Existence, stability and numerical results for a Timoshenko beam with thermodiffusion effects

Abstract

In this paper, we consider a new Timoshenko beam model with thermal and mass diffusion effects. Heat and mass exchange with the environment during thermodiffusion in Timoshenko beam. Firstly, by the $${C}_{0}$$ -semigroup theory, we prove the well posedness of the considered problem with Dirichlet or Neumann boundary conditions. Then we show, without assuming the well-known equal wave speeds condition, the lack of exponential stability for the Neumann problem, meanwhile one linear frictional damping is strong enough to guarantee the exponential stability for the Dirichlet problem. Then, we introduce a finite element approximation and we prove that the associated discrete energy decays. Finally, we obtain some a priori error estimates assuming additional regularity on the solution and we present some numerical results which demonstrate the accuracy of the approximation and the behaviour of the solution.