In this work, accurate solutions to linear and nonlinear diffusion equations were introduced. A polynomial-based differential quadrature scheme in space and a strong stability preserving Runge-Kutta scheme in time have been combined for solving these equations. This scheme needs less storage space, as opposed to conventional numerical methods, and causes less accumulation of numerical errors. The results computed by this way have been compared with the exact solutions to show the accuracy of the method. The approximate solutions to the nonlinear equation have been computed without transforming the equation and without using the linearization. The present results are seen to be a very reliable alternative method to the existing techniques for the problems. In order to obtain physical models much closer to the nature, this procedure has a potential to be used to other nonlinear partial differential equations. Copyright (C) 2009 John Wiley & Sons, Ltd.