Role of the sample boundaries in the problem of dissipative magnetization dynamics

The Landau-Lifshitz or the Landau-Lifshitz-Gilbert equation of motion for the magnetization M(r,t) a partial integro-differential equation in time and space, for which, in general, an initial value condition and a boundary condition for the field have to be prescribed. This, however, is only true for analytic solutions. It is shown that a unique albeit approximate numerical solution of M(r,t) in a finite sample with surfaces, for which the position dependent form of the effective field occurring in these equations with position dependent material parameters is used, no boundary condition of the magnetization is required. The analytical boundary conditions nevertheless play an important role also in numerical simulations, since they provide valuable estimates required for the accurate calculation of the exchange field at the surfaces.