Fractal calculus approach to diffusion on fractal combs

• Published in: Chaos, solitons and fractals Vol. 175; p. 114021
• Main Authors: Khalili Golmankhaneh, Alireza, Ontiveros, Lilián Aurora Ochoa
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.10.2023
• Subjects: Combs’ staircase function; Diffusion on fractal; Fractal calculus; Fractal comb; Mean square displacement; Diffusion on fractal; Fractal calculus; Mean square displacement; Combs’ staircase function; Fractal comb
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2023.114021

In this paper, we present a generalization of diffusion on fractal combs using fractal calculus. We introduce the concept of a fractal comb and its associated staircase function. To handle functions supported on these combs, we define derivatives and integrals using the staircase function. We then derive the Fokker–Planck equation for a fractal comb with dimension α, incorporating fractal time, and provide its solution. Additionally, we explore α-dimensional and (2α)-dimensional Brownian motion on fractal combs with drift and fractal time. We calculate the corresponding fractal mean square displacement for these processes. Furthermore, we propose and solve the heat equation on an α-dimensional fractal comb space. To illustrate our findings, we include graphs that showcase the specific details and outcomes of our results. •The diffusion on fractal combs using fractal calculus is presented.•The derivatives and integrals on fractal combs are defined.•The Fokker–Planck equation for a fractal comb is derived, and its solution provided.•The Brownian motion on fractal combs with drift and fractal time is explored.