Carlo Alberini | Searcharxiv

arXiv2020

On the time fractional heat equation with obstacle

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 $α\in(0,1)$ to the same stationary state, the solution of the classic elliptic obstacle problem. The only thing which changes with $α$ 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 $α=1$.

math.AP↗

arXiv2020

A numerical study of an Heaviside function driven degenerate diffusion equation

We analyze a nonlinear degenerate parabolic problem whose diffusion coefficient is the Heaviside function of the distance of the solution itself from a given target function. We show that this model behaves as an evolutive variational inequality having the target as an obstacle: under suitable hypotheses, starting from an initial state above the target the solution evolves in time towards an asymptotic solution, eventually getting in contact with part of the target itself. We also study a finite difference approach to the solution of this problem, using the exact Heaviside function or a regular approximation of it, showing the results of some numerical tests.

math.AP↗