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

Yang Zhao,De-Fu Cheng,Carlo Cattani,Dumitru Baleanu,Xiao-Jun Yang

doi:10.1155/2013/686371

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.

Highlights

Nondifferentiability, complexity, and similarity represent the basic properties of the nature
In [27], the Navier-Stokes equations on Cantor sets based on local fractional vector calculus were proposed
We recall the basic definitions and theorems for local fractional vector calculus, which are used throughout the paper

Summary

Introduction

Nondifferentiability, complexity, and similarity represent the basic properties of the nature. 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 were proposed in [15]. Based on the Hausdorff derivative, fractal continuum electrodynamics in time-space Ω4α ⊂ Ω4 was proposed [18]. In [27], the Navier-Stokes equations on Cantor sets based on local fractional vector calculus were proposed. 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.

Fundaments

Local Fractional Integral Forms of Maxwell’s Equations on Cantor Sets

Local Fractional Differential Forms of Maxwell’s Equations on Cantor Sets

Conclusions