Time-fractional porous medium equation: Erdélyi–Kober integral equations, compactly supported solutions, and numerical methods

The time-fractional porous medium equation is an important model for 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 a 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 Erdélyi–Kober fractional operator. Using a backward shooting method, we show that there exists a unique solution to our problem.