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.

中文翻译：

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空间分数艾伦-卡恩方程及其在相分离中的应用：数值研究


非局域性和空间异质性现象非常复杂，使用分数阶微分方程为理解这些特征提供了一种稳健的建模方法。另一方面，对此类现象进行数值近似既耗时又具有挑战性。在本文中，我们采用谱方法求解空间分数艾伦-卡恩方程进行数值研究。 Allen–Cahn 是广泛用于相分离动力学的反应扩散方程。谱方法最吸引人的特点是它能够提供分数算子的完全对角表示。进行数值模拟以验证所提出方案的有效性、准确性和功效，并探索分数阶如何影响解决方案的动态。对于不同的分数阶值，观察到解分布趋于平衡的趋势。可以看出，解的稳定速率受到分数阶值的显着影响，较大的值导致较快的速率，较小的值导致较慢的速率。此外，据观察，随着分数阶的增加，界面轮廓表现出平滑和扩散的行为，但随着分数阶的降低，界面轮廓变得尖锐。此外，据观察，所有方案都是能量稳定的，而对于较小的分数阶值，能量分布更加耗散。