantor sets based on the local fractional vector calculus.Employing the local fractional divergence and curl of the vector function, we deduced Maxwell's equations on Cantor sets.The local fractional differential forms of Maxwell's equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates were discussed.Finding a formulation of Maxwell's equations on Cantor set within local fractional operators is the first step towards generalizing a simple field equation which allows the unification of Maxwell's equations to the standard model with the dynamics of cold dark matter.We noticed that the classical case was debated in [32].

Figure 1 :
Figure 1: Graph for comparison of the measuring structures of time in fractal, fractional, and classical electrodynamics.