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Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013, Article ID 686371, 6 pageshttp://dx.doi.org/10.1155/2013/686371

Research ArticleMaxwell’s Equations on Cantor Sets: A LocalFractional Approach

Yang Zhao,1,2 Dumitru Baleanu,3,4,5 Carlo Cattani,6 De-Fu Cheng,1 and Xiao-Jun Yang7

1 College of Instrumentation & Electrical Engineering, Jilin University, Changchun 130061, China2 Electronic and Information Technology Department, Jiangmen Polytechnic, Jiangmen 529090, China3Department 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, Turkey5 Institute of Space Sciences, Magurele, 077125 Bucharest, Romania6Department of Mathematics, University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy7 Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou,Jiangsu 221008, China

Correspondence should be addressed to De-Fu Cheng; [email protected]

Received 3 October 2013; Accepted 7 November 2013

Academic Editor: Gongnan Xie

Copyright © 2013 Yang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1. Introduction

Nondifferentiability, complexity, and similarity represent thebasic properties of the nature. Fractals [1] are the basiccharacteristics of nature, which are that fractal geometryof substances generalizes to noninteger dimensions. Micro-physics reveals the fractal behaviors of matter distribution inthe universe [2] and soft materials [3].

Fractal time was used to describe the transport of chargesand defects in the condensedmatter [4]. In fractal space-time,the geometric analogue of relativistic quantum mechanicswas presented in [5–8]. In fractal-Cantorian space-timeΩ4𝛼

⊂ 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 Ω3𝛼+1 ⊂ Ω4 were

developed in [14] and fractional Maxwell’s equations wereproposed in [15]. In [16, 17], the concept of static fractional

electric potential was developed. Recently, based on theHausdorff derivative, fractal continuum electrodynamics intime-space Ω

4𝛼⊂ Ω4 was proposed [18]. The fractional

differential form of Maxwell’s equations on fractal sets wassuggested in [19]. In [20], theMaxwell equations of fractionalelectrodynamics in time-space Ω3+𝛼 ⊂ Ω

4 were considered.The Maxwell equations on anisotropic fractal media in time-space Ω3𝛼+1 ⊂ Ω

4 were developed in [21].The local fractional calculus theory [22, 23] was applied

to model some dynamics systems with nondifferentiablecharacteristics. In [22–26], the heat-conduction equationon Cantor sets was considered. In [27], the Navier-Stokesequations on Cantor sets based on local fractional vectorcalculus were proposed. Helmholtz and diffusion equationsvia local fractional vector calculus were reported in [28]. TheFokker-Planck equation with local fractional space derivativewas suggested in [29]. In [30], Lagrangian and Hamiltonian

2 Advances in High Energy Physics

mechanics with local fractional space derivative was pre-sented.Themeasuring 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 equationson Cantor sets from the local fractional calculus theory [23,27, 28] point of view. This paper is structured as follows. InSection 2, we introduce the basic definitions and theoremsfor local fractional vector calculus. In Section 3, Maxwell’sequations onCantor sets in the local fractional vector integralform are presented.Maxwell’s equations on Cantor sets in theCantorian and Cantor-type cylindrical coordinates are givenin Section 4. Finally, Section 5 is devoted to conclusions.

2. Fundaments

In this section, we recall the basic definitions and theoremsfor local fractional vector calculus, which are used through-out the paper.

Local fractional gradient of the scale function 𝜙 is definedas [23, 27]

∇𝛼𝜙 = lim𝑑𝑉(𝛾)→0

(1

𝑑𝑉(𝛾)∯𝑆(𝛽)

𝜙𝑑S(𝛽)) , (1)

where S(𝛽) is its bounding fractal surface,𝑉 is a small fractalvolume enclosing 𝑃, and the local fractional surface integralis given by [23, 27, 28]

∬𝑢(𝑟𝑃) 𝑑S(𝛽) = lim

𝑁→∞

𝑁

∑

𝑃=1

𝑢 (𝑟𝑃)n𝑃Δ𝑆(𝛽)

