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We propose a discontinuous Galerkin method for convection-subdiffusion equations with a fractional operator of order alpha [1,2] defined through the fractional Laplacian. The fractional operator of order alpha is expressed as a composite of first order derivatives and fractional integrals of order 2 − alpha, and the fractional convection-diffusion problem is expressed as a system of low order differential/integral equations and a local discontinuous Galerkin method scheme is derived for the equations. We prove stabilityand optimal order of convergence O(h^(k+1)) for subdiffusion, and an order of convergence of O(h^(k+1/2)) is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.