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

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.