Fractal calculus and its geometrical explanation

By Ji-Huan He. Results in Physics, 2018

457 citations recorded by the source corpus.

Abstract Fractal calculus is very simple but extremely effective to deal with phenomena in hierarchical or porous media. Its operation is almost same with that by the advanced calculus, making it much accessible to all non-mathematicians. This paper begins with the basic concept of fractal gradient of temperature, i.e., the temperature change between two points in a fractal medium, to reveal the basic properties of fractal calculus. The fractal velocity and fractal material derivative are then introduced to deduce laws for fluid mechanics and heat conduction in fractal space. Conservation of mass in a fractal space is geometrically explained, and an approximate transform of a fractal space on a smaller scale into its continuous partner on a larger scale is illustrated by a nanofiber membrane, which is smooth on any observable scales, but its air permeability has to studied in a nano scale, under such a small scale, the nanofiber membrane becomes a porous one. Finally an example is given to explain cocoon’s heat-proof property, which cannot be unveiled by advanced calculus.

Subfields: Mathematical Physics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d0d7d9bdd41a0c7cda1cdcd6a7dee1fb4e6ac5e6

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Citations in the corpus

On two-scale dimension and its applications (2019, unknown): “where  is the two-scale dimension, T/x  – the fractal derivative defined [11, 12]:”
The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators (2019, unknown): “…in case the Duffing equation’s frequency is known u00 þ uþ eu3 þ p½fðuÞ u eu3 ¼ 0 (37) or u00 þ uþ ðe0 þ e1pþ e2p2 þ Þu3 þ p fðuÞ u eu3 ¼ 0 (38) where e0 þ e1 þ e2 þ ¼ e (39) Li and He18 suggested a modification of the homotopy perturbation method coupled with the enhanced perturbation method,…”
New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle (2020, background): “Before proceeding further, we first give some definitions and theorems on fractal calculus for easy understanding[7,61-64].”
The simplest approach to nonlinear oscillators (2019, unknown): “The method can also extend to fractal oscillators [9,10] and the variational approach to nonliner oscillators [11,12].”