𝑃, (2)

with 𝑁 elements of area with a unit normal local fractionalvector n

𝑃, Δ𝑆(𝛽)𝑃

→ 0 as𝑁 → ∞ 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 = lim𝑑𝑉(𝛾)→0

(1

𝑑𝑉(𝛾)∯𝑆(2𝛼)

u ⋅ 𝑑S(𝛽)) , (3)

where the local fractional surface integral is suggested by [23,27, 28]

∬ u (𝑟𝑃) ⋅ 𝑑S(𝛽) = lim

𝑁→∞

𝑁

∑

𝑃=1

u (𝑟𝑃) ⋅ n𝑃Δ𝑆(𝛽)

𝑃, (4)

with 𝑁 elements of area with a unit normal local fractionalvector n

𝑃, Δ𝑆(𝛽)𝑃

→ 0 as𝑁 → ∞ 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

(1

𝑑𝑆(𝛽)∮𝑙(𝛼)

u ⋅ 𝑑l(𝛼))n𝑃, (5)

where the local fractional line integral of the function u alonga fractal line 𝑙𝛼 is given by [23]

∫𝑙(𝛼)

u (𝑥𝑃, 𝑦𝑃, 𝑧𝑃) ⋅ 𝑑l(𝛼)

= lim𝑁→∞

𝑁

∑

𝑃=1

u (𝑥𝑃, 𝑦𝑃, 𝑧𝑃) ⋅ Δl(𝛼)𝑃,

(6)

Fractal spaceFractional space Euclidean space

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ta

t

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

with the elements of line Δl(𝛼)𝑃

requiring that all |Δ𝑙(𝛼)𝑃| → 0

as𝑁 → ∞ 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 iswritten as [23]

∭ u (𝑟𝑃) 𝑑𝑉(𝛾)

= lim𝑁→∞

𝑁

∑

𝑃=1

u (𝑟𝑃) Δ𝑉(𝛾)

𝑃, (8)

with the elements of volume Δ𝑉(𝛾)𝑃

→ 0 as 𝑁 → ∞ and𝛾 = (3/2)𝛽 = 3𝛼.

The local fractional Stokes theorem of the fractal fieldstates that [23, 27]

∮𝑙(𝛼)

u ⋅ 𝑑l(𝛼) = ∬𝑆(𝛽)

(∇𝛼× u) ⋅ 𝑑S(𝛽). (9)

The Reynolds transport theorem of the local fractional vectorfield states that [27]𝐷𝛼

𝐷𝑡𝛼∭𝑉(𝛾)

𝐹 (𝑥, 𝑡) 𝑑𝑉(𝛾)

= ∭𝑉(𝛾)

𝜕𝛼

𝜕𝑡𝛼𝐹 (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 ofMaxwell’s Equations on Cantor Sets

According to fractional complex transform method [31], thefact that the classical differential equations always transform

Advances in High Energy Physics 3

into the local fractional differential equations leads to the ideaof yieldingMaxwell’s equations on Cantor sets using the localfractional vector calculus.

3.1. Charge Conservations in Local Fractional Field. Let usconsider the total charge, which is described as follows:

𝑄 = ∭𝑉(𝛾)

𝜌 (𝑟, 𝑡) 𝑑𝑉(𝛾), (11)

and the total electrical current is as follows

𝐼 = ∬𝑆(𝛽)

𝐽 (𝑟, 𝑡) ⋅ 𝑑S(𝛽), (12)

where 𝜌(𝑟, 𝑡) is the fractal electric charge density and 𝐽(𝑟, 𝑡) isthe fractal electric current density.

The Reynolds transport theorem in the fractal field gives

𝐷𝛼𝑄

𝐷𝑡𝛼=

𝐷𝛼

𝐷𝑡𝛼∭𝑉(𝛾)

𝜌 (𝑟, 𝑡) 𝑑𝑉(𝛾)

= ∭𝑉(𝛾)

𝜕𝛼𝜌 (𝑟, 𝑡)

𝜕𝑡𝛼𝑑𝑉(𝛾)

+∯𝑆(𝛽)

𝜌 (𝑟, 𝑡) 𝜐 ⋅ 𝑑S(𝛽)

= 0,

(13)

which leads to

∭𝑉(𝛾)



