The time-fractional porous medium equation is an important model of many hydrological, physical, and chemical flows. We study its self-similar solutions, which make up the profiles of many important experimentally measured situations. We prove that there is a unique solution to the general initial-boundary value problem in the one-dimensional setting. When supplemented with boundary conditions from the physical models, the problem exhibits a self-similar solution described with the use of the Erd\'elyi-Kober fractional operator. Using a backward shooting method, we show that there exists a unique solution to our problem. The shooting method is not only useful in deriving the theoretical results. We utilize it to devise an efficient numerical scheme to solve the governing problem along with two ways of discretizing the Erd\'elyi-Kober fractional derivative. Since the latter is a nonlocal operator, its numerical realization has to include some truncation. We find the correct truncation regime and prove several error estimates. Furthermore, the backward shooting method can be used to solve the main problem, and we provide a convergence proof. The main difficulty lies in the degeneracy of the diffusivity. We overcome it with some regularization. Our findings are supplemented with numerical simulations that verify the theoretical findings.

翻译：时间分数阶多孔介质方程是诸多水文、物理及化学流动中的重要模型。本文研究其自相似解，这些解构成了众多关键实验测量场景的剖面。我们证明了一维情形下一般初边值问题存在唯一解。当补充物理模型中的边界条件时，该问题呈现出自相似解，该解可通过Erdélyi-Kober分数阶算子进行描述。利用逆向打靶法，我们证明问题存在唯一解。该打靶法不仅有助于推导理论结果，还可用于设计高效数值方案以求解控制方程，同时结合两种离散化Erdélyi-Kober分数阶导数的方法。由于后者为非局部算子，其数值实现需引入截断处理。我们确定了正确的截断机制，并证明了若干误差估计。此外，逆向打靶法可用于求解主问题，我们提供了收敛性证明。主要难点在于扩散项的退化性，我们通过正则化方法加以克服。数值模拟结果验证了理论发现。