An original mixed finite-difference/spectral method based upon a Fokker-Planck equation is proposed for the numerical simulation of non-homogeneous flows of FENE-type fluids. Since the configuration distribution function psi varies as a function of time and in both physical and configuration space careful attention has been paid to the discretization scheme for derivatives of psi in these variables, the domains of definition of the physical and configuration coordinates being inter-dependent. Numerical results for start-up planar Poiseuille flow are in excellent quantitative agreement with those of a stochastic simulation and, for comparable levels of accuracy, are much less CPU expensive. Theoretical results for the polymeric stress fields under equilibrium conditions are verified numerically and generalize earlier results for Hookean dumbbells by Brunn and Grisafi [P.O. Brunn, S. Grisafi, Chem. Eng. Commun. 36 (1985) 367-383]. (C) 2004 Elsevier B.V. All rights reserved.