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This study aims to investigate the well‐posedness and stability of a thermoelastic Timoshenko system with non‐Fourier heat conduction. Specifically, we analyze the system using the dual‐phase‐lag (DPL) model, which incorporates two thermal relaxation times, τq$$ {\tau}_q $$ and τθ$$ {\tau}_{\theta } $$, to model non‐instantaneous heat propagation. Applying the semigroup approach, we demonstrate the existence and uniqueness of the solutions. Subsequently, we introduce a novel stability parameter ϰ$$ \varkappa $$ using the multiplier method. Exponential decay is proven for the case of ϰ=0$$ \varkappa =0 $$ with 2τθ>τq$$ 2{\tau}_{\theta }>{\tau}_q $$. Using Gearhart–Prüss theorem, we show the lack of exponential stability when ϰ≠0$$ \varkappa \ne 0 $$ and 2τθ=τq$$ 2{\tau}_{\theta }={\tau}_q $$. Numerically, we present a fully discrete approximation using the finite element method and the backward Euler scheme, and we provide some numerical simulations to show the discrete energy decay and the behavior of solutions. [ABSTRACT FROM AUTHOR]

This study aims to investigate numerically and theoretically a thermoelastic Bresse system where the heat conduction given by the Green and Naghdi theories. Theoretically, we establish the existence and uniqueness of solutions as well as the exponential stability regardless of the system's parameters. Numerically, we present a finite element approximation, and we show the discrete energy decay. At the end, we provide some numerical results with an error estimate based on additional regularity of the solution. [ABSTRACT FROM AUTHOR]