  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.
