A new algorithm available for the numerical inversion of Laplace transforms using multi-precision computational environment provides controlled accuracy, that is the inversion can be carried out to yield any pre-specified number of significant digits. One important application is characterizing theological phenomena, because the modeler is no longer restricted to a narrow class of "analytically treatable" constitutive equations, and the numerical inversion provides de facto "analytic" results for any reasonable model formulated in the Laplace domain. The advantages of Laplace domain models are discussed. In addition, idealized but important abstract flows (such as the Rayleigh or Stokes first problem) are calculated. As an illustration the fractional-power model of Low Density Polyethylene (LDPE) at 150degreesC (for small shear strains) is considered and the arrival of a sinusoidal wave is calculated. (C) 2003 Elsevier B.V. All rights reserved.