A numerical scheme for time-fractional Allen-Cahn equation with application in phase separation

The present study proposes a numerical method to solve the time-fractional Allen-Cahn equation. The Grunwald-Letnikov formula is employed for discretizing the time-fractional derivative, whereas the finite difference scheme is used for spatial approximation. Theoretical analysis indicates that the proposed method is stable in the discrete $ L_{2} $ L2-norm and the numerical results satisfy the energy dissipation property. Numerical simulations are conducted for validation of the proposed method. It has been observed that the solution profile approaches equilibrium for various fractional-order values in the range of $ (0,1) $ (0,1). Moreover, the fractional order values have a significant effect on the solution stabilization rate which is faster for larger values and slower for the smaller values.