Time–space fabric underlying anomalous diffusion

Abstract

This study unveils the time-space transforms underlying anomalous diffusion process. PACS numbers: 61.43. Hv, 47.27.Qb, 65.40.Fb, 05.40.Ca, 43.20.Bi Anomalous diffusion is one of the most important concepts in modern physics [1][2][3][4] and is present in extremely diverse engineering fields such as charges transport in amorphous semiconductor, 5 vibration and acoustic dissipation in soft matter, 6 magnetic plasma, 7 polymer dynamics, 8 turbulence 9 and quantum processes 10 among many other problems. 4 However, it is noted that the anomaly is mostly introduced in a descriptive level of statistical representation or phenomenological modeling. The purpose of this

Modelling long-term deformation of granular soils incorporating the concept of fractional calculus

“…It is indeed a memory-intensive and path-dependent phenomenon which might be mathematically represented by using the concept of fractional calculus [24][25][26] . By using the fractional calculus theory [27][28][29], Yin et al [25,26] successfully proposed a framework for modelling strain hardening and softening of geomaterials. The model could be easily incorporated in engineering-oriented finite element method due to its explicit expression.…”

Axiomatic Local Metric Derivatives for Low-Level Fractionality with Mittag-Leffler Eigenfunctions

“…By employing the local fractional differential operators in connection with the Hausdorff derivative [12], we can write that:…”

“…The local differential equation The differential equation, with the local Hausdorff derivative proposed in Ref. [12], reads:…”

Generalized Maxwell Relations in Thermodynamics with Metric Derivatives

“…Interconnections and equivalence of the metric derivatives was presented in Reference [1]particularly between definitions of fractal derivatives, called Hausdorff derivative in Reference [25], q-derivatives in Reference [26], and conformable derivatives in Reference [23]. Therein, a simple scale change in the variable was used to show that for differentiable functions, the conformable derivatives are equivalent to Hausdorff derivatives.…”

“…, one of the authors shows [1] that Equation (2) is nothing but the Hausdorff derivative [1,[3][4][5]19,25] up to a constant, and is valid for differentiable functions.…”