In this paper, we focus on the numerical simulation for two-dimensional nonlinear fractional sub-diffusion equations in the presence of time delay. Firstly, we investigate the existence, uniqueness and regularity of the solution for such problems. The theoretical result implies that the solution at is smoother than that at , where is a constant time delay, and this is an improvement for the work (Tan et al., 2022). Secondly, a high-order difference scheme based on method is constructed. For the sake of repairing the convergence order in temporal direction and improving the computational efficiency, an efficient time two-grid algorithm based on nonuniform meshes is first developed. The convergence order of the two-grid scheme reaches , where and represent the number of the fine and coarse grids respectively, while and are the space-step sizes. Furthermore, stability and convergence analysis of the proposed scheme are carefully verified by energy method. Finally, numerical experiments are carried out to show the validity of theoretical statements.

中文翻译：

---

变步长[公式略]结合时间二网格算法解决二维非线性时滞分式方程多奇异性问题


在本文中，我们重点研究存在时滞的二维非线性分数次扩散方程的数值模拟。首先考察此类问题解的存在性、唯一性和规律性。理论结果意味着 处的解比 处的解更平滑，其中 是恒定的时间延迟，这是对工作的改进（Tan et al., 2022）。其次，构造了基于方法的高阶差分格式。为了修复时间方向上的收敛顺序并提高计算效率，首先提出了一种基于非均匀网格的高效时间二网格算法。双网格方案的收敛阶数达到 ，其中 和 分别表示细网格和粗网格的数量，而 和 是空间步长。此外，所提出方案的稳定性和收敛性分析通过能量方法进行了仔细验证。最后，进行数值实验以证明理论陈述的有效性。