Electromagnetic Fields on Fractals

TL;DR

This paper explores the behavior of electromagnetic fields on fractal structures by employing fractional calculus to generalize Maxwell's equations, demonstrating that fractals can be modeled as unique media with fractional integrals.

Contribution

It introduces a fractional calculus approach to Maxwell's equations on fractals, providing a new framework for understanding electromagnetic phenomena on complex geometries.

Findings

Fractals can be modeled as specific media using fractional integrals.

Fractional Maxwell equations approximate electromagnetic behavior on fractals.

Fractals are characterized as measurable metric sets with non-integer Hausdorff dimensions.

Abstract

Fractals are measurable metric sets with non-integer Hausdorff dimensions. If electric and magnetic fields are defined on fractal and do not exist outside of fractal in Euclidean space, then we can use the fractional generalization of the integral Maxwell equations. The fractional integrals are considered as approximations of integrals on fractals. We prove that fractal can be described as a specific medium.