Numerical method for the time-fractional porous medium equation

TL;DR

This paper develops and analyzes a finite difference scheme for solving the complex, nonlocal, and nonlinear time-fractional porous medium equation, demonstrating convergence and providing practical examples.

Contribution

It introduces a novel finite difference approach for the time-fractional porous medium equation, including convergence analysis and a specific midpoint quadrature method.

Findings

Convergent finite difference schemes are identified for a broad parameter range.

The method effectively handles nonlocal and nonlinear features of the equation.

Numerical examples validate the theoretical convergence results.

Abstract

This papers deals with a construction and convergence analysis of a finite difference scheme for solving time-fractional porous medium equation. The governing equation exhibits both nonlocal and nonlinear behaviour making the numerical computations challenging. Our strategy is to reduce the problem into a single one-dimensional Volterra integral equation for the self-similar solution and then to apply the discretization. The main difficulty arises due to the non-Lipschitzian behaviour of the equation's nonlinearity. By the analysis of the recurrence relation for the error we are able to prove that there exists a family of finite difference methods that is convergent for a large subset of the parameter space. We illustrate our results with a concrete example of a method based on the midpoint quadrature.