The ordering process in a nearly one-dimensional anisotropic ferromagnetic system is studied by utilizing the modified Ginzburg-Landau equation. In the early stages the process is dominated by the formation of Bloch walls. Then, annihilation of domains follows, which promotes the growth of magnetic domains. By deriving the equation of motion for the assembly of Bloch-wall positions, we study the temporal evolution of the scattering function both from the numerical and the theoretical standpoint. It is found that the average domain size increases logarithmically in time and the scattering function asymptotically takes the exponential form in the later stages. This can be quantitatively well explained by the kink dynamics. Furthermore, the temporal evolution of the distribution function for domain sizes is numerically obtained. We find that it can be well approximated fundamentally by the exponential law.