  In this paper, we propose a time-fractional molecular beam epitaxy (MBE)
model with slope selection and its efficient, accurate, full discrete, linear
numerical approximation. The numerical scheme utilizes the fast algorithm for
the Caputo fractional derivative operator in time discretization and Fourier
spectral method in spatial discretization. Refinement tests are conducted to
verify the $2-\alpha$ order of time convergence, with $\alpha \in (0, 1]$ the
fractional order of derivative. Several numerical simulations are presented to
demonstrate the accuracy and efficiency of our newly proposed scheme. By
exploring the fast algorithm calculating the Caputo fractional derivative, our
numerical scheme makes it practice for long time simulation of MBE coarsening,
which is essential for MBE model in practice. With the proposed fractional MBE
model, we observe that the scaling law for the energy decays as $
O(t^{-\frac{\alpha}{3}})$ and the roughness increases as
$O(t^{\frac{\alpha}{3}})$, during the coarsening dynamics with random initial
condition. That is to say, the coarsening rate of MBE model could be
manipulated by the fractional order $\alpha$, and it is linearly proportional
to $\alpha$. This is the first time in literature to report/discover such
scaling correlation. It provides a potential application field for fractional
differential equations. Besides, the numerical approximation strategy proposed
in this paper can be readily applied to study many classes of time-fractional
and high dimensional phase field models.
