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

Abstract

Maxwell’s equations on Cantor sets are derived from the local fractional vector calculus. It is shown that Maxwell’s equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Local fractional differential forms of Maxwell’s equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates are obtained. Maxwell's equations on Cantor set with local fractional operators are the first step towards a unified theory of Maxwell’s equations for the dynamics of cold dark matter.

1. 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].

Fractal time was used to describe the transport of charges and defects in the condensed matter [4]. In fractal space-time, the geometric analogue of relativistic quantum mechanics was presented in [5–8]. In fractal-Cantorian space-time Ω^4𝛼⊂Ω⁴ [9, 10], the unified field theory, quantum physics, cosmology, and chaotic systems were discussed in [11–13].

Based on the fractal distribution of charged particles, the electric and magnetic fields in time-space Ω^3𝛼+1⊂Ω⁴ 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^4𝛼⊂Ω⁴ 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+𝛼⊂Ω⁴ were considered. The Maxwell equations on anisotropic fractal media in time-space Ω^3𝛼+1⊂Ω⁴ were developed in [21].

The local fractional calculus theory [22, 23] was applied to model some dynamics systems with nondifferentiable characteristics. In [22–26], 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.

Figure 1 

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

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.

2. 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 𝜙 is defined as [23, 27] ∇^𝛼𝜙=lim_{𝑑𝑉(𝛾)→0}1𝑑𝑉^{(𝛾)𝑆(𝛽)}𝜙𝑑𝐒^(𝛽), (1) where 𝐒^(𝛽) is its bounding fractal surface, 𝑉 is a small fractal volume enclosing 𝑃, and the local fractional surface integral is given by [23, 27, 28] 𝑢𝑟_𝑃𝑑𝐒^(𝛽)=lim^𝑁_{𝑁→∞𝑃=1}𝑢𝑟_𝑃𝐧_𝑃Δ𝑆_𝑃^(𝛽), (2) with 𝑁 elements of area with a unit normal local fractional vector 𝐧_𝑃, Δ𝑆_𝑃^(𝛽)→0 as 𝑁→∞ for 𝛾=(3/2)𝛽=3𝛼, and ∇^𝛼 is denoted as the local fractional Laplace operator [22, 23].

The local fractional divergence of the vector function 𝐮 is defined through [23] ∇^𝛼⋅𝐮=lim_{𝑑𝑉(𝛾)→0}1𝑑𝑉^{(𝛾)𝑆(2𝛼)}𝐮⋅𝑑𝐒^(𝛽), (3) where the local fractional surface integral is suggested by [23, 27, 28] 𝐮𝑟_𝑃⋅𝑑𝐒^(𝛽)=lim^𝑁_{𝑁→∞𝑃=1}𝐮𝑟_𝑃⋅𝐧_𝑃Δ𝑆_𝑃^(𝛽), (4) with 𝑁 elements of area with a unit normal local fractional vector 𝐧_𝑃, Δ𝑆_𝑃^(𝛽)→0 as 𝑁→∞ for 𝛾=(3/2)𝛽=3𝛼.

The local fractional curl of the vector function 𝐮 [23] is defined as follows: ∇^𝛼×𝐮=lim_{𝑑𝑆(𝛽)→0}1𝑑𝑆^{(𝛽)𝑙(𝛼)}𝐮⋅𝑑𝐥^(𝛼)𝐧_𝑃, (5) where the local fractional line integral of the function 𝐮 along a fractal line 𝑙^𝛼 is given by [23] _𝑙(𝛼)𝐮𝑥_𝑃,𝑦_𝑃,𝑧_𝑃⋅𝑑𝐥^(𝛼)=lim^𝑁_{𝑁→∞𝑃=1}𝐮𝑥_𝑃,𝑦_𝑃,𝑧_𝑃⋅Δ𝐥_𝑃^(𝛼), (6) with the elements of line Δ𝐥_𝑃^(𝛼) requiring that all |Δ𝑙_𝑃^(𝛼)|→0 as 𝑁→∞ and 𝛽=2𝛼.

