A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive law

We consider a fractional order viscoelasticity problem modelled by power-law type stress relaxation function. This viscoelastic is Volterra integral equation of the second kind with weakly singular kernel where convolution corresponds to differentiation/integration. use spatial finite element method and difference scheme in time. Due weak singularity, integration time managed approximately linear interpolation so that we can formulate fully discrete problem. In this paper, present stability bound as well priori error estimates. Furthermore, carry out numerical experiments varying regularity exact solutions at end.