From PDEs on standard domains to self-similar particle systems on fractals

TL;DR

This paper develops a framework to construct PDEs on fractal domains from classical equations on the interval, and derives explicit particle system approximations, enabling numerical analysis of nonlocal fractal models.

Contribution

It introduces a novel method combining measure-preserving isometries, nonlocal-to-local operator approximations, and Galerkin discretization to analyze PDEs on fractals.

Findings

Constructed transported PDEs on fractals from reference equations on [0,1]

Derived explicit self-similar interacting particle systems approximating fractal PDEs

Outlined extensions to local charts and nonlocal equations on fractals.

Abstract

We construct transported PDEs on self-similar fractal domains from reference equations posed on the unit interval, and derive explicit self-similar interacting particle systems that approximate the resulting dynamics. The construction combines a measure-preserving isometry between $L^2$-spaces on $[0,1]$ and on the fractal \cite{Med2026}, a nonlocal-to-local approximation of differential operators \cite{PauTre2025}, and a Galerkin discretization on the canonical self-similar partitions. This yields a two-parameter approximation scheme whose error separates a nonlocal consistency term from a Galerkin network term. We work out the transport, Burgers, and heat equations, discuss the relation with intrinsic operators on fractals, and outline extensions to local charts and to pullbacks of nonlocal equations on fractal domains. Moreover, the reverse mapping transforms a nonlocal evolution equation on a fractal domains into the evolution equation on the unit interval, where the methods of classical numerical analysis can be applied. This suggests a promising direction for the development of numerical methods for nonlocal models on fractals, including fractional heat equation, fractal scattering, and related models.