Pré-Publication, Document De Travail Année : 2026

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

Résumé

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.

Fichier principal
Vignette du fichier
preprint.pdf (1.12 Mo) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
licence

Dates et versions

hal-05601143 , version 1 (23-04-2026)

Licence

Identifiants

Citer

Georgi Medvedev, Emmanuel Trélat. From PDEs on standard domains to self-similar particle systems on fractals. 2026. ⟨hal-05601143⟩
0 Consultations
0 Téléchargements

Altmetric

Partager

  • More