  In this paper, a class of linear parabolic singularly perturbed second order
differential equations of reaction-diffusion type with initial and Robin
boundary conditions is considered. The solution u of this equation is smooth,
whereas the first derivative in the space variable exhibits parabolic boundary
layers. A numerical method composed of a classical finite difference scheme on
a piecewise uniform Shishkin mesh is suggested. This method is proved to be
first order convergent in time and essentially first order convergent in the
space variable in the maximum norm uniformly in the perturbation parameters.
