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

TL;DR

This paper develops a second order numerical scheme for solving a time-fractional porous medium equation on the half-line, addressing memory effects in nonlinear systems and demonstrating improved accuracy and convergence over traditional methods.

Contribution

The paper introduces a novel second order numerical scheme for a time-fractional porous medium equation, proving its convergence and outperforming standard finite difference methods.

Findings

The scheme achieves higher accuracy in numerical solutions.

Convergence of the proposed methods is rigorously proven.

Numerical examples validate the effectiveness of the approach.

Abstract

Many physical, biological, and economical systems exhibit various memory effects due to which their present state depends on the history of the whole evolution. Combined with the nonlinearity of the process these phenomena pose serious difficulties in both analytical and numerical treatment. We investigate a time-fractional porous medium equation that has proved to be important in many applications, notably in hydrology and material sciences. We show that solutions of the free boundary Dirichlet, Neumann, and Robin problems on the half-line satisfy a Volterra integral equation with a non-Lipschitz nonlinearity. Based on this result we prove existence, uniqueness, and construct a family of numerical methods that solve these equations outperforming the usual finite difference approach. Moreover, we prove the convergence of these methods and support the theory with several numerical examples.