This study focused on a coupled nonlinear wave equation system featuring fractional damping and polynomial source terms within a bounded domain. We demonstrated that, under specific conditions, the solutions exhibited blow-up in a finite time-frame.

This paper analyzes an Euler-Bernoulli beam equation in a bounded domain with a boundary control condition involving a fractional derivative. By utilizing the semigroup theory of linear operators and building on the results of Borichev and Tomilov, the stability properties of the system are examined. Additionally, a numerical scheme is developed to reproduce various decay rate behaviors. The numerical simulations confirm the theoretical stability results regarding the energy decay rate and demonstrate exponential decay for specific configurations of initial data.