+∯𝑆(𝛽)

𝜌 (𝑟, 𝑡) 𝜐 ⋅ 𝑑S(𝛽) = 0. (14)

From (14), we have

∭𝑉(𝛾)



+∯𝑆(𝛽)

𝐽 (𝑟, 𝑡) ⋅ 𝑑S(𝛽)

= ∭𝑉(𝛾)

(𝜕𝛼𝜌 (𝑟, 𝑡)

𝜕𝑡𝛼+ ∇𝛼⋅ 𝐽 (𝑟, 𝑡)) 𝑑𝑉

(𝛾)

= 0,

(15)

where 𝐽 = 𝜌𝜐 represents the current density in the fractalfield.

Hence, from (15), we get


𝜕𝑡𝛼+ ∇𝛼⋅ 𝐽 (𝑟, 𝑡) = 0. (16)

By analogy with electric charge density in the fractalfield, we obtain the conservation of fractal magnetic charge,namely,

𝜕𝛼𝜌𝑚(𝑟, 𝑡)

𝜕𝑡𝛼+ ∇𝛼⋅ 𝐽𝑚(𝑟, 𝑡) = 0, (17)

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. Wenow derive Maxwell’s equations on Cantor set based on thelocal fractional vector calculus.

3.2.1. Gauss’s Law for the Fractal Electric Field. From (3), theelectric charge density can be written as

∇𝛼⋅ 𝐷 = lim

𝑑𝑉(𝛾)→0

(1


𝐷 ⋅ 𝑑S(𝛽)) = 𝜌, (18)

where𝐷 is electric displacement in the fractal electric field.From (7), (18) becomes

∭𝑉(𝛾)

∇𝛼⋅ 𝐷 𝑑𝑉

(𝛾)= ∯𝑆(𝛽)

𝐷 ⋅ 𝑑S(𝛽),

∭𝑉(𝛾)

∇𝛼⋅ 𝐷 𝑑𝑉

(𝛾)= ∭

𝑉(𝛾)

𝜌 𝑑𝑉(𝛾).

(19)

Hence, we obtain Gauss’s law for the fractal electric fieldin the form

∯𝑆(𝛽)

𝐷 ⋅ 𝑑S(𝛽) = ∭𝑉(𝛾)

𝜌𝑑𝑉(𝛾). (20)

3.2.2. Ampere’s Law in the Fractal Magnetic Field. Mathe-matically, Ampere’s law in the fractal magnetic field can besuggested as [23]

∮𝑙(𝛼)

𝐻 ⋅ 𝑑l(𝛼) = 𝐼, (21)

where𝐻 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

(1


𝐻 ⋅ 𝑑l(𝛼))n𝑃= 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)

(𝐽𝑎+ 𝐽𝑏) ⋅ 𝑑S(𝛽), (24)

where 𝐽𝑎is the conductive current and

𝐽𝑏=𝜕𝛼𝐷

𝜕𝑡𝛼, (25)

which satisfies the following condition:

𝐼𝑏= ∯𝑆(𝛽)

𝐽𝑏⋅ 𝑑S(𝛽)

= ∯𝑆(𝛽)

𝜕𝛼

𝜕𝑡𝛼𝐷 ⋅ 𝑑S(𝛽)

=𝜕𝛼

𝜕𝑡𝛼∯𝑆(𝛽)

𝐷 ⋅ 𝑑S(𝛽).

(26)

Hence, Ampere’s law in the fractal field is expressed asfollows:

∮𝑙(𝛼)

𝐻 ⋅ 𝑑l(𝛼) = ∬𝑆(𝛽)

(𝐽𝑎+𝜕𝛼𝐷

𝜕𝑡𝛼) ⋅ 𝑑S(𝛽). (27)


3.2.3. Faraday’s Law in the Fractal Electric Field. Mathemat-ically, Faraday’s law in the local fractional field is expressedas

𝐸 = −𝜕𝛼𝜑

𝜕𝑡𝛼, (28)

where 𝜑 is the magnetic potential in the fractal field and 𝐸 isthe electrical field strength in the fractal field.

