Abstract In this paper, an efficient finite difference scheme is developed for solving the time-fractional Cahn–Hilliard equations which is the well-known representative of phase-field models. The time Caputo derivative is approximated by the popular L 1 formula. The stability and convergence of the finite difference scheme in the discrete L 2 -norm are proved by the discrete energy method. To compare and observe the phenomenon of solution, a generalized difference scheme based on the graded mesh in time is also given. The dynamics of the solution and accuracy of the schemes are verified numerically. Numerical experiments show that the solution of the time-fractional Cahn-Hilliard equation always tends to be in an equilibrium state with the increase of time for different values of order α ∈ ( 0 , 1 ) , which is consistent with the phase separation phenomenon.

中文翻译：

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时间分数阶 Cahn-Hilliard 方程的隐式差分格式

摘要 本文开发了一种有效的有限差分格式来求解时间分数阶 Cahn-Hilliard 方程，该方程是相场模型的著名代表。时间 Caputo 导数近似于流行的 L 1 公式。离散能量法证明了有限差分格式在离散L 2 -范数下的稳定性和收敛性。为了比较和观察解的现象，还给出了一种基于时间梯度网格的广义差分方案。数值验证了求解的动力学和方案的准确性。数值实验表明，对于不同阶α ∈ ( 0 , 1 ) ，时间分数阶Cahn-Hilliard方程的解随着时间的增加总是趋于平衡状态，