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

by   Belen López, et al.

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 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 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 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.


page 1

page 2

page 3

page 4


On Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions

In this paper, we are interested in the study of a problem with fraction...

Error estimates for backward fractional Feynman-Kac equation with non-smooth initial data

In this paper, we are concerned with the numerical solution for the back...

Second order scheme for self-similar solutions of a time-fractional porous medium equation on the half-line

Nonlocality in time is an important property of systems in which their p...

Error Estimates for a Linearized Fractional Crank-Nicolson FEM for Kirchhoff type Quasilinear Subdiffusion Equation with Memory

In this paper, we develop a linearized fractional Crank-Nicolson-Galerki...

Numerical schemes for reconstructing profiles of moving sources in (time-fractional) evolution equations

This article is concerned with the derivation of numerical reconstructio...

Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

We present an exponentially convergent numerical method to approximate t...

Controllability properties from the exterior under positivity constraints for a 1-D fractional heat equation

We study the controllability to trajectories, under positivity constrain...

Please sign up or login with your details

Forgot password? Click here to reset