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

Authors: Yang Zhao; Dumitru Baleanu; Carlo Cattani; De-Fu Cheng; Xiao-Jun Yang
Journal: Advances in High Energy Physics
 (2013)

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## 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.


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## Body

## 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 Ω4α⊂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-spaceΩ3α+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 Ω4α⊂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 Ω3+α⊂Ω4 were considered. The Maxwell equations on anisotropic fractal media in time-space Ω3α+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 functionϕ is defined as [23, 27]
(1)∇αϕ=limdV(γ)→0(1dV(γ)∯S(β)ϕdS(β)),
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]
(2)∬u(rP)dS(β)=limN→∞∑P=1Nu(rP)nPΔSP(β),
with N elements of area with a unit normal local fractional vector nP, ΔSP(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 functionu is defined through [23]
(3)∇α·u=limdV(γ)→0(1dV(γ)∯S(2α)u·dS(β)),
where the local fractional surface integral is suggested by [23, 27, 28]
(4)∬u(rP)·dS(β)=limN→∞∑P=1Nu(rP)·nPΔSP(β),
with N elements of area with a unit normal local fractional vector nP, ΔSP(β)→0 as N→∞ for γ=(3/2)β=3α.The local fractional curl of the vector functionu [23] is defined as follows:
(5)∇α×u=limdS(β)→0(1dS(β)∮l(α)u·dl(α))nP,
where the local fractional line integral of the function u along a fractal line lα is given by [23]
(6)∫l(α)u(xP,yP,zP)·dl(α)=limN→∞∑P=1Nu(xP,yP,zP)·ΔlP(α),
with the elements of line ΔlP(α) requiring that all |ΔlP(α)|→0 as N→∞ and β=2α.The local fractional Gauss theorem of the fractal vector field states that [23, 27]
(7)∭V(γ)∇α·udV(γ)=∯S(β)u·dS(β),
where the local fractional volume integral of the function u is written as [23]
(8)∭u(rP)dV(γ)=limN→∞∑P=1Nu(rP)ΔVP(γ),
with the elements of volume ΔVP(γ)→0 as N→∞ and γ=(3/2)β=3α.The local fractional Stokes theorem of the fractal field states that [23, 27]
(9)∮l(α)u·dl(α)=∬S(β)(∇α×u)·dS(β).
The Reynolds transport theorem of the local fractional vector field states that [27]
(10)DαDtα∭V(γ)F(x,t)dV(γ)=∭V(γ)∂α∂tαF(x,t)dV(γ)+∯S(β)F(x,t)υ·dS(β),
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:(11)Q=∭V(γ)ρ(r,t)dV(γ),
and the total electrical current is as follows(12)I=∬S(β)J(r,t)·dS(β),
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(13)DαQDtα=DαDtα∭V(γ)ρ(r,t)dV(γ)=∭V(γ)∂αρ(r,t)∂tαdV(γ)+∯S(β)ρ(r,t)υ·dS(β)=0,
which leads to
(14)∭V(γ)∂αρ(r,t)∂tαdV(γ)+∯S(β)ρ(r,t)υ·dS(β)=0.From (14), we have
(15)∭V(γ)∂αρ(r,t)∂tαdV(γ)+∯S(β)J(r,t)·dS(β)=∭V(γ)(∂αρ(r,t)∂tα+∇α·J(r,t))dV(γ)=0,
where J=ρυ represents the current density in the fractal field.Hence, from (15), we get
(16)∂αρ(r,t)∂tα+∇α·J(r,t)=0.By analogy with electric charge density in the fractal field, we obtain the conservation of fractal magnetic charge, namely,(17)∂αρm(r,t)∂tα+∇α·Jm(r,t)=0,
where ρm(r,t) is the fractal magnetic charge density and Jm(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
(18)∇α·D=limdV(γ)→0(1dV(γ)∯S(2α)D·dS(β))=ρ,
where D is electric displacement in the fractal electric field.From (7), (18) becomes
(19)∭V(γ)∇α·DdV(γ)=∯S(β)D·dS(β),∭V(γ)∇α·DdV(γ)=∭V(γ)ρdV(γ).Hence, we obtain Gauss’s law for the fractal electric field in the form(20)∯S(β)D·dS(β)=∭V(γ)ρdV(γ).


