We investigate a coupled hyperbolic system with memory effects and a time-dependent singular damping term of the form \(\frac{\sigma }{(t+\epsilon )^q}\) . The study focuses on the interaction between the singular dissipation, the memory kernel, and a fractional diffusion operator \(\mathcal {M}^\ell \) , employing both analytical and numerical methods. Energy techniques and spectral analysis are used to derive a polynomial decay estimate, highlighting the influence of parameters q and \(\ell \) on the long-term behavior of the system. Numerical simulations are conducted to validate the theoretical findings and provide additional insights into the impact of singular damping and memory effects on wave attenuation. These results are pertinent to applications in viscoelasticity, smart structures, and wave propagation in complex media with fading memory.
