A physically based connection between fractional calculus and fractal geometry

Abstract

We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry. © 2014 Elsevier Inc

Fatigue-Induced Cole Electrical Impedance Model Changes of Biceps Tissue Bioimpedance

“…Instead, these models assume that the behaviour is captured in the fractal structure of the tissue [16,17]. Recent research has shown the relationship between fractals and fractional calculus, based on physical and geometric considerations [21]. It was noted in [17] that muscle fibers, tendon, and nerve fibers exhibit patterns that support the dynamics of these multiscale structures expressed by fractional-order models.…”

Fractional calculus in pharmacokinetics

“…It has been shown that differential equations with fractional derivatives (FDEs) describe experimental data of anomalous diffusion more accurately. Although fractional-order derivatives were first introduced as a novel mathematical concept with unclear physical meaning, nowadays, a clear connection between diffusion over fractal spaces (such as networks of capillaries) and fractional-order dynamics has been established [15][16][17][18]. In particular, in anomalous diffusion the standard assumption that the mean square displacement x 2 is proportional to time does not hold.…”

Analogues to Lie Method and Noether’s Theorem in Fractal Calculus

“…There is a lot of research in this direction, such as fractional Fokker-Planck equations, fractional diffusion equations, fractional master equations, stochastic fractional equations, and so on [25][26][27][28][29][30][31][32]. Fractal features and macroscopic anomalous exploits of systems are connected to the order of fractional derivatives [33][34][35][36][37][38][39][40]. In addition, memory effects in physical models are presented by using fractional derivatives because of their non-local character [41].…”