The local fractional Gauss theorem of the fractal vector field states that [23, 27] _𝑉(𝛾)∇^𝛼⋅𝐮𝑑𝑉^(𝛾)=_𝑆(𝛽)𝐮⋅𝑑𝐒^(𝛽), (7) where the local fractional volume integral of the function 𝐮 is written as [23] 𝐮𝑟_𝑃𝑑𝑉^(𝛾)=lim^𝑁_{𝑁→∞𝑃=1}𝐮𝑟_𝑃Δ𝑉_𝑃^(𝛾), (8) with the elements of volume Δ𝑉_𝑃^(𝛾)→0 as 𝑁→∞ and 𝛾=(3/2)𝛽=3𝛼.

The local fractional Stokes theorem of the fractal field states that [23, 27] _𝑙(𝛼)𝐮⋅𝑑𝐥^(𝛼)=_𝑆(𝛽)∇^𝛼×𝐮⋅𝑑𝐒^(𝛽). (9) The Reynolds transport theorem of the local fractional vector field states that [27] 𝐷^𝛼𝐷𝑡^𝛼_𝑉^(𝛾)𝐹(𝑥,𝑡)𝑑𝑉^(𝛾)=_𝑉(𝛾)𝜕^𝛼𝜕𝑡^𝛼𝐹(𝑥,𝑡)𝑑𝑉^(𝛾)+_𝑆(𝛽)𝐹(𝑥,𝑡)𝝊⋅𝑑𝐒^(𝛽), (10) where 𝝊 is the fractal fluid velocity.

3. 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.

3.1. Charge Conservations in Local Fractional Field

Let us consider the total charge, which is described as follows: 𝑄=_𝑉(𝛾)𝜌(𝑟,𝑡)𝑑𝑉^(𝛾), (11) and the total electrical current is as follows𝐼=_𝑆(𝛽)𝐽(𝑟,𝑡)⋅𝑑𝐒^(𝛽), (12) where 𝜌(𝑟,𝑡) is the fractal electric charge density and 𝐽(𝑟,𝑡) is the fractal electric current density.

The Reynolds transport theorem in the fractal field gives 𝐷^𝛼𝑄𝐷𝑡^𝛼=𝐷^𝛼𝐷𝑡^𝛼_𝑉^(𝛾)𝜌(𝑟,𝑡)𝑑𝑉^(𝛾)=_𝑉(𝛾)𝜕^𝛼𝜌(𝑟,𝑡)𝜕𝑡^𝛼𝑑𝑉^(𝛾)+_𝑆(𝛽)𝜌(𝑟,𝑡)𝝊⋅𝑑𝐒^(𝛽)=0, (13) which leads to _𝑉(𝛾)𝜕^𝛼𝜌(𝑟,𝑡)𝜕𝑡^𝛼𝑑𝑉^(𝛾)+_𝑆(𝛽)𝜌(𝑟,𝑡)𝝊⋅𝑑𝐒^(𝛽)=0. (14)

From (14), we have _𝑉(𝛾)𝜕^𝛼𝜌(𝑟,𝑡)𝜕𝑡^𝛼𝑑𝑉^(𝛾)+_𝑆(𝛽)𝐽(𝑟,𝑡)⋅𝑑𝐒^(𝛽)=_𝑉(𝛾)𝜕^𝛼𝜌(𝑟,𝑡)𝜕𝑡^𝛼+∇^𝛼⋅𝐽(𝑟,𝑡)𝑑𝑉^(𝛾)=0, (15) where 𝐽=𝜌𝝊 represents the current density in the fractal field.

Hence, from (15), we get 𝜕^𝛼𝜌(𝑟,𝑡)𝜕𝑡^𝛼+∇^𝛼⋅𝐽(𝑟,𝑡)=0. (16)

By analogy with electric charge density in the fractal field, we obtain the conservation of fractal magnetic charge, namely, 𝜕^𝛼𝜌_𝑚(𝑟,𝑡)𝜕𝑡^𝛼+∇^𝛼⋅𝐽_𝑚(𝑟,𝑡)=0, (17) where 𝜌_𝑚(𝑟,𝑡) is the fractal magnetic charge density and 𝐽_𝑚(𝑟,𝑡) is the fractal magnetic current density in the fractal field.

