Etude mathématique et numérique de la stabilité de certains systèmes thermoélastiques couplés paraboliques-hyperbolique
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2024
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Abstract
This thesis presents a comprehensive study of some partial diferential equations systems in context of vibrations with efect of thermoelasticity, specifcally the Timoshenko and Bresse systems. Through these models, we address issues pertaining to existence and uniqueness of solutions using semigroup theory,kthe stability behavior of these solutions via the energy method, as well as the numerical approximation and a priori error analysis. An in-depth exploration of the thermoelastic Bresse system reveals fundamental insights into its solution properties and stability characteristics under thermal efects. The integration of damping and thermal dissipation mechanisms based on Green and Naghdi theories signifcantly enhances stability. Furthermore, novel conditions for exponential stability of the dual phase lag (DPL) thermoelastic Timoshenko system address defciencies in the classic equal wave propagation velocity assumption. Extending these fndings, by generalizing the analysis of the DPL thermoelasticity to the Bresse system confrms that the stability conditions identifed for the imoshenko system also apply. Additionally, numerical experiments using fnite element and fnite diference approximations demonstrate behavior and stability of the solutions, deepening our understanding of these systems dynamic.