Energy dissipation for hereditary and energy conservation for non-local fractional wave equations

TL;DR

This paper proves energy dissipation for hereditary fractional wave equations and energy conservation for non-local fractional wave equations using a priori energy estimates, encompassing a broad class of models.

Contribution

It introduces a unified approach to analyze energy behavior in hereditary and non-local fractional wave equations, including general distributed-order models.

Findings

Energy dissipation established for hereditary fractional wave equations.

Energy conservation demonstrated for non-local fractional wave equations.

Applicable to a wide range of fractional and integer order models.

Abstract

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.