The extreme eigenvalues of the Chebyshev pseudospectral differentiation operator are O(N2), where N is the number of grid points. As a result of this, the allowable time step in an explicit time marching algorithm is O(N−2) which, in many cases, is much below the time step dictated by the physics of the PDE. In this paper we introduce a new differentiation operator whose eigenvalues are O (N) and the allowable time step is O (N−1). The new algorithm is based on interpolating at the zeroes of a parameter dependent, nonperiodic trigonometric function. The properties of the new algorithm are similar to those of the Fourier method but in addition it provides highly accurate solution for nonperiodic boundary value problems.