Search results for "self-similarity."
showing 10 items of 71 documents
Self-similarity and response of fractional differential equations under white noise input
2022
Self-similarity, fractal behaviour and long-range dependence are observed in various branches of physical, biological, geological, socioeconomics and mechanical systems. Self-similarity, also termed self-affinity, is a concept that links the properties of a phenomenon at a certain scale with the same properties at different time scales as it happens in fractal geometry. The fractional Brownian motion (fBm), i.e. the Riemann-Liouville fractional integral of the Gaussian white noise, is self-similar; in fact by changing the temporal scale t -> at (a > 0), the statistics in the new time axis (at) remain proportional to those calculated in the previous axis (t). The proportionality coeffi…
Fractional differential equations solved by using Mellin transform
2014
In this paper, the solution of the multi-order differential equations, by using Mellin Transform, is proposed. It is shown that the problem related to the shift of the real part of the argument of the transformed function, arising when the Mellin integral operates on the fractional derivatives, may be overcame. Then, the solution may be found for any fractional differential equation involving multi-order fractional derivatives (or integrals). The solution is found in the Mellin domain, by solving a linear set of algebraic equations, whose inverse transform gives the solution of the fractional differential equation at hands.
On boundaries of attractors in dynamical systems
2021
Abstract Fractal geometry is one of the beautiful and challenging branches of mathematics. Self similarity is an important property, exhibited by most of the fractals. Several forms of self similarity have been discussed in the literature. Iterated Function System (IFS) is a mathematical scheme to generate fractals. There are several variants of IFSs such as condensation IFS, countable IFS, etc. In this paper, certain properties of self similar sets, using