An unconditionally stable numerical algorithm for the modified Korteweg-de Vries equation based on the B-spline finite element method is described. The algorithm is validated through a single soliton simulation. In further numerical experiments forced boundary conditions (u=U_{0}) are applied at the end (x=0) and the generated states of solitary waves are studied. By long impulse experiments these are shown to be generated periodically with period (\Delta T_{B}) proportional to (U_{0}^{-3}) and to have a limiting amplitude proportional to (U_{0}). This limit is achieved by all waves, after the first, provided the experiment proceeds long enough. The temporal development of the derivatives (U^{\prime}(0, t), U^{\prime \prime}(0, t)) and (U^{\prime \prime \prime}(0, t)) is also periodic, with period (\Delta T_{B}). The effect of negative forcing is to generate a train of negative waves. The solitary wave states generated by applying a positive impulse followed immediately by an negative impulse, of equal amplitude and duration, is dependent on the period of forcing. The solitary waves generated by these various forcing functions possess many of the attributes of free solitons. (c) 1994 Academic Press, Inc.