Fractal differential equations on the Sierpiński gasket
The authors consider the bounded Sierpiński gasket and its approximating prefractal graphs. Using the self-similarity of the fractal and the decimation properties of its Laplacian they derive several exact recursive algorithms to compute harmonic and biharmonic functions as well as Dirichlet eigenfunctions on the graphs and on lines. The proofs are based on well-known decimation properties of harmonic and eigenfunctions. The algorithms are illustrated by 43 pages of graphics and used to prove some theorems on specific features of these functions. Solutions to the heat and wave equation are approximated by finite difference schemes. This results in 7 further pages of graphics and the remarkable observation that there is strong evidence for the wave propagator to have infinite speed.
scientific article; zbMATH DE number 1310354