On the time fractional heat equation with obstacle

TL;DR

This paper investigates a Caputo time fractional degenerate diffusion equation, establishing its equivalence to a fractional parabolic obstacle problem, analyzing convergence to stationary states, and comparing numerical methods for various fractional orders.

Contribution

It extends previous results to all fractional orders in (0,1), demonstrating convergence behavior and providing numerical comparisons for the fractional heat equation with obstacle.

Findings

Solutions converge to the classical obstacle problem stationary state.

Convergence speed varies with the fractional order .

Numerical methods show consistent results across different fractional orders.

Abstract

We study a Caputo time fractional degenerate diffusion equation which we prove to be equivalent to the fractional parabolic obstacle problem, showing that its solution evolves for any $\alpha\in(0,1)$ to the same stationary state, the solution of the classic elliptic obstacle problem. The only thing which changes with $\alpha$ is the convergence speed. We also study the problem from the numerical point of view, comparing some finite different approaches, and showing the results of some tests. These results extend what recently proved in [1] for the case $\alpha=1$.