The asymptotic form of shocks in relaxing media (Q1114475)

The Laplace transform is used to analyze the asymptotic solutions \((\alpha \to \pm \infty)\) of a class of integrodifferential equations encountered in the study of nonlinear viscoelastic problems. It is found that the step-up behaviour from the smaller equilibrium value (\(\alpha\) \(\to -\infty)\) is exponential, while the approach to the larger value (\(\alpha\) \(\to \infty)\) is dependent on the form of the relaxation function being considered. A perturbation solution to one of the equations is also given. The form of the lowest-order solution shows how the oscillatory nature of some of the previously presented solutions arises.