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Electrodynamics of Fractal Distributions of Charges and Fields

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Fractional Dynamics

Part of the book series: Nonlinear Physical Science ((NPS,volume 0))

Abstract

Joseph Liouville was a pioneer in development of fractional calculus to electrodynamics (Lutzen, 1985). The theory of fractional derivatives and integrals (Kilbas et al., 2006; Samko et al., 1993) can be applied to several specific electromagnetic problems (see, for example, (Engheta, 1997; Zelenyi and Milovanov, 2004; Milovanov, 2009; Potapov, 2005; Tarasov, 2008, 2009; Bogolyubov et al., 2009)). In this chapter, we consider electrodynamics of fractal distribution of charges and fields in the framework of fractional continuous model (Tarasov, 2005a,b, 2006a,b).

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References

  • A.N. Bogolyubov, A.A. Potapov, S.Sh. Rehviashvili, 2009, An approach to introducing fractional integro-differentiation in classical electrodynamics, Moscow University Physics Bulletin, 64, 365–368.

    Article  ADS  MATH  Google Scholar 

  • G. Calcagni, 2010, Quantum Field Theory, Gravity and Cosmology in a Fractal Universe, E-print: arXiv: 1001.0571.

    Google Scholar 

  • R.M. Christensen, 2005, Mechanics of Composite Materials, Dover, New York.

    Google Scholar 

  • S.R. De Groot, L.G. Suttorp, 1972, Foundation of Electrodynamics, North-Holland, Amsterdam.

    Google Scholar 

  • N. Engheta, 1997, On the role of fractional calculus in electromagnetic theory, Antennas and Propagation Magazine, 39, 35–46.

    Article  ADS  Google Scholar 

  • J.D. Jackson, 1998, Classical Electrodynamics, 3rd ed., Wiley, New York.

    Google Scholar 

  • A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.

    MATH  Google Scholar 

  • A.G. Kulikovskiy, G.A. Lyubimov, 1965, Magnetohydrodynamics, Addison Wesley, Massachusetts; Translated from Russian: Nauka, Moscow, 1964.

    Google Scholar 

  • J. Lutzen, 1985, Liouville’s differential calculus of arbitrary order and its electrodynamical origin, in Proc. 19th Nordic Congress Mathenzaticians, Icelandic Mathematical Society, Reykjavik, 149–160.

    Google Scholar 

  • M. Materassi, G. Consolini, 2007, Magnetic reconnection rate in space plasmas: A fractal approach, Physical Review Letters, 99, 175002.

    Article  ADS  Google Scholar 

  • A.V. Milovanov, 2009, Pseudochaos and low-frequency percolation scaling for turbulent diffusion in magnetized plasma, Physical Rewiew E, 79, 046403.

    Article  ADS  Google Scholar 

  • A.A. Potapov, 2005, Fractals in Radiophysics and Radiolocation, 2nd ed., Universitetskaya Kniga, Moscow. In Russian.

    Google Scholar 

  • S.G. Samko, A.A. Kilbas, O.I. Marichev, 1993, Integrals and Derivatives of Fractional Order and Applications, Nauka i Tehnika, Minsk, 1987, in Russian; and Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York, 1993.

    Google Scholar 

  • V.E. Tarasov, 2005a, Electromagnetic field of fractal distribution of charged particles, Physics of Plasmas, 12, 082106.

    Article  ADS  Google Scholar 

  • V.E. Tarasov, 2005b, Multipole moments of fractal distribution of charges, Modern Physics Letters B, 19, 1107–1118.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • V.E. Tarasov, 2006a, Magnetohydrodynamics of fractal media, Physics of Plasmas, 13, 052107.

    Article  MathSciNet  ADS  Google Scholar 

  • V.E. Tarasov, 2006b, Electromagnetic fields on fractals, Modern Physics Letters A, 21, 1587–1600.

    Article  ADS  MATH  Google Scholar 

  • V.E. Tarasov, 2006c, Gravitational field of fractal distribution of particles, Celestial Mechanics and Dynamical Astronomy, 94, 1–15.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • V.E. Tarasov, 2008, Fractional vector calculus and fractional Maxwell’s equations, Annals of Physics, 323, 2756–2778.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • V.E. Tarasov, 2009, Fractional integro-differential equations for electromagnetic waves in dielectric media, Theoretical and Mathematical Physics, 158, 355–359.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • L.M. Zelenyi, A.V. Milovanov, 2004, Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics, Physics Uspekhi, 47, 749–788.

    Article  Google Scholar 

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Tarasov, V.E. (2010). Electrodynamics of Fractal Distributions of Charges and Fields. In: Fractional Dynamics. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14003-7_4

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