Space fractional Allen–Cahn equation and its applications in phase separation: A numerical study

The phenomena of non-locality and spatial heterogeneity are intricate, and using fractional differential equations provides a robust modeling approach for understanding these characteristics. On the other hand, approximating such phenomena numerically is time-consuming and challenging. In this article, we conduct a numerical study employing the spectral method for solving the space fractional Allen–Cahn equation. The Allen–Cahn is a reaction–diffusion equation widely used for phase separation dynamics. The most appealing characteristic of the spectral method is its ability to provide a completely diagonal representation of the fractional operator. Numerical simulations are performed for the validity, accuracy, and efficacy of the proposed schemes as well as to explore how the fractional order influences the dynamics of the solution. The tendency of the solution profile towards equilibrium has been observed for different fractional order values. It is observed that the solution stabilization rate is significantly influenced by the fractional order values, with larger values resulting in a faster rate and smaller values leading to a slower rate. Furthermore, it has been observed that the interface profile exhibits smooth and diffusive behavior as the fractional order increases, but it becomes sharp as the fractional order decreases. Additionally, it is observed that all the schemes are energy-stable while the energy profile is more dissipative for smaller values of the fractional order.