Fractal calculus approach to diffusion on fractal combs

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 a, incorporating fractal time, and provide its solution. Additionally, we explore a-dimensional and (2a)-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 a-dimensional fractal comb space. To illustrate our findings, we include graphs that showcase the specific details and outcomes of our results.

Automated summary

This paper presents a generalization of diffusion on fractal combs using fractal calculus, including the concepts, definitions of derivatives and integrals, derivation and solution of the Fokker-Planck equation, study of Brownian motion, and proposal and solution of the heat equation on fractal combs.