Scaling properties of fractional continuous growth equations

The fractional Langevin equation is obtained by generalization of the Edward–Wilkinson equation in order to describe various growth processes. In order to solve the fractional Langevin equation analytically, due to the linearity, Fourier transforms and the Riesz derivative is implemented. It is shown that the surface width in very short-times behaves as W∼t(α−d)/2αW∼t(α−d)/2α and in long-times behaves as W∼L(α−d)/2W∼L(α−d)/2. The mean-square surface width, the height–height correlation function, and the auto-height correlation function are calculated and discussed for arbitrary dimensions. Also, the results where checked for known cases: the Edwards–Wilkinson equation (α=2)(α=2) and the Mullins–Herring equation (α=4)(α=4) justifying the calculations and results.