The fractional Laplacian in R d, which we write as (− Δ) α/2 with α∈(0, 2), has multiple
equivalent characterizations. Moreover, in bounded domains, boundary conditions must be …

This chapter surveys recent numerical advances in the phase field method for geometric
surface evolution and related geometric nonlinear partial differential equations (PDEs) …

For the time-fractional phase-field models, the corresponding energy dissipation law has not
been well studied on both the continuous and the discrete levels. In this work, we address …

We derive a fractional Cahn--Hilliard equation (FCHE) by considering a gradient flow in the
negative order Sobolev space H^-α, α∈0,1, where the choice α=1 corresponds to the …

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 …

In this work, we consider a time-fractional Allen–Cahn equation, where the conventional first
order time derivative is replaced by a Caputo fractional derivative with order α ∈ (0, 1) …

In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order
equation with nonlinear source terms. More specifically, we consider the time-fractional …

The fractional Laplacian in R^ d has multiple equivalent characterizations. Moreover, in
bounded domains, boundary conditions must be incorporated in these characterizations in …

Whether high order temporal integrators can preserve the maximum principle of Allen-Cahn
equation has been an open problem in recent years. This work provides a positive answer …

We construct and analyze the variable-step scheme to efficiently solve the space-time
fractional Cahn–Hilliard equation in two dimensions. The associated variational energy …