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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.

Nondifferentiability, complexity, and similarity represent the basic properties of the nature. Fractals [

Fractal time was used to describe the transport of charges and defects in the condensed matter [

Based on the fractal distribution of charged particles, the electric and magnetic fields in time-space

The local fractional calculus theory [

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 [

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

The local fractional divergence of the vector function

The local fractional curl of the vector function

The local fractional Gauss theorem of the fractal vector field states that [

The local fractional Stokes theorem of the fractal field states that [

According to fractional complex transform method [

Let us consider the total charge, which is described as follows:

The Reynolds transport theorem in the fractal field gives

From (

Hence, from (

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

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

From (

From (

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

Mathematically, Ampere’s law in the fractal magnetic field can be suggested as [

The current density in the fractal field can be written as

The total current in the fractal fried reads

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

Mathematically, Faraday’s law in the local fractional field is expressed as

From (

In view of (

So, from (

From (

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

Similar to the constitutive relations in fractal continuous medium mechanics [

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 [

Making use of (

From (

In view of (

Using (

From (

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 [

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