New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

Abstract

Any physical laws are scale-dependent, the same phenomenon might lead to debating theories if observed using different scales. The two-scale thermodynamics observes the same phenomenon using two different scales, one scale is generally used in the conventional continuum mechanics, and the other scale can reveal the hidden truth beyond the continuum assumption, and fractal calculus has to be adopted to establish governing equations. Here basic properties of fractal calculus are elucidated, and the relationship between the fractal calculus and traditional calculus is revealed using the two-scale transform, fractal variational principles are discussed for 1-D fluid mechanics. Additionally planet distribution in the fractal solar system, dark energy in the fractal space, and a fractal ageing model are also discussed.

New aspects on the fractional Euler-Lagrange equation with non-singular kernels

“…Additionally, the initial findings of this paper are the inclusion of the dissipative term as shown in the obtained fractionally-modified Euler-Lagrange equation which is far different to the ones obtained using the Caputo operator, that contains a singular kernel function [13,[18][19][20]25], and for the conformable operator [22]. Applications of the concepts found in this paper are an essential extension of this current work, such as the works done in [26,27], the invariant conditions, Noether's theorem, and the DuBois-Reymond condition. Lastly, the continuation of this work will focus on finding the energy function, which can describe the origin of the found dissipative term on the fractionally-modified Euler-Lagrange equation and to further extend with their physical implications.…”

“…Remark 1 Definition (10) is the fractional Euler-Lagrange equation in the framework of Caputo-Fabrizio operator, and the dissipative term is given on the right handside of the equation (27). The classical Euler-Lagrange equation is recovered when α → 1.…”

A powerful and simple frequency formula to nonlinear fractal oscillators

“…The geometric illustration of the fractal two-scale transform method was given in He and Ain. 23 The advantage of the fractal two-scale transform method is that it can be adopted convert approximately a fractal space into its continuous partner.…”

He's frequency formulation for fractal nonlinear oscillator arising in a microgravity space

“…In this article, the fractal variational principle of fractal nonlinear oscillator equation is obtained via the fractal semi-inverse transform method [29,30] and He's frequency formulation [31,32], and two-scale transform method [33][34][35][36] is successfully used to find its approximate frequency in microgravity space. By comparison, we can clearly observe that the method proposed in this article is simple, efficient, and accurate for dealing with nonlinear oscillation problems.…”