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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Anomalous Relaxation in Dielectrics with Hilfer Fractional Derivative

Journal of Applied Nonlinear Dynamics 10(3) (2021) 479--491 | DOI:10.5890/JAND.2021.09.009

A. R. G 'omez Plata , E. Capelas de Oliveira and Ester C. A. F. Rosa

Department of Mathematics, Universidad Militar Nueva Granada Cajic'a-Zipaquir'a, 250247, Colombia Departament of Applied Mathematics, Imecc--Unicamp Campinas, 13083-859, SP, Brazil

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Abstract

We introduce a new relaxation function depending on an arbitrary parameter as a solution of a kinetic equation in the same way as the relaxation function introduced empirically by Debye, Cole-Cole, Davidson-Cole and Havriliak-Negami regarding, anomalous relaxation in dielectrics, which are recovered as particular cases. We propose a differential equation introducing a fractional operator written in terms of the Hilfer fractional derivative of order $\xi$, with $0 < \xi \leq 1$ and type $\eta$, with $0 \leq \eta \leq 1$. To discuss the solution of the fractional differential equation, the methodology of Laplace transform is required. As a by product we mention particular cases where the solution is completely monotone. %Some graphics show the behaviour of this completely monotone function. Finally, the empirical models are recovered as particular cases.

References

  1. [1]  de Oliveira, E.C., Mainardi, F., and Vaz Jr., J. (2011), {Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics}, ph {The European Physical Journal}, {193}, 161-171.
  2. [2]  de Oliveira, E.C., Mainardi, F., and Vaz Jr., J. (2014), {Fractional models of anomalous relaxation based on the Kilbas and Saigo function}, ph {Meccanica}, {49}(9), 2049-2060.
  3. [3]  de Oliveira, E.C. and Tenreiro Machado, J.A. (2014), {A review of definitions for fractional derivatives and integrals}, ph {Math. Prob. Ing.}, {2014}, ID 238459.
  4. [4]  Teodoro, G.S., Machado, J.A.T., and De Oliveira, E.C. (2019), {A review of definition of fractional derivatives and others operator}, ph {J. Comput. Phys.}, {388}, 195-209.
  5. [5]  Tarasov, V.E. (2013), {No violation of the Leibniz rule. No fractional derivative}, ph {Comm. Nonlinear Sci. Num. Simulat.}, {18}, 2945-2948.
  6. [6]  Tarasov, V.E. (2018), {No nonlocality. No fractional derivative}, ph {Comm. Nonlinear Sci. Num. Simulat.}, {62}, 157-163.
  7. [7]  Akkurt, A., Yildirim, M.E., and Yildirim, H. (2017), {A new generalized fractional derivative and integral}, ph {Konuralp J. Math.}, {5}, 248-259.
  8. [8]  Caputo, M. and Fabrizio, M.A. (2015), ph{New definition of fractional derivative without singular kernel}, { Prog. Fract. Diff. Appl.}, {1}, 73-85.
  9. [9]  Losada, J. and Nieto, J.J. (2015), {Properties of a new fractional derivative without singular kernel}, {1}, 87-92.
  10. [10]  B\"ohmer, R., Ngai, K., Angell, C., and Plazeck, D. (1993), {Nonexponential relaxations in strong and fragile glass formers}, ph {J. Chem. Phys.}, {99}, 4201.
  11. [11]  Hilfer, R. (2003), {On fractional relaxation}, ph {Fractals}, {11}, 251-257.
  12. [12]  Sabatier, J., Agrawal, O.P., Machado, J.A.T. (Editors) (2007), ph{Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering}, Springer, The Netherlands.
  13. [13]  Mainardi, F. (2010), ph{Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models}, Imperial College Press.
  14. [14]  Debye, P. (1929), ph{Polar Molecules}, Dover, New York.
  15. [15]  Kohlrausch, R. (1854), {Theorie des elektrischen R\"uckstandes in der Leidener Flasche}, ph {Pogg. Ann. Phys. Chem.}, {91}, 179-214.
  16. [16]  Williams, G. and Watts, D.C. (1970), {Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function}, ph {Trans. Faraday Soc.}, {66}, 80-85.
