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In this paper we consider the fully discrete local discontinuous Galerkin method,where the third order explicit Runge-Kutta time marching is coupled.For the one-dimensional time-dependent singularly perturbed problem with a boundary layer,we shall prove that the resulted scheme is not only of good behavior at the local stability,but also has the double-optimal local error estimate.It is to say,the convergence rate is optimal in both space and time,and the width of the cut-off subdomain is also nearly optimal,if the boundary condition at each intermediate stage is given in a proper way.Numerical experiments are also given.