An exact solution to the Frenkel-Kontorova equation for a one-dimensional dislocation under stress is obtained by numerical calculations, and is compared with some of approximate solutions which are calculated from a continuum model, a successive approach, the usual Peierls barrier model and the two-parabola approximation to a sinusoidal potential. The exact solution shows that i) the Peierls force is not an oscillatory but a monotonically decreasing function of the thickness of the boundary layer, and ii) of the above approximations, the usual Peierls potential barrier model is best, while the two-parabola approximation is worst. As examples of the Peierls force, an effective shear stress of the coherent twin boundary and an intrinsic coercive force of 180° Bloch domain wall are discussed.