A polynomial approximation method for calculating state profiles for plug-flow reactors is extended to one-dimensional reactor models that include axial dispersion. The method is based on the conservation of reactor state profile moments along the spatial dimension. The moments are then transformed analytically into a polynomial approximation at each timestep. The boundary conditions of the parabolic partial differential equation are given special attention. It is shown that the Danckwerts boundary conditions are an appropriate set of boundary conditions for flow problems with axial dispersion in closed-closed geometries. A significant feature of the present method is that boundary conditions of the partial differential equation model to be solved are implicitly satisfied via the moment transformation, while the polynomial profile in the numerical approximation does not have to satisfy the boundary conditions exactly. The method is tested in two cases: startup of a tubular reactor and fixed-bed adsorber involving axial dispersion. (c) 2010 Elsevier Ltd. All rights reserved.