  The exhaustive group classification of a class of variable coefficient
generalized KdV equations is presented, which completes and enhances results
existing in the literature. Lie symmetries are used for solving an initial and
boundary value problem for certain subclasses of the above class. Namely, the
found Lie symmetries are applied in order to reduce the initial and boundary
value problem for the generalized KdV equations (which are PDEs) to an initial
value problem for nonlinear third-order ODEs. The latter problem is solved
numerically using the finite difference method. Numerical solutions are
computed and the vast parameter space is studied.
