One-dimensional wave equations defined by fractal Laplacians

TL;DR

This paper investigates wave equations governed by fractal Laplacians derived from complex measures, employing numerical methods to approximate solutions and proving convergence despite the measures' intricate fractal structure.

Contribution

It introduces a numerical framework for solving wave equations with fractal Laplacians defined by overlapping iterated function systems, including convergence proofs.

Findings

Numerical solutions converge to the weak solution.

Rate of convergence estimates are established.

Applicable to measures generated by overlaps, like Bernoulli convolutions.

Abstract

We study one-dimensional wave equations defined by a class of fractal Laplacians. These Laplacians are defined by fractal measures generated by iterated function systems with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio and the 3-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the post-critically finite condition or the open set condition. By using second-order self-similar identities introduced by Strichartz et al., we discretize the equations and use the finite element and central difference methods to obtain numerical approximations to the weak solutions. We prove that the numerical solutions converge to the weak solution, and obtain estimates for the rate of convergence.