A parabolic equation on domains with random boundaries

TL;DR

This paper investigates a heat equation on domains with uncertain boundaries, approximating statistical moments of solutions using shape derivatives and boundary integral methods, with rigorous proofs of existence.

Contribution

It introduces a rigorous framework for analyzing heat equations on random domains and develops methods to compute statistical moments of solutions using shape derivatives.

Findings

Statistical moments of solutions are effectively approximated.

Existence of the shape derivative is rigorously proven.

Boundary integral methods facilitate computation of moments.

Abstract

A heat equation with uncertain domains is thoroughly investigated. Statistical moments of the solution is approximated by the counterparts of the shape derivative. A rigorous proof for the existence of the shape derivative is presented. Boundary integral equation methods are used to compute statistical moments of the shape derivative.