From (28), we have

∇𝛼× 𝐸 = lim

𝑑𝑆(𝛽)→0

(1


𝐸 ⋅ 𝑑l(𝛼))n𝑃

= −𝜕𝛼

𝜕𝑡𝛼(∇𝛼× 𝜑) = −

𝜕𝛼𝐵

𝜕𝑡𝛼,

(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

𝜕𝑡𝛼) ⋅ 𝑑S(𝛽)

= −𝜕𝛼

𝜕𝑡𝛼∬𝑆(𝛽)

𝐵 ⋅ 𝑑S(𝛽).

(30)

So, from (30), Faraday’s law in the fractal field reads as

∮𝑙(𝛼)

𝐸 ⋅ 𝑑l(𝛼) + 𝜕𝛼

𝜕𝑡𝛼∬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 themagnetic induction in the fractal field, namely,

∇𝛼⋅ 𝐵 = lim𝑑𝑉(𝛾)→0

(1


𝐵 ⋅ 𝑑S(𝛽)) = 0. (32)

Furthermore, the magnetic Gauss’ law for the fractalmagnetic field reads as

∯𝑆(𝛽)

𝐵 ⋅ 𝑑S(𝛽) = 0. (33)

3.2.5. The Constitutive Equations in the Fractal Field. Similarto the constitutive relations in fractal continuous mediummechanics [23], the constitutive relationships in fractal elec-tromagnetic can be written as

𝐷 = 𝜀𝑓𝐸,

𝐻 = 𝜇𝑓𝐵,

(34)

where 𝜀𝑓is the fractal dielectric permittivity and 𝜇

𝑓is the

fractal magnetic permeability.

4. Local Fractional Differential Forms ofMaxwell’s Equations on Cantor Sets

In this section, we investigate the local fractional differentialforms of Maxwell’s equations on Cantor sets.

The Cantor-type cylindrical coordinates can be written asfollows [26]:

𝑥𝛼= 𝑅𝛼cos𝛼𝜃𝛼,

𝑦𝛼= 𝑅𝛼sin𝛼𝜃𝛼,

𝑧𝛼= 𝑧𝛼,

(35)

with 𝑅 ∈ (0, +∞), 𝑧 ∈ (−∞, +∞), 𝜃 ∈ (0, 𝜋], and 𝑥2𝛼 + 𝑦2𝛼

=

𝑅2𝛼.Making use of (35), we have

∇𝛼⋅ r = 𝜕

𝛼𝑟𝑅

𝜕𝑅𝛼+

1

𝑅𝛼

𝜕𝛼𝑟𝜃

𝜕𝜃𝛼+

𝑟𝑅

𝑅𝛼+𝜕𝛼𝑟𝑧

𝜕𝑧𝛼,

∇𝛼× r = (

1

𝑅𝛼

𝜕𝛼𝑟𝜃

𝜕𝜃𝛼−𝜕𝛼𝑟𝜃

𝜕𝑧𝛼) e𝛼𝑅

+ (𝜕𝛼𝑟𝑅

𝜕𝑧𝛼−𝜕𝛼𝑟𝑧

𝜕𝑅𝛼) e𝛼𝜃

+ (𝜕𝛼𝑟𝜃

𝜕𝑅𝛼+

𝑟𝑅

𝑅𝛼−

1

𝑅𝛼

𝜕𝛼𝑟𝑅

𝜕𝜃𝛼) e𝛼𝑧,

(36)

where

r = 𝑅𝛼cos𝛼𝜃𝛼e𝛼1+ 𝑅𝛼sin𝛼𝜃𝛼e𝛼2+ 𝑧𝛼e𝛼3

= 𝑟𝑅e𝛼𝑅+ 𝑟𝜃e𝛼𝜃+ 𝑟𝑧e𝛼𝑧.

