Two-scale mathematics and fractional calculus for thermodynamics

Abstract

A three dimensional problem can be approximated by either a two-dimensional or one-dimensional case, but some information will be lost. To reveal the lost information due to the lower dimensional approach, two-scale mathematics is needed. Generally one scale is established by usage where traditional calculus works, and the other scale is for revealing the lost information where the continuum assumption might be forbidden, and fractional calculus or fractal calculus has to be used. The two-scale transform can approximately convert the fractional calculus into its traditional partner, making the two-scale thermodynamics much promising.

Robust passivity analysis of Markov‐type Lotka–Volterra model with time‐varying delay and uncertain mode transition rates

“…3 The fractional calculus and fractional order models have been studied in depth as in He, 14 He, 15 and Khan et al 16 and were widely applied to deal with discontinuous problems. 17,18 Liu et al suggested a fractional LV model, and its oscillation property is studied. 7 A delayed model was established by introducing disease into the predator group and the dynamical characteristics of various equilibrium points are investigated.…”

“…22 In general, the uncertainty may cause oscillation, divergence, or instability. For some research based on fractal calculus to deal with the uncertain problem that arises from using different observation scales, we can refer to Liu et al 7,18 and He and Ji. 7,18 The authors have also investigated the LV model with interval uncertainty, uncertain impulsive, switching, polytopic uncertainty, and so on based on the integral calculus and ordinary differential equations as in other studies.…”

“…For some research based on fractal calculus to deal with the uncertain problem that arises from using different observation scales, we can refer to Liu et al 7,18 and He and Ji. 7,18 The authors have also investigated the LV model with interval uncertainty, uncertain impulsive, switching, polytopic uncertainty, and so on based on the integral calculus and ordinary differential equations as in other studies. [22][23][24][25][26] As we know, the system would switch/jump from one environmental regime to another one that is governed by a Markovn chain.…”

Fractal calculus for analysis of wool fiber: Mathematical insight of its biomechanism

“…In this article, the fractal structure of wool fiber is investigated using the two-scale dimension method. 8,9 Fractal calculus of wool fiber on the two-scale dimension Like most natural things, the hierarchic structure of wool fiber does not obey rigorous self-similarity. Fortunately, the two-scale dimension proposed by He and colleagues 8,9 provides an effective tool to dealing such a fractal problem that lacks a self-similar structure.…”

A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative

“…FDEs also provide marvellous tools for the depiction of memory and inherited properties of many materials and processes. In view of recent developments, one can consequently conclude that FDEs have emerged significant achievements in the last couple of decades [10][11][12][13][14][15][16]. The study of integral equations in the scope of non-integer-order equations has been in the spotlight in the recent years.…”