This paper introduces a new numerical method for solving space-fractional partial differential equations (PDEs) on non-uniform adaptive finite difference meshes, considering a fractional order in one dimension. The fractional Laplacian in PDE is computed by using Riemann–Liouville (R–L) derivatives, incorporating a boundary condition of the form in . The proposed approach extends the L2 method to non-uniform meshes for calculating the R–L derivatives. The spatial mesh generation employs adaptive moving finite differences, offering adaptability at each time step through grid reallocation based on previously calculated solutions. The chosen mesh movement technique, moving mesh PDE-5 (MMPDE-5), demonstrates rapid and efficient mesh movement. The numerical solutions are obtained by applying the non-uniform L2 numerical scheme and the MMPDE-5 method for moving meshes automatically. Two numerical experiments focused on the space-fractional heat equation validate the convergence of the proposed scheme. The study concludes by exploring patterns in equations involving the fractional Laplacian term within the Gray–Scott system. It reveals self-replication, travelling wave, and chaotic patterns, along with two distinct evolution processes depending on the order : from self-replication to standing waves and from travelling waves to self-replication.

中文翻译：

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一维空间分数阶 Gray-Scott 系统的自适应非均匀 L2 离散化


本文介绍了一种新的数值方法，用于求解非均匀自适应有限差分网格上的空间分数阶偏微分方程 (PDE)，并考虑一维分数阶。偏微分方程中的分数拉普拉斯算子是通过使用黎曼-刘维尔 (R-L) 导数计算的，并结合了 中形式的边界条件。所提出的方法将 L2 方法扩展到非均匀网格来计算 R-L 导数。空间网格生成采用自适应移动有限差分，通过基于先前计算的解决方案的网格重新分配在每个时间步提供适应性。所选择的网格移动技术，移动网格 PDE-5 (MMPDE-5)，展示了快速高效的网格移动。通过应用非均匀L2数值格式和自动移动网格的MMPDE-5方法获得数值解。两个针对空间分数热方程的数值实验验证了所提出方案的收敛性。该研究的结论是探索了格雷-斯科特系统中涉及分数拉普拉斯项的方程模式。它揭示了自我复制、行波和混沌模式，以及取决于顺序的两个不同的演化过程：从自我复制到驻波和从行波到自我复制。