The normalized time-fractional Cahn-Hilliard equation

Abstract

We present a normalized time-fractional Cahn–Hilliard (TFCH) equation by incorporating time-fractional derivatives to model memory effects in phase separation processes. We use a normalized time-fractional derivative, which is a form of the Caputo fractional derivative, to improve the flexibility and physical interpretation of the model. This normalization allows for a more consistent interpretation of fractional orders, which enables fair comparisons across different orders of the derivative. To solve the normalized TFCH equation, we use an efficient computational scheme based on the Fourier spectral method, which ensures high accuracy and computational efficiency. Furthermore, we conduct a thorough investigation into the dynamic behavior of the normalized TFCH equation and focus on how varying the fractional-order time derivative influences the evolution and morphology of phase domains. Numerical simulations demonstrate the versatility and effectiveness of the proposed method in modeling complex phase separation dynamics. © 2025 Elsevier Ltd