#### 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]
(21)∮l(α)H·dl(α)=I,
where H is the magnetic field strength in the fractal field.The current density in the fractal field can be written as(22)∇α×H=limdS(β)→0(1dS(β)∮l(α)H·dl(α))nP=J,
which leads to
(23)∮l(α)H·dl(α)=∬S(β)(∇α×H)·dS(β),∬S(β)J·dS(β)=∬S(β)(∇α×H)·dS(β).The total current in the fractal fried reads(24)I=∬S(β)(Ja+Jb)·dS(β),
where Ja is the conductive current and
(25)Jb=∂αD∂tα,
which satisfies the following condition:
(26)Ib=∯S(β)Jb·dS(β)=∯S(β)∂α∂tαD·dS(β)=∂α∂tα∯S(β)D·dS(β).Hence, Ampere’s law in the fractal field is expressed as follows:(27)∮l(α)H·dl(α)=∬S(β)(Ja+∂αD∂tα)·dS(β).


#### 3.2.3. Faraday’s Law in the Fractal Electric Field

Mathematically, Faraday’s law in the local fractional field is expressed as(28)E=-∂αφ∂tα,
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
(29)∇α×E=limdS(β)→0(1dS(β)∮l(α)E·dl(α))nP=-∂α∂tα(∇α×φ)=-∂αB∂tα,
where B is the magnetic induction in the fractal field.In view of (9), we rewrite (29) as
(30)∮l(α)E·dl(α)=∬S(β)(∇α×E)·dS(β),∬S(β)(∇α×E)·dS(β)=∬S(β)(-∂αB∂tα)·dS(β)=-∂α∂tα∬S(β)B·dS(β).So, from (30), Faraday’s law in the fractal field reads as
(31)∮l(α)E·dl(α)+∂α∂tα∬S(β)B·dS(β)=0.


#### 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,
(32)∇α·B=limdV(γ)→0(1dV(γ)∯S(2α)B·dS(β))=0.Furthermore, the magnetic Gauss’ law for the fractal magnetic field reads as(33)∯S(β)B·dS(β)=0.


#### 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
(34)D=εfE,H=μfB,
where εf is the fractal dielectric permittivity and μf is the fractal magnetic permeability.


## 3.1. Charge Conservations in Local Fractional Field

Let us consider the total charge, which is described as follows:(11)Q=∭V(γ)ρ(r,t)dV(γ),
and the total electrical current is as follows(12)I=∬S(β)J(r,t)·dS(β),
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(13)DαQDtα=DαDtα∭V(γ)ρ(r,t)dV(γ)=∭V(γ)∂αρ(r,t)∂tαdV(γ)+∯S(β)ρ(r,t)υ·dS(β)=0,
which leads to
(14)∭V(γ)∂αρ(r,t)∂tαdV(γ)+∯S(β)ρ(r,t)υ·dS(β)=0.From (14), we have
(15)∭V(γ)∂αρ(r,t)∂tαdV(γ)+∯S(β)J(r,t)·dS(β)=∭V(γ)(∂αρ(r,t)∂tα+∇α·J(r,t))dV(γ)=0,
where J=ρυ represents the current density in the fractal field.Hence, from (15), we get
(16)∂αρ(r,t)∂tα+∇α·J(r,t)=0.By analogy with electric charge density in the fractal field, we obtain the conservation of fractal magnetic charge, namely,(17)∂αρm(r,t)∂tα+∇α·Jm(r,t)=0,
where ρm(r,t) is the fractal magnetic charge density and Jm(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
(18)∇α·D=limdV(γ)→0(1dV(γ)∯S(2α)D·dS(β))=ρ,
where D is electric displacement in the fractal electric field.From (7), (18) becomes
(19)∭V(γ)∇α·DdV(γ)=∯S(β)D·dS(β),∭V(γ)∇α·DdV(γ)=∭V(γ)ρdV(γ).Hence, we obtain Gauss’s law for the fractal electric field in the form(20)∯S(β)D·dS(β)=∭V(γ)ρdV(γ).