(37)

From (7) and (20), the local fractional differential form ofGauss’s law for the fractal electric field is expressed by

∇𝛼⋅ 𝐷 = 𝜌, (38)

which leads to

𝜕𝛼𝐷𝑅

𝜕𝑅𝛼+

1

𝑅𝛼

𝜕𝛼𝐷𝜃

𝜕𝜃𝛼+𝐷𝑅

𝑅𝛼+𝜕𝛼𝐷𝑧

𝜕𝑧𝛼= 𝜌 (𝑅, 𝜃, 𝑧, 𝑡) , (39)

where𝐷 = 𝐷𝑅e𝛼𝑅+ 𝐷𝜃e𝛼𝜃+ 𝐷𝑧e𝛼𝑧.

In view of (9) and (27), we can present the local fractionaldifferential forms of Ampere’s law in the fractalmagnetic field

∇𝛼× 𝐻 = 𝐽

𝑎+𝜕𝛼𝐷

𝜕𝑡𝛼,

(1

𝑅𝛼

𝜕𝛼𝐻𝜃

𝜕𝜃𝛼−𝜕𝛼𝐻𝜃

𝜕𝑧𝛼) e𝛼𝑅+ (

𝜕𝛼𝐻𝑅

𝜕𝑧𝛼−𝜕𝛼𝐻𝑧


+ (𝜕𝛼𝐻𝜃

𝜕𝑅𝛼+𝐻𝑅

𝑅𝛼−

1

𝑅𝛼

𝜕𝛼𝐻𝑅

𝜕𝜃𝛼) e𝛼𝑧

= 𝐽𝑎(𝑅, 𝜃, 𝑧, 𝑡) +

𝜕𝛼𝐷 (𝑅, 𝜃, 𝑧, 𝑡)

𝜕𝑡𝛼,

(40)

where𝐻 = 𝐻𝑅e𝛼𝑅+ 𝐻𝜃e𝛼𝜃+ 𝐻𝑧e𝛼𝑧.

Advances in High Energy Physics 5

Using (9) and (31), the local fractional differential formsof Faraday’s law in the fractal electric field can be written as

∇𝛼× 𝐸 = −

𝜕𝛼𝐵

𝜕𝑡𝛼,

(1

𝑅𝛼

𝜕𝛼𝐸𝜃

𝜕𝜃𝛼−𝜕𝛼𝐸𝜃

𝜕𝑧𝛼) e𝛼𝑅+ (

𝜕𝛼𝐸𝑅

𝜕𝑧𝛼−𝜕𝛼𝐸𝑧


+ (𝜕𝛼𝐸𝜃

𝜕𝑅𝛼+𝐸𝑅

𝑅𝛼−

1

𝑅𝛼

𝜕𝛼𝐸𝑅

𝜕𝜃𝛼) e𝛼𝑧

= −𝜕𝛼𝐵 (𝑅, 𝜃, 𝑧, 𝑡)

𝜕𝑡𝛼,

(41)

where 𝐸 = 𝐸𝑅e𝛼𝑅+ 𝐸𝜃e𝛼𝜃+ 𝐸𝑧e𝛼𝑧.

From (7) and (33), the local fractional differential formsof the magnetic Gauss law for the fractal magnetic field readas follows:

∇𝛼⋅ 𝐵 = 0,

𝜕𝛼𝐵𝑅

𝜕𝑅𝛼+

1

𝑅𝛼

𝜕𝛼𝐵𝜃

𝜕𝜃𝛼+𝐵𝑅

𝑅𝛼+𝜕𝛼𝐵𝑧

𝜕𝑧𝛼= 0,

(42)

where 𝐵 = 𝐵𝑅e𝛼𝑅+ 𝐵𝜃e𝛼𝜃+ 𝐵𝑧e𝛼𝑧.

5. Conclusions

In this work, we proposed the local fractional approachfor Maxwell’s equations on Cantor sets based on the localfractional vector calculus. Employing the local fractionaldivergence and curl of the vector function, we deducedMaxwell’s equations on Cantor sets. The local fractionaldifferential forms of Maxwell’s equations on Cantor sets inthe Cantorian and Cantor-type cylindrical coordinates werediscussed. Finding a formulation of Maxwell’s equations onCantor set within local fractional operators is the first steptowards generalizing a simple field equation which allows theunification ofMaxwell’s equations to the standardmodel withthe dynamics of cold darkmatter.Wenoticed that the classicalcase was debated in [32].

Conflict of Interests

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

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