3.2. Formulation of Maxwell’s Equations on Cantor Sets

We now derive Maxwell’s equations on Cantor set based on the local fractional vector calculus.

3.2.1. Gauss’s Law for the Fractal Electric Field

From (3), the electric charge density can be written as ∇^𝛼⋅𝐷=lim_{𝑑𝑉(𝛾)→0}1𝑑𝑉^{(𝛾)𝑆(2𝛼)}𝐷⋅𝑑𝐒^(𝛽)=𝜌, (18) where 𝐷 is electric displacement in the fractal electric field.

From (7), (18) becomes _𝑉(𝛾)∇^𝛼⋅𝐷𝑑𝑉^(𝛾)=_𝑆(𝛽)𝐷⋅𝑑𝐒^(𝛽),_𝑉(𝛾)∇^𝛼⋅𝐷𝑑𝑉^(𝛾)=_𝑉(𝛾)𝜌𝑑𝑉^(𝛾). (19)

Hence, we obtain Gauss’s law for the fractal electric field in the form _𝑆(𝛽)𝐷⋅𝑑𝐒^(𝛽)=_𝑉(𝛾)𝜌𝑑𝑉^(𝛾). (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] _𝑙(𝛼)𝐻⋅𝑑𝐥^(𝛼)=𝐼, (21) where 𝐻 is the magnetic field strength in the fractal field.

The current density in the fractal field can be written as ∇^𝛼×𝐻=lim_{𝑑𝑆(𝛽)→0}1𝑑𝑆^{(𝛽)𝑙(𝛼)}𝐻⋅𝑑𝐥^(𝛼)𝐧_𝑃=𝐽, (22) which leads to _𝑙(𝛼)𝐻⋅𝑑𝐥^(𝛼)=_𝑆(𝛽)∇^𝛼×𝐻⋅𝑑𝐒^(𝛽),_𝑆(𝛽)𝐽⋅𝑑𝐒^(𝛽)=_𝑆(𝛽)∇^𝛼×𝐻⋅𝑑𝐒^(𝛽). (23)

The total current in the fractal fried reads 𝐼=_𝑆(𝛽)𝐽_𝑎+𝐽_𝑏⋅𝑑𝐒^(𝛽), (24) where 𝐽_𝑎 is the conductive current and 𝐽_𝑏=𝜕^𝛼𝐷𝜕𝑡^𝛼, (25) which satisfies the following condition: 𝐼_𝑏=_𝑆(𝛽)𝐽_𝑏⋅𝑑𝐒^(𝛽)=_𝑆(𝛽)𝜕^𝛼𝜕𝑡^𝛼𝐷⋅𝑑𝐒^(𝛽)=𝜕^𝛼𝜕𝑡^𝛼_𝑆^(𝛽)𝐷⋅𝑑𝐒^(𝛽). (26)

Hence, Ampere’s law in the fractal field is expressed as follows: _𝑙(𝛼)𝐻⋅𝑑𝐥^(𝛼)=_𝑆(𝛽)𝐽_𝑎+𝜕^𝛼𝐷𝜕𝑡^𝛼⋅𝑑𝐒^(𝛽). (27)

3.2.3. Faraday’s Law in the Fractal Electric Field

Mathematically, Faraday’s law in the local fractional field is expressed as 𝜕𝐸=−^𝛼𝜑𝜕𝑡^𝛼, (28) where 𝜑 is the magnetic potential in the fractal field and 𝐸 is the electrical field strength in the fractal field.