### 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]
(21)∮l(α)H·dl(α)=I,
where H is the magnetic field strength in the fractal field.The current density in the fractal field can be written as(22)∇α×H=limdS(β)→0(1dS(β)∮l(α)H·dl(α))nP=J,
which leads to
(23)∮l(α)H·dl(α)=∬S(β)(∇α×H)·dS(β),∬S(β)J·dS(β)=∬S(β)(∇α×H)·dS(β).The total current in the fractal fried reads(24)I=∬S(β)(Ja+Jb)·dS(β),
where Ja is the conductive current and
(25)Jb=∂αD∂tα,
which satisfies the following condition:
(26)Ib=∯S(β)Jb·dS(β)=∯S(β)∂α∂tαD·dS(β)=∂α∂tα∯S(β)D·dS(β).Hence, Ampere’s law in the fractal field is expressed as follows:(27)∮l(α)H·dl(α)=∬S(β)(Ja+∂αD∂tα)·dS(β).


### 3.2.3. Faraday’s Law in the Fractal Electric Field

Mathematically, Faraday’s law in the local fractional field is expressed as(28)E=-∂αφ∂tα,
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
(29)∇α×E=limdS(β)→0(1dS(β)∮l(α)E·dl(α))nP=-∂α∂tα(∇α×φ)=-∂αB∂tα,
where B is the magnetic induction in the fractal field.In view of (9), we rewrite (29) as
(30)∮l(α)E·dl(α)=∬S(β)(∇α×E)·dS(β),∬S(β)(∇α×E)·dS(β)=∬S(β)(-∂αB∂tα)·dS(β)=-∂α∂tα∬S(β)B·dS(β).So, from (30), Faraday’s law in the fractal field reads as
(31)∮l(α)E·dl(α)+∂α∂tα∬S(β)B·dS(β)=0.


### 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,
(32)∇α·B=limdV(γ)→0(1dV(γ)∯S(2α)B·dS(β))=0.Furthermore, the magnetic Gauss’ law for the fractal magnetic field reads as(33)∯S(β)B·dS(β)=0.


### 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
(34)D=εfE,H=μfB,
where εf is the fractal dielectric permittivity and μf is the fractal magnetic permeability.


## 3.2.1. Gauss’s Law for the Fractal Electric Field

From (3), the electric charge density can be written as
(18)∇α·D=limdV(γ)→0(1dV(γ)∯S(2α)D·dS(β))=ρ,
where D is electric displacement in the fractal electric field.From (7), (18) becomes
(19)∭V(γ)∇α·DdV(γ)=∯S(β)D·dS(β),∭V(γ)∇α·DdV(γ)=∭V(γ)ρdV(γ).Hence, we obtain Gauss’s law for the fractal electric field in the form(20)∯S(β)D·dS(β)=∭V(γ)ρdV(γ).


## 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]
(21)∮l(α)H·dl(α)=I,
where H is the magnetic field strength in the fractal field.The current density in the fractal field can be written as(22)∇α×H=limdS(β)→0(1dS(β)∮l(α)H·dl(α))nP=J,
which leads to
(23)∮l(α)H·dl(α)=∬S(β)(∇α×H)·dS(β),∬S(β)J·dS(β)=∬S(β)(∇α×H)·dS(β).The total current in the fractal fried reads(24)I=∬S(β)(Ja+Jb)·dS(β),
where Ja is the conductive current and
(25)Jb=∂αD∂tα,
which satisfies the following condition:
(26)Ib=∯S(β)Jb·dS(β)=∯S(β)∂α∂tαD·dS(β)=∂α∂tα∯S(β)D·dS(β).Hence, Ampere’s law in the fractal field is expressed as follows:(27)∮l(α)H·dl(α)=∬S(β)(Ja+∂αD∂tα)·dS(β).


