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Title: Time-fractional porous medium equation: Erdélyi–Kober integral equations, compactly supported solutions, and numerical methods
Authors: López, Belén 
Okrasińska-Płociniczak, Hanna
Płociniczak, Łukasz
Rocha, Juan 
UNESCO Clasification: 12 Matemáticas
120219 Ecuaciones diferenciales ordinarias
Keywords: Erdélyi–Kober fractional operator
Numerical method
Time-fractional porous medium equation
Issue Date: 2024
Journal: Communications in Nonlinear Science and Numerical Simulation 
Abstract: 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. The shooting method is not only useful for deriving theoretical results. We use it to devise an efficient numerical scheme to solve the governing problem along with two ways to discretize the Erdélyi–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.
ISSN: 1007-5704
DOI: 10.1016/j.cnsns.2023.107692
Source: Communications in Nonlinear Science and Numerical Simulation [ISSN 1007-5704], v. 128, (Enero 2024)
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