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2015 Relaxation Equations: Fractional Models
C.F.A.E. Rosa, E. Capelas de Oliveira
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J. Phys. Math. 6(2): 1-7 (2015). DOI: 10.4172/2090-0902.1000146


The relaxation functions introduced empirically by Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami are, each of them, solutions to their respective kinetic equations. In this work, we propose a generalization of such equations by introducing a fractional differential operator written in terms of the Riemann-Liouville fractional derivative of order $γ$, $0\ltγ\le1$. In order to solve the generalized equations, the Laplace transform methodology is introduced and the corresponding solutions are then presented, in terms of Mittag-Leffler functions. In the case in which the derivative’s order is $γ = 1$, the traditional relaxation functions are recovered. Finally, we presented some 2D graphs of these function.


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C.F.A.E. Rosa. E. Capelas de Oliveira. "Relaxation Equations: Fractional Models." J. Phys. Math. 6 (2) 1 - 7, 2015.


Published: 2015
First available in Project Euclid: 31 August 2017

Digital Object Identifier: 10.4172/2090-0902.1000146

Keywords: Cole-Cole , complex susceptibility , Davidson-Cole , Debye , dielectric relaxation , Fractional differential equations , Havriliak-Negami , Laplace transform , Mittag-Leffler functions , relaxation function , response function , Riemann-Liouville derivative

Rights: Copyright © 2015 Ashdin Publishing (2009-2013) / OMICS International (2014-2016)

Vol.6 • No. 2 • 2015
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