Quantum mechanics

Phase (matter)

Pure mathematics

Statistics

Applied mathematics

Field (mathematics)

Energy (signal processing)

Economics

Computer science

Integer (computer science)

Scaling

Statistical physics

Physics

Mathematics

Allen–Cahn equation

Phase field models

Mathematical analysis

Geometry

Programming language

Partial differential equation

Finance

Cahn–Hilliard equation

Fractional calculus

Order (exchange)

Computers & Mathematics with Applications

We study (time) fractional Allen–Cahn and Cahn–Hilliard phase-field models to account for the anomalously subdiffusive transport behavior in heterogeneous porous materials or memory effect of certain materials. We develop an efficient finite difference scheme and a Fourier spectral scheme to effectively treat the significantly increased memory requirement and computational complexity, which arise due to the nonlocal behavior of the time-fractional models. For time fractional Cahn–Hilliard model, we observe from the numerical results that the bigger the fractional order α is, the faster the energy decays. However, for time fractional Allen–Cahn model, we derived an opposite conclusion. Moreover, we also study the coarsening dynamics for time fractional Cahn–Hilliard model, numerical results reveal that the scaling law for the energy decays as O(t−α3), which is consistent with the well-known result O(t−13) for integer order Cahn–Hilliard model.

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