Accurate relationships between fractals and fractional integrals: New approaches and evaluations

Abstract

In this paper the accurate relationships between the averaging procedure of a smooth function over 1D-fractal sets and the fractional integral of the RL-type are established. The numerical verifications are realized for confirmation of the analytical results and the physical meaning of these obtained formulas is discussed. Besides, the generalizations of the results for a combination of fractal circuits having a discrete set of fractal dimensions were obtained. We suppose that these new results help understand deeper the intimate links between fractals and fractional integrals of different types. These results can be used in different branches of the interdisciplinary physics, where the different equations describing the different physical phenomena and containing the fractional derivatives and integrals are used.MSC 2010 : Primary 28A80, 26A33; Secondary: 60G18, 26A30, 28A78 Key Words and Phrases: fractal object, self-similar object, spatial fractional integral, averaging of smooth functions on spatial fractal sets, Cantor set c

Atrial Rotor Dynamics Under Complex Fractional Order Diffusion

“…Although, there is a general agreement about a relation between both theories, the formal mathematical arguments supporting this relation are still being developed. Important advances in this regard have been made in the last two decades (Sornette, 1998 ; Nigmatullin and Le Mehaute, 2005 ; Nigmatullin and Baleanu, 2013 ; Calcagni, 2017 ; Nigmatullin et al, 2017 ). Therefore, this work contextualizes this theoretical frame and situates it within the scope of cardiac electrophysiological systems.…”

“…The biophysical interpretation of the complex derivative order is build on the mathematical theory of smoothed functions averaged over fractal sets (Nigmatullin and Le Mehaute, 2005 ). This problem has been solved for fractional temporal derivatives and the relation with the complex dimension has been stablished (Nigmatullin et al, 2017 ). This theory is supported by experimental results (Nigmatullin et al, 2007 , 2018 ).…”

Fractal Calculus of Functions on Cantor Tartan Spaces

“…The fractional derivatives have non-local property which are suitable to model the process with the memory effect, nonconservative systems [22,23,30,39,41,24,27,28,33,18]. The fractional calculus, which involves derivatives with arbitrary orders, has applied on the process with fractals structures [37,29,9]. The anomalous diffusion on fractals was formulated which included sub-and supper diffusion in view of different random walks [40,11,38].…”

The Complex-Order Fractional Laplacian: Stable Operators, Spectral Methods, and Applications to Heterogeneous Media

“…In particular, the real-order fractional Laplacian has been extensively exploited in the context of superdiffusion [11]. Importantly, recent theoretical results have established a reciprocity between complex-order fractional operators and the averaging of smooth functions over fractal sets [12][13][14]. Such results could represent a powerful methodology to account for inhomogeneities at a fractal (micro-)scale, while still preserving a mesoscale description of heterogeneous media.…”

“…In this work, we introduce suitable definitions for the complex-order fractional Laplacian, fully consistent with the aforementioned theory of averaging over fractal sets [12][13][14], and propose tailored spectral methods for their numerical treatment. Numerical simulations of diffusion processes driven by the complex-order fractional Laplacian demonstrate that the proposed operator attains the intended connection between diffusion and diffraction phenomena, with solutions exhibiting a dual particle-wave behaviour where wave-like features arise depending on the magnitude of the complex part of the fractional order.…”