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Meccanica

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01.09.2014 | New Trends in Fluid and Solid Mechanical Models

Fractional models of anomalous relaxation based on the Kilbas and Saigo function

verfasst von: Edmundo Capelas de Oliveira, Francesco Mainardi, Jayme Vaz Jr.

Erschienen in: Meccanica | Ausgabe 9/2014

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Abstract

We revisit the Kilbas and Saigo functions of the Mittag-Leffler type of a real variable $t$, with two independent real order-parameters. These functions, subjected to the requirement to be completely monotone for $t>0$, can provide suitable models for the responses and for the corresponding spectral distributions in anomalous (non–Debye) relaxation processes, found e.g. in dielectrics. Our analysis includes as particular cases the classical models referred to as Cole–Cole (the one-parameter Mittag-Leffler function) and to as Kohlrausch (the stretched exponential function). After some remarks on the Kilbas and Saigo functions, we discuss a class of fractional differential equations of order $\alpha \in (0,1]$ with a characteristic coefficient varying in time according to a power law of exponent $\beta $, whose solutions will be presented in terms of these functions. We show 2D plots of the solutions and, for a few of them, the corresponding spectral distributions, keeping fixed one of the two order-parameters. The numerical results confirm the complete monotonicity of the solutions via the non-negativity of the spectral distributions, provided that the parameters satisfy the additional condition $0<\alpha +\beta \le 1$, assumed by us.

Vorheriger Artikel Hele-Shaw flow with a small obstacle

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$$\begin{aligned} u(t) = \int \limits_0^\infty \hbox {e}^{-rt} \, \hbox {d}\mu (r), \end{aligned}$$

where $\mu (t)$ is non-decreasing and the integral converges for $ 0 < t < \infty $. In other words $u(t)$ is required to the real Laplace transform of a non negative measure, in particular

$$\begin{aligned} u(t) = \int \limits _0^\infty \hbox {e}^{-rt} \, K(r)\, \hbox {d} r, \quad K(r) \ge 0, \end{aligned}$$

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Titel: Fractional models of anomalous relaxation based on the Kilbas and Saigo function
verfasst von: Edmundo Capelas de Oliveira
Francesco Mainardi
Jayme Vaz Jr.
Publikationsdatum: 01.09.2014
Verlag: Springer Netherlands
Erschienen in: Meccanica / Ausgabe 9/2014
Print ISSN: 0025-6455
Elektronische ISSN: 1572-9648
DOI: https://doi.org/10.1007/s11012-014-9930-0

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