Time-fractional porous medium equation: Erd\'elyi-Kober integral equations, compactly supported solutions, and numerical methods

Abstract

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.