The Fractional Diffusion Equation (FDE) is a mathematical model that describes anomalous transport phenomena characterized by non-local and long-range dependencies which deviate from the traditional behavior of diffusion. Solving this equation numerically is challenging due to the need to discretize complicated integral operators which increase the computational costs. These complexities are exacerbated by nonlinear source terms, nonsmooth data and irregular domains. In this study, we propose a second order Exponential Time Differencing Finite Element Method (ETD-RDP-FEM) to efficiently solve nonlinear FDE, posed in irregular domains. This approach discretizes matrix exponentials using a rational function with real and distinct poles, resulting in an L-stable scheme that damps spurious oscillations caused by non-smooth initial data. The method is shown to outperform existing second-order methods for FDEs with a higher accuracy and faster computational time.

翻译：分数阶扩散方程（FDE）是一种描述异常输运现象的数学模型，其特征表现为非局域性和长程依赖性，偏离了传统扩散行为。由于需要离散化复杂的积分算子，导致计算成本增加，因此对该方程进行数值求解颇具挑战性。非线性源项、非光滑数据以及不规则区域等因素进一步加剧了这些复杂性。在本研究中，我们提出了一种二阶指数时间差分有限元方法（ETD-RDP-FEM），用于高效求解定义在不规则区域上的非线性FDE。该方法利用具有实部和不同极点的有理函数对矩阵指数进行离散化，从而形成一种L-稳定格式，能够抑制由非光滑初始数据引起的虚假振荡。实验表明，该方法在求解分数阶扩散方程时，相比现有二阶方法具有更高的精度和更快的计算速度。