From (28), we have ∇^𝛼×𝐸=lim_{𝑑𝑆(𝛽)→0}1𝑑𝑆^{(𝛽)𝑙(𝛼)}𝐸⋅𝑑𝐥^(𝛼)𝐧_𝑃𝜕=−^𝛼𝜕𝑡^𝛼∇^𝛼𝜕×𝜑=−^𝛼𝐵𝜕𝑡^𝛼, (29) where 𝐵 is the magnetic induction in the fractal field.

In view of (9), we rewrite (29) as _𝑙(𝛼)𝐸⋅𝑑𝐥^(𝛼)=_𝑆(𝛽)∇^𝛼×𝐸⋅𝑑𝐒^(𝛽),_𝑆(𝛽)∇^𝛼×𝐸⋅𝑑𝐒^(𝛽)=_𝑆(𝛽)−𝜕^𝛼𝐵𝜕𝑡^𝛼⋅𝑑𝐒^(𝛽)𝜕=−^𝛼𝜕𝑡^𝛼_𝑆^(𝛽)𝐵⋅𝑑𝐒^(𝛽). (30)

So, from (30), Faraday’s law in the fractal field reads as _𝑙(𝛼)𝐸⋅𝑑𝐥^(𝛼)+𝜕^𝛼𝜕𝑡^𝛼_𝑆^(𝛽)𝐵⋅𝑑𝐒^(𝛽)=0. (31)

3.2.4. 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, ∇^𝛼⋅𝐵=lim_{𝑑𝑉(𝛾)→0}1𝑑𝑉^{(𝛾)𝑆(2𝛼)}𝐵⋅𝑑𝐒^(𝛽)=0. (32)

Furthermore, the magnetic Gauss’ law for the fractal magnetic field reads as _𝑆(𝛽)𝐵⋅𝑑𝐒^(𝛽)=0. (33)

3.2.5. 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 𝐷=𝜀_𝑓𝐸,𝐻=𝜇_𝑓𝐵, (34) where 𝜀_𝑓 is the fractal dielectric permittivity and 𝜇_𝑓 is the fractal magnetic permeability.

4. 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]: 𝑥^𝛼=𝑅^𝛼cos_𝛼𝜃^𝛼,𝑦^𝛼=𝑅^𝛼sin_𝛼𝜃^𝛼,𝑧^𝛼=𝑧^𝛼, (35) with 𝑅∈(0,+∞), 𝑧∈(−∞,+∞), 𝜃∈(0,𝜋], and 𝑥^2𝛼+𝑦^2𝛼=𝑅^2𝛼.

Making use of (35), we have ∇^𝛼𝜕⋅𝐫=^𝛼𝑟_𝑅𝜕𝑅^𝛼+1𝑅^𝛼𝜕^𝛼𝑟_𝜃𝜕𝜃^𝛼+𝑟_𝑅𝑅^𝛼+𝜕^𝛼𝑟_𝑧𝜕𝑧^𝛼,∇^𝛼1×𝐫=𝑅^𝛼𝜕^𝛼𝑟_𝜃𝜕𝜃^𝛼−𝜕^𝛼𝑟_𝜃𝜕𝑧^𝛼𝐞^𝛼_𝑅+𝜕^𝛼𝑟_𝑅𝜕𝑧^𝛼−𝜕^𝛼𝑟_𝑧𝜕𝑅^𝛼𝐞^𝛼_𝜃+𝜕^𝛼𝑟_𝜃𝜕𝑅^𝛼+𝑟_𝑅𝑅^𝛼−1𝑅^𝛼𝜕^𝛼𝑟_𝑅𝜕𝜃^𝛼𝐞^𝛼_𝑧, (36) where 𝐫=𝑅^𝛼cos_𝛼𝜃^𝛼𝐞^𝛼₁+𝑅^𝛼sin_𝛼𝜃^𝛼𝐞^𝛼₂+𝑧^𝛼𝐞^𝛼₃=𝑟_𝑅𝐞^𝛼_𝑅+𝑟_𝜃𝐞^𝛼_𝜃+𝑟_𝑧𝐞^𝛼_𝑧. (37)

