Semigroup Theory and Numerical Approximation for Equations in Linear Viscoelasticity

The following abstract integro-differential equation \[ u(t) + A[ {Eu(t) - ∫- τ^0 {g(s)u(t + s)ds} } ] = f(t)\] is considered on a Hilbert space. Such equations arise in the modeling of linear viscoelastic beams. The equation is reformulated as an abstract Cauchy problem, and several approximation schemes are discussed. Well-posedness and convergence results are given in the context of linear semigroup theory. Results of numerical eigenvalue calculations for various approximation schemes are discussed.