Approximation of random diffusion equation by nonlocal diffusion equation in free boundary problems of one space dimension

TL;DR

This paper demonstrates how a free boundary problem with random diffusion in one dimension can be approximated by a nonlocal diffusion problem, providing error estimates and a method for the approximation.

Contribution

It introduces a novel approximation method for Stefan type free boundary problems with random diffusion using nonlocal diffusion with explicit error bounds.

Findings

Successful approximation of random diffusion free boundary problems by nonlocal models

Development of auxiliary free boundary problems for analysis

Error estimates proportional to a positive power of epsilon

Abstract

We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [4,8]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form $J_\epsilon(x)=\frac 1\epsilon J(\frac x\epsilon)$ for small $\epsilon>0$, where $J(x)$ has compact support. We also give an estimate of the error term of the approximation by some positive power of $\epsilon$.