From (7) and (20), the local fractional differential form of Gauss’s law for the fractal electric field is expressed by ∇^𝛼⋅𝐷=𝜌, (38) which leads to 𝜕^𝛼𝐷_𝑅𝜕𝑅^𝛼+1𝑅^𝛼𝜕^𝛼𝐷_𝜃𝜕𝜃^𝛼+𝐷_𝑅𝑅^𝛼+𝜕^𝛼𝐷_𝑧𝜕𝑧^𝛼=𝜌(𝑅,𝜃,𝑧,𝑡), (39) where 𝐷=𝐷_𝑅𝐞^𝛼_𝑅+𝐷_𝜃𝐞^𝛼_𝜃+𝐷_𝑧𝐞^𝛼_𝑧.

In view of (9) and (27), we can present the local fractional differential forms of Ampere’s law in the fractal magnetic field ∇^𝛼×𝐻=𝐽_𝑎+𝜕^𝛼𝐷𝜕𝑡^𝛼,1𝑅^𝛼𝜕^𝛼𝐻_𝜃𝜕𝜃^𝛼−𝜕^𝛼𝐻_𝜃𝜕𝑧^𝛼𝐞^𝛼_𝑅+𝜕^𝛼𝐻_𝑅𝜕𝑧^𝛼−𝜕^𝛼𝐻_𝑧𝜕𝑅^𝛼𝐞^𝛼_𝜃+𝜕^𝛼𝐻_𝜃𝜕𝑅^𝛼+𝐻_𝑅𝑅^𝛼−1𝑅^𝛼𝜕^𝛼𝐻_𝑅𝜕𝜃^𝛼𝐞^𝛼_𝑧=𝐽_𝑎()+𝜕𝑅,𝜃,𝑧,𝑡^𝛼𝐷(𝑅,𝜃,𝑧,𝑡)𝜕𝑡^𝛼, (40) where 𝐻=𝐻_𝑅𝐞^𝛼_𝑅+𝐻_𝜃𝐞^𝛼_𝜃+𝐻_𝑧𝐞^𝛼_𝑧.

Using (9) and (31), the local fractional differential forms of Faraday’s law in the fractal electric field can be written as ∇^𝛼𝜕×𝐸=−^𝛼𝐵𝜕𝑡^𝛼,1𝑅^𝛼𝜕^𝛼𝐸_𝜃𝜕𝜃^𝛼−𝜕^𝛼𝐸_𝜃𝜕𝑧^𝛼𝐞^𝛼_𝑅+𝜕^𝛼𝐸_𝑅𝜕𝑧^𝛼−𝜕^𝛼𝐸_𝑧𝜕𝑅^𝛼𝐞^𝛼_𝜃+𝜕^𝛼𝐸_𝜃𝜕𝑅^𝛼+𝐸_𝑅𝑅^𝛼−1𝑅^𝛼𝜕^𝛼𝐸_𝑅𝜕𝜃^𝛼𝐞^𝛼_𝑧𝜕=−^𝛼𝐵(𝑅,𝜃,𝑧,𝑡)𝜕𝑡^𝛼, (41) where 𝐸=𝐸_𝑅𝐞^𝛼_𝑅+𝐸_𝜃𝐞^𝛼_𝜃+𝐸_𝑧𝐞^𝛼_𝑧.

From (7) and (33), the local fractional differential forms of the magnetic Gauss law for the fractal magnetic field read as follows: ∇^𝛼𝜕⋅𝐵=0,_𝛼𝐵_𝑅𝜕𝑅^𝛼+1𝑅^𝛼𝜕^𝛼𝐵_𝜃𝜕𝜃^𝛼+𝐵_𝑅𝑅^𝛼+𝜕^𝛼𝐵_𝑧𝜕𝑧^𝛼=0, (42) where 𝐵=𝐵_𝑅𝐞^𝛼_𝑅+𝐵_𝜃𝐞^𝛼_𝜃+𝐵_𝑧𝐞^𝛼_𝑧.

5. 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].

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.