Electromagnetism on anisotropic fractal media.

*(English)*Zbl 1276.78002The basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows for a generalization of the Green-Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, the authors derive the Faraday and Ampere laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, they employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell’s electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity. This work is very interesting. It is an excellent article, and is deserved to be read.

Reviewer: Guanggan Chen (Chengdu)

##### MSC:

78A25 | Electromagnetic theory (general) |

28A80 | Fractals |

35Q61 | Maxwell equations |

35Q60 | PDEs in connection with optics and electromagnetic theory |

##### Keywords:

fractal media; anisotropy electromagnetism; Maxwell’s equations; Faraday’s law; Ampère’s law; variational principle; electromagnetic energy; stress
PDF
BibTeX
XML
Cite

\textit{M. Ostoja-Starzewski}, Z. Angew. Math. Phys. 64, No. 2, 381--390 (2013; Zbl 1276.78002)

**OpenURL**

##### References:

[1] | Ostoja-Starzewski, M.; Li, J., Fractal materials, beams and fracture mechanics, ZAMP, 60, 1194-1205, (2009) · Zbl 1319.74002 |

[2] | Li, J., Ostoja-Starzewski, J.: Fractal solids, product measures and fractional wave equations. Proc. R. Soc. A 465, 2521-2536 (2009); Errata (2010) · Zbl 1186.74011 |

[3] | Ignaczak J., Ostoja-Starzewski M.: Thermoelasticity with Finite Wave Speeds. Oxford University Press, Oxford (2009) · Zbl 1183.80001 |

[4] | Demmie, P.N.; Ostoja-Starzewski, M., Waves in fractal media, J. Elast., 104, 187-204, (2011) · Zbl 1311.74057 |

[5] | Li, J.; Ostoja-Starzewski, M., Micropolar continuum mechanics of fractal media, Int. J. Eng. Sci., 49, 1302-1310, (2011) · Zbl 1423.74040 |

[6] | Joumaa, H.; Ostoja-Starzewski, M., On the wave propagation in isotropic fractal media, ZAMP, 62, 1117-1129, (2011) · Zbl 1291.74094 |

[7] | Tarasov, V.E., Electromagnetic fields on fractals, Mod. Phys. Lett. A, 21, 1587-1600, (2006) · Zbl 1097.78003 |

[8] | Tarasov, V.E., Fractional vector calculus and fractional maxwell’s equations, Ann. Phys., 323, 2756-2778, (2008) · Zbl 1180.78003 |

[9] | Tarasov V.E.: Fractional Dynamics. Springer, HEP, Berlin, Beijing (2010) · Zbl 1214.81004 |

[10] | Falconer K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (2003) · Zbl 1060.28005 |

[11] | Seliger, R.L.; Whitham, G.B., Variational principles in continuum mechanics, Proc. R. Soc. A, 305, 1-25, (1968) · Zbl 0198.57601 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.