Asymptotic Behavior of a transmission Heat/Piezoelectric smart material with internal fractional dissipation law

Ibtissam Issa; Abdelaziz Soufyane; Octavio Vera Villagran

doi:10.3934/eect.2024049

Article Contents

2025, Volume 14, Issue 1: 157-184. Doi: 10.3934/eect.2024049

This issue Previous Article Stochastic optimal switching and systems of variational inequalities with interconnected obstacles Next Article The quadratic tracking problem for systems with persistent memory in $ \mathbb{R}^d $

Asymptotic Behavior of a transmission Heat/Piezoelectric smart material with internal fractional dissipation law

1.
Università degli studi dell'Aquila, Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, L'Aquila, Italy
2.
University of Sharjah, College of Science, Department of Mathematics, Ash Shariqah, UAE
3.
Universidad de Tarapaca, Departamento de Matemática, Arica, Chile

^*Corresponding author: Ibtissam Issa
^*Corresponding author: Ibtissam Issa

Received: June 2024

Revised: August 2024

Early access: September 2024

Published: February 2025

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Abstract

In this paper, we analyze the stability of a system involving a heat-conducting copper rod and a magnetizable piezoelectric beam, where fractional damping influences the longitudinal displacement of the beam's centerline. The coupled dynamics are governed by partial differential equations that incorporate heat diffusion in the copper rod and piezoelectric effects in the beam, including both mechanical and electrical interactions. Previous research has extensively studied piezoelectric systems and heat transfer dynamics separately, but the combined effect with fractional damping presents a novel challenge. Our investigation employs semi-group theory and multiplier methods to establish a polynomial stability result that is dependent on the order of the fractional derivative, offering insights into the interplay between heat transfer and piezoelectric behavior under fractional damping, which is critical for developing robust and efficient energy harvesting devices and structural control mechanisms.

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Mathematics Subject Classification: Primary: 74D05, 47D03, 74M05; Secondary: 93D23.

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Table 1. Table

Parameter	Corresponding Significance
$ \chi $	The elastic stiffness coefficient
$ \kappa $	The thermal diffusivity constant
$ \gamma $	Piezoelectric coefficient
$ \beta $	The beam coefficient of impermeability
$ \rho $	Mass density per unit volume,
$ \mu $	Magnetic permeability of beam

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