## 3.2.3. Faraday’s Law in the Fractal Electric Field

Mathematically, Faraday’s law in the local fractional field is expressed as(28)E=-∂αφ∂tα,
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
(29)∇α×E=limdS(β)→0(1dS(β)∮l(α)E·dl(α))nP=-∂α∂tα(∇α×φ)=-∂αB∂tα,
where B is the magnetic induction in the fractal field.In view of (9), we rewrite (29) as
(30)∮l(α)E·dl(α)=∬S(β)(∇α×E)·dS(β),∬S(β)(∇α×E)·dS(β)=∬S(β)(-∂αB∂tα)·dS(β)=-∂α∂tα∬S(β)B·dS(β).So, from (30), Faraday’s law in the fractal field reads as
(31)∮l(α)E·dl(α)+∂α∂tα∬S(β)B·dS(β)=0.


## 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,
(32)∇α·B=limdV(γ)→0(1dV(γ)∯S(2α)B·dS(β))=0.Furthermore, the magnetic Gauss’ law for the fractal magnetic field reads as(33)∯S(β)B·dS(β)=0.


## 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
(34)D=εfE,H=μfB,
where εf is the fractal dielectric permittivity and μf 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]:
(35)xα=Rαcosαθα,yα=Rαsinαθα,zα=zα,
with R∈(0,+∞), z∈(-∞,+∞), θ∈(0,π], and x2α+y2α=R2α.Making use of (35), we have
(36)∇α·r=∂αrR∂Rα+1Rα∂αrθ∂θα+rRRα+∂αrz∂zα,∇α×r=(1Rα∂αrθ∂θα-∂αrθ∂zα)eRα+(∂αrR∂zα-∂αrz∂Rα)eθα+(∂αrθ∂Rα+rRRα-1Rα∂αrR∂θα)ezα,
where
(37)r=Rαcosαθαe1α+Rαsinαθαe2α+zαe3α=rReRα+rθeθα+rzezα.From (7) and (20), the local fractional differential form of Gauss’s law for the fractal electric field is expressed by
(38)∇α·D=ρ,
which leads to
(39)∂αDR∂Rα+1Rα∂αDθ∂θα+DRRα+∂αDz∂zα=ρ(R,θ,z,t),
where D=DReRα+Dθeθα+Dzezα.In view of (9) and (27), we can present the local fractional differential forms of Ampere’s law in the fractal magnetic field
(40)∇α×H=Ja+∂αD∂tα,(1Rα∂αHθ∂θα-∂αHθ∂zα)eRα+(∂αHR∂zα-∂αHz∂Rα)eθα+(∂αHθ∂Rα+HRRα-1Rα∂αHR∂θα)ezα=Ja(R,θ,z,t)+∂αD(R,θ,z,t)∂tα,
where  H=HReRα+Hθeθα+Hzezα.Using (9) and (31), the local fractional differential forms of Faraday’s law in the fractal electric field can be written as
(41)∇α×E=-∂αB∂tα,(1Rα∂αEθ∂θα-∂αEθ∂zα)eRα+(∂αER∂zα-∂αEz∂Rα)eθα+(∂αEθ∂Rα+ERRα-1Rα∂αER∂θα)ezα=-∂αB(R,θ,z,t)∂tα,
where E=EReRα+Eθeθα+Ezezα.From (7) and (33), the local fractional differential forms of the magnetic Gauss law for the fractal magnetic field read as follows:
(42)∇α·B=0,∂αBR∂Rα+1Rα∂αBθ∂θα+BRRα+∂αBz∂zα=0,
where B=BReRα+Bθeθα+Bzezα.


## 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].



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