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Differential Quadrature Method (DQM) to integrate the one-dimensional Advection–diffusion Equation (ADE) is presented. This method was applied to two examples and the results were compared with the performance of the Explicit Finite Difference Method (EFDM) and Implicit Finite Differences Method (IFDM). Based on the comparison with the exact solution, and both the explicit and implicit finite difference solutions, it was concluded that the DQM provides similar results but less grid points; besides the results are converged quickly. A numerical comparison for a case in which the exact solution is known, DQM gives closer results to the exact values than EFDM and IFDM.