Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation

Abstract

In this paper, we presented two time-fractional phase-field models to account for the memory effect of certain material or the transport process in heterogeneously distributed media. The energy dissipation of the system has been observed for both the time fractional Allen-Cahn equation and the time fractional Cahn-Hilliard equation in one and two space dimension. More precisely, the energy dissipation of the time fractional Allen-Cahn equation is faster than the that for integer order case, which seems not consistent with the definition of the fractional operator physically. However, the energy dissipation of the time fractional Cahn-Hilliard equation is proportional to the time fractional order. Namely, the larger fractional order is, the faster the energy decreases, which is more conform to the practice. Besides, we have observed that the time fractional Cahn-Hilliard equation also mass conservative. Sensitivity analysis on the tunable decay behavior for the one dimensional time fractional Allen-Cahn equation show that the decay behavior is more sensitive to the mobility and the time-fractional differential operator, especially the interface thickness.