AbstractA fully Sinc–Galerkin method for solving advection–diffusion equations subject to arbitrary radiation boundary conditions is presented. This procedure gives rise to a discretization, which has its most natural representation in the form of a Sylvester system where the coefficient matrix for the temporal discretization is full. The word “full” often implies a computationally more complex method compared to, for example, temporal marching. In a comparison of time‐marching versus this sinc‐temporal procedure, the Sylvester formulation defines a common framework within which these procedures can be evaluated. This framework has been included in the introduction to illustrate an efficiency measure for either method. Similar remarks with regard to fullness versus sparseness in the Sylvester formulation apply when the spatial discretization is spectral or, for example, differencing. Although it is indicated how this sinc‐temporal method can be combined with alternative spatial discretizations, the natural affinity between sinc methods for space and time discretizations motivate carrying out the numerical illustrations using the sinc basis in each. © 1995 John Wiley & Sons, Inc.