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Fractal Fract 2017, 1(1), 13; doi:10.3390/fractalfract1010013

The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials

1
College of Geoscience, Texas A & M University, College Station, TX 77843, USA
2
Department of Mathematics, University of Bologna, Piazza Porta San Donato 5, 40129 Bologna, Italy
*
Author to whom correspondence should be addressed.
Received: 24 October 2017 / Revised: 13 November 2017 / Accepted: 13 November 2017 / Published: 21 November 2017
(This article belongs to the Special Issue Fractional Dynamics)
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Abstract

The extension of the fractional order derivative to the distributed order fractional derivative (DOFD) is somewhat simple from a formal point of view, but it does not yet have a simple, obvious analytic form that allows its fast numerical calculation, which is necessary when solving differential equations with DOFD. In this paper, we supply a simple analytic kernel for the Caputo DOFD and the Caputo-Fabrizio DOFD, which may be used for numerical calculation in cases where the weight function is unity. This, in turn, could potentially allow faster solution of differential equations containing DOFD. Utilizing an analytical formulation of simple physical systems with phenomenological equations that include a DOFD, we show the relevant differences between the Caputo DOFD and the Caputo-Fabrizio DOFD. Finally, we propose a model based on DOFD for modeling composed materials that comprise different constituents, and show its compatibility with thermodynamics. View Full-Text
Keywords: fractional derivatives; distributed order; numerical calculation; memory; complex materials fractional derivatives; distributed order; numerical calculation; memory; complex materials
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Caputo, M.; Fabrizio, M. The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials. Fractal Fract 2017, 1, 13.

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