The sinc-Galerkin method lies on the use of orthogonal basis S(k,h)(x)\(\equiv \sin c(x-kh/h)\), \(x\in {\mathbb{R}}\), \(h>0\), \(k\in {\mathbb{Z}}\), where the sinc function is given by sinc(x)\(\equiv \sin (\pi x)/\pi x\), \(x\in {\mathbb{R}}\). Corresponding numerical methods including the sinc- Galerkin method have been analyzed by F. Stenger during the last ten years. The main property is an exponential convergence rate even in the presence of singularities. The aim of this paper is to extend such sinc-Galerkin methods to the steady state problem in three dimensions and the time (hyperbolic and parabolic) dependent problems in at least two dimensions. For the time dependent problem, the Galerkin method is used to approximate both space and time coordinates. In this paper, the analysis is limited to rectangular domains and the extension to more general geometries should not be so easy. Numerical examples are given including one Poisson problem in three dimensions, two second order hyperbolic problems in two space dimensions (one example exhibits singularities) and two parabolic problems in two space dimensions (once again, one example exhibits singularities).