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Fractional Wave Equation for Dielectric Medium with Havriliak–Negami Response

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Fractional Dynamics and Control

Abstract

The fractional generalizations of the relaxation equation and the wave equation in dielectrics with the response function of the Havriliak–Negami type are considered. The obtained fractional wave equation is concordant with the asymptotical equations derived by Tarasov VE (J Phys Condens Matter 20:145212, 2008) from Jonscher’s universal law. The explicit expression for the fractional operator in this equation is obtained and the Monte Carlo algorithm for calculation of actions of this operator and of the inverse one is constructed.

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References

  1. Fröhlich H (1958) Theory of dielectrics, 2nd edn. Oxford University Press, Oxford

    MATH  Google Scholar 

  2. Debye P (1954) Polar molecules. Dover, New York

    Google Scholar 

  3. Jonscher AK (1977) The “universal” dielectric response. Nature 267:673

    Article  Google Scholar 

  4. Ramakrishnan TV, Raj Lakshmi M (1987) Non-debye relaxation in condensed matter. World Scientific, Singapore

    Google Scholar 

  5. Cole KS, Cole RH (1941) Dispersion and absorption in dielectrics. J Chem Phys 9:341

    Article  Google Scholar 

  6. Davidson DW, Cole RH (1951) Dielectric relaxation in glycerol, propylene glycol, and n-propanol. J Chem Phys 19:1484

    Article  Google Scholar 

  7. Havriliak S, Negami S (1966) A complex plane analysis of α-dispersions in some polymer systems. J Polymer Sci 14:99

    Google Scholar 

  8. Novikov VV, Wojciechowski KW, Komkova OA, Thiel T (2005) Anomalous relaxation in dielectrics. Equations with fractional derivatives. Mater Sci Poland 23:977

    Google Scholar 

  9. Nigmatullin RR, Ryabov YaE (1997) Cole-Davidson dielectric relaxation as a self-similar relaxation process. Phys Solid State 39:87

    Article  Google Scholar 

  10. Jonscher AK (1996) Universal relaxation law. Chelsea-Dielectrics Press, London

    Google Scholar 

  11. Weron K (1991) How to obtain the universal response law in the Jonscher screened hopping model for dielectric relaxation. Phys Condens Matter 3:221

    Article  Google Scholar 

  12. Weron K, Kotulski M (1996) On the equivalence of the parallel channel and the correlated cluster relaxation models. J Stat Phys 88:1241

    Article  Google Scholar 

  13. Nigmatullin RR (1984) To the theoretical explanation of the “universal response”. Phys Stat Sol b 123:739–745

    Article  Google Scholar 

  14. Glöckle WG, Nonnenmacher TF (1993) Fox function representation of non-Debye relaxation processes. J Stat Phys 71:741

    Article  MATH  Google Scholar 

  15. Jurlewicz A, Weron K (2000) Relaxation dynamics of the fastest channel in multichannel parallel relaxation mechanism. Chaos Solitons Fractals 11:303

    Article  MathSciNet  MATH  Google Scholar 

  16. Coffey WT, Kalmykov YuP, Titov SV (2002) Anomalous dielectric relaxation in the context of the Debye model of noninertial rotational diffusion. J Chem Phys 116:6422

    Article  Google Scholar 

  17. Déjardin J-L (2003) Fractional dynamics and nonlinear harmonic responses in dielectric relaxation of disordered liquids. Phys Rev E 68:031108

    Article  Google Scholar 

  18. Aydiner E (2005) Anomalous rotational relaxation: A fractional Fokker-Planck equation approach. Phys Rev E 71:046103

    Article  Google Scholar 

  19. Weron K, Jurlewicz A, Magdziarz M (2005) Havriliak–Negami response in the framework of the continuous-time random walk. Acta Physica Polonica B 36:1855–1868

    Google Scholar 

  20. Bochner S (1949) Diffusion equation and stochastic processes. Proc Nat Acad Sci USA 35:368–370

    Article  MathSciNet  MATH  Google Scholar 

  21. Phillips RS (1952) On the generation of semigroups of linear operators. Pacific J Math 2:343–369

    MathSciNet  MATH  Google Scholar 

  22. Yosida K (1980) Functional analysis. Springer, New York

    MATH  Google Scholar 

  23. Tarasov VE (2008) Universal electromagnetic waves in dielectric. J Phys Condens Matter 20:175223

    Article  Google Scholar 

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Acknowledgment

The authors are grateful to the Russian Foundation for Basic Research (grant 10-01-00618) for financial support.

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Authors and Affiliations

  1. Ulyanovsk State University, 42 Leo Tolstoy street, Ulyanovsk, Russia

    R. T. Sibatov, V. V. Uchaikin & D. V. Uchaikin

Authors
  1. R. T. Sibatov
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  2. V. V. Uchaikin
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  3. D. V. Uchaikin
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Corresponding author

Correspondence to R. T. Sibatov .

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Editors and Affiliations

  1. Faculty of Art and Sciences, Mathematics and Computer Sciences, Cankaya University, Ankara, Turkey

    Dumitru Baleanu

  2. Institute of Engineering, Department of Electrical Engineering, Polytechnic Institute of Porto, Rua Dr António Bernadino Almeida 431, Porto, 4200-072, Portugal

    José António Tenreiro Machado

  3. Edwardsville, School of Engineering, Southern Illinois University, Room: EB2064, Edwardsville, 62026-1805, USA

    Albert C. J. Luo

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Sibatov, R.T., Uchaikin, V.V., Uchaikin, D.V. (2012). Fractional Wave Equation for Dielectric Medium with Havriliak–Negami Response. In: Baleanu, D., Machado, J., Luo, A. (eds) Fractional Dynamics and Control. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0457-6_25

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  • DOI: https://doi.org/10.1007/978-1-4614-0457-6_25

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  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-0456-9

  • Online ISBN: 978-1-4614-0457-6

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