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In this paper, we propose a time-fractional molecular beam epitaxy (MBE)model with slope selection and its efficient, accurate, full discrete, linearnumerical approximation. The numerical scheme utilizes the fast algorithm forthe Caputo fractional derivative operator in time discretization and Fourierspectral method in spatial discretization. Refinement tests are conducted toverify the $2-\alpha$ order of time convergence, with $\alpha \in (0, 1]$ thefractional order of derivative. Several numerical simulations are presented todemonstrate the accuracy and efficiency of our newly proposed scheme. Byexploring the fast algorithm calculating the Caputo fractional derivative, ournumerical scheme makes it practice for long time simulation of MBE coarsening,which is essential for MBE model in practice. With the proposed fractional MBEmodel, 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 initialcondition. That is to say, the coarsening rate of MBE model could bemanipulated by the fractional order $\alpha$, and it is linearly proportionalto $\alpha$. This is the first time in literature to report/discover suchscaling correlation. It provides a potential application field for fractionaldifferential equations. Besides, the numerical approximation strategy proposedin this paper can be readily applied to study many classes of time-fractionaland high dimensional phase field models.

An Accurate and Efficient Algorithm for The Time-fractional Molecular Beam Epitaxy Model with Slope Selection