In this paper, we present a numerical scheme for the approximate solutions of the one-dimensional parabolic convection-diffusion model problems. This method is based on the Bessel collocation method used for some problems of ordinary differential equations. In fact, the approximate solution of the problem in the truncated Bessel series form is obtained by this method. By substituting truncated Bessel series solution into the problem and by using the matrix operations and the collocation points, the suggested scheme reduces the problem to a linear algebraic equation system. By solving this equation system, the unknown Bessel coefficients can be computed. An error estimation technique is given for the considered problem and the method. To show the accuracy and the efficiency of the method, numerical examples are implemented and the comparisons are given by the other methods. © 2013 Elsevier Inc. All rights reserved.