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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.

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