Authors: Zorica, Dušan 
Oparnica, Ljubica
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Energy dissipation for hereditary and energy conservation for non-local fractional wave equations
Journal: Philosophical Transactions of the Royal Society. A: Mathematical, Physical and Engineering Sciences
Volume: 378
Issue: 2172
First page: 20190295
Issue Date: 29-May-2020
Rank: M21
ISSN: 1364-503X
DOI: 10.1098/rsta.2019.0295
Using the method of a priori energy estimates, energy dissipation is proved for the class of hereditary fractional wave equations, obtained through the system of equations consisting of equation of motion, strain and fractional order constitutive models, that include the distributed-order constitutive law in which the integration is performed from zero to one generalizing all linear constitutive models of fractional and integer orders, as well as for the thermodynamically consistent fractional Burgers models, where the orders of fractional differentiation are up to the second order. In the case of non-local fractional wave equations, obtained using non-local constitutive models of Hooke- and Eringen-type in addition to the equation of motion and strain, a priori energy estimates yield the energy conservation, with the reinterpreted notion of the potential energy. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.
Keywords: energy dissipation and conservation | fractional wave equation | hereditary and non-local fractional constitutive equations
Publisher: The Royal Society Publishing
Project: Serbian Ministry of Education, Science and Technological Development, Grant no. 451-03-68/2020-14/200125
Methods of Functional and Harmonic Analysis and PDE with Singularities 
Provincial Secretariat for Higher Education and Scientific Research, Grant no. 142-451-2102/2019
FWO Odysseus, Grant no. G.0H94.18N: Analysis and Partial Differential Equations

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