  17. [17]  Cardona, M., Chamberlin, R.V., and Marx, W. (2007), {Comment on the history of the stretched exponential function}, ph {Ann. Phys. }(Leipzig) {16}, 842-845.
  18. [18]  Cole, K.S. and Cole, R.H. (1941), {Dispersion and absorption in dielectrics I. Alternating current characteristics}, ph {J. Chem. Phys.}, {9}, 341-351.
  19. [19]  Davidson, D.W. and Cole, R.H. (1950), {Dielectric relaxation in glycerine}, (Letter to the Editor), ph {J. Chem. Phys.}, {18}, 1417. ph{Dielectric relaxation in glycerol, propylene glycol, and $n$-propanol}, J. Chem. Phys., {19}, 1484-1490 (1951).
  20. [20]  Havriliak, S. and Negami, S. (1967), {A complex plane representation of dielectric and mechanical relaxation processes in some polymers}, ph {Polymer}, {8}, 161-210.
  21. [21]  Garrappa, R. (2016), {Gr\"unwald-Letnikov operators for fractional relaxation in Havriliak-Negami models}, ph {Comm. Non. Sci. Num. Simulat.}, {38}, 178-191.
  22. [22]  Bia, P., Caratelli, D., Mescia, L., Cicchetti, R., Maione, G., and Prudenzano, F. (2015), {A novel FDTD formulation based on fractional derivatives for dispersive Havriliak-Negami media}, ph {Signal Processing}, {107}, 312-318.
  23. [23]  Mainardi, F. (2018),{Fractional Calculus}: {Theorey and Applications}, ph{ Mathematica}, MDPI, Basel.
  24. [24]  Shibatov, R.T., Uchaikin, V.V., and Uchaikin, D.V. (2012), ph{Fractional wave equation for dielectric medium with Havriliak-Negami response}, in {\sf Fractional Dynamics and Control}, D. Baleanu, J. A. Tenreiro Machado, A. C. J. Luo (Editors), Springer, New York.
  25. [25]  Casalini, R., Paluch, M., and Roland, C.M. (2002), {Correlation between the $\alpha$-relaxation and the excess wing for polychlorinated biphenyls and glycerol}, ph {J. Thermal Anal. Calorimetry}, {69}, 947-952.
  26. [26]  Rosa, E.C.F.A. and de Oliveira, E.C. (2015), {Relaxation equations: fractional models}, ph {J. Phys. Math.}, 6, 146. doi:10.4172/2090-0902.1000146.
  27. [27]  Mainardi, F. and Garrappa, R. (2015), {On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics}, ph {J. Comp. Phys.}, {293}, 70-80.
  28. [28]  Garrappa, R., Mainardi, F., and Maione, G. (2016), {Models of dielectric relaxation based on completely monotone functions}, ph {Fract. Cal. $\&$ Appl. Anal.}, {19}, 1105-116.
  29. [29]  Rosa, E.C.F.A. (2017), ph{Fractional Kinetic Relaxation Functions}, (in Portuguese) Doctoral thesis, Unicamp, Campinas-SP.
  30. [30]  Machado, J.A.T., Mainardi, F. and Kiryakova, V. (2015), {Fractional calculus}: {Quo vadimus}? ({Where are we going}?) (Contributions to the roundtable held in ICFDA 2014), ph {Fract. Cal. $\&$ Appl. Anal.}, {18}, 495-526.
  31. [31]  Machado, J.A.T., Mainardi, F., Kiryakova, V., and Atanackovi\c, T. (2016), {Fractional calculus:} {Dou venons-nous}? {Que sommes-nous}? {O\`u allons-nous}? (Contributions to the roundtable held in ICFDA 2016), ph {Fract. Cal. $\&$ Appl. Anal.}, {19}, 1074-1104.
  32. [32]  Giusti, A. and Colombaro, I. (2018), {Prabhakar-like fractional viscoelasticity}, ph {Comm. Nonlinear Sci. Num. Simulat.}, {56}, 138-143.
  33. [33]  Chang, A., Sun, H., Zheng, C., Lu, B., Lu, C., Ma, R., and Zhang, Y. (2018), {A time fractional convection-diffusion equation to model gas transport through heterogeneous soil and gas reservoirs}, ph {Phys. A: Stat. Mech $\&$ Appl.}, {502}, 356-369.
  34. [34]  Ha, S.Y. and Jung, J. (2018), {Remarks on the slow relaxation for the fractional Kuramoto model for synchronization}, ph {J. Math. Phys.}, {59}, 032702.
  35. [35]  G\orska, K., Horzela, A., Bratek, L., Dattoli, G., and Penson, K.A. (2018), {The Havriliak-Negami and its relatives}: {the response, relaxation and probability density functions}, ph {J. Phys. A: Math. $\&$ Theor.}, {51}, 135202.
  36. [36]  Sun, H.G., Zhang, Y., Baleanu, D., Chen, W., and Chen, Y.Q. (2018), {A new collection of real world applications of fractional calculus in science and engineering}, ph {Comm. Nonlinear Sci. Numer. Simulat.}, {64}, 213-231.
  37. [37]  de Oliveira, E.C., Jarosz, S., and Vaz. Jr., J. (2019), {Fractional calculus via Laplace transform and its application in relaxation processes}, ph {Comm. Nonlinear Sci. Numer. Simulat.}, {69}, 58-72.
  38. [38]  Mori, H. (1965), {A continued-fractionl representation of the time correlation function}, ph {Prog. Theor. Phys.}, {30}, 399-416.
  39. [39]  Zwanzig, R. (1961), ph{Lectures in Theoretical Physics}, Interscience, New York.
  40. [40]  Hilfer, R. (2000), in ph{Applications of Fractional Calculus in Physics}, ed. R. Hilfer (World Scientific, Singapore, 2000), p.87.
  41. [41]  Novikov, V.V., Wojciechowski, K.W., Komkova, O.A., and Thiels, T. (2005), ph{Anomalous relaxation in dielectrics. Equations with fractional derivatives}, Materials Science-Poland, {23}, 977-984.
  42. [42]  Weron, K., Jurlewicz, A., and Magdziark, M. (2005), {Havriliak-Negami response in the framework of the continuous-time random walk}, ph {Acta Phys. Pol. B}, {36}, 1855-1868.
  43. [43]  Jurlewicz, A., Weron, K., and Teuerle, M. (2008), {Generalized Mittag-Leffler relaxation}: ph{Clustering-jump continuous-time random walk approach}, ph {Phys. Rev. E}, {78}, 011103.
  44. [44]  Khamzin, A.A., Nigmatullin, R.R., and Popov, I.I. (2014), {Justification of the empirical laws of the anomalous dielectric relaxation in the framework of the memory function formalism}, ph {Frac. Cal. $\&$ Appl. Anal.}, {17}, 247-258.
  45. [45]  Pandey, S.C. (2017), {The Lorenzo-Hartleys function for fractional calculus and its applications pertaining to fractional order modelling of anomalous relaxation in dielectrics}, ph {Comp. Appl. Math.}, DOI. 10.1007/s40314-017-0472-7.
  46. [46]  Gorenflo, R., Kilbas, A.A., Mainardi, F., and Rogosin, S.V. (2014), ph{Mittag-Leffler Functions, Related Topics and Applications}, Springer, Heildelberg.
  47. [47]  Camargo, R.F. and de Oliveira, E.C. (2015), ph{Fractional Calculus}, (in Portuguese), Editora Livraria da F\{\i}sica, S\~ao Paulo.
  48. [48]  Gelfand, I.M. and Shilov, G.E. (1964), ph{Generalized Functions}, Vol. 1, Academic Press, New York.
  49. [49]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), {Theory and Applications of Fractional Differential Equations}, ph {Elsevier}, Amsterdam.
  50. [50]  Mittag-Leffler, G.M. (1903), {Sur la nouvelle fonction $E_{\alpha}(x)$}, ph {C. R. Acad. Sci.}, {137}, 554-558.
  51. [51]  Bottcher, C.J.F. and Bordewijk, P. (1978), {Theory of Electric Polarization}, ph {Elsevier Sci. Publ.}, Amsterdam.
  52. [52]  Hanyga, A. and Seredynska, A. (2008), {On a mathematical framework for the constitutive equations of anisotropic dielectric relaxation}, ph {J. Stat. Phys.}, {131}, 269-303.
  53. [53]  Havriliak Jr., S. and Negami, S. (1966), {A complex plane analysis of $\alpha$-dispersion in some polymer systems}, ph {J. Polymers Sci.}, {14}, 99-117.
  54. [54]  Frohlich, H. (1958), ph{Theory of Dielectrics}, Oxford University Press, London.
  55. [55]  Manning, M.F. and Bell, M.E. (1940), {Electrical conduction and related phenomena in solid dielectrics}, ph {Rev. Mod. Phys.}, {12}, 215-257.
  56. [56]  Willians, G. (1972), {Use of the dipole correlation function in dielectric relaxation}, ph {J. Chem. Rev.}, {77}, 55-69.
  57. [57]  Boon, J.P. and Yip, S. (1980), ph{Molecular Hydrodynamic}, Dover, New York.
  58. [58]  Giraldo, R.G.R., G\omez, A.R., and de Oliveira, E.C. (2020), ph{On the complete monotonicity of fractional relaxation functions}, to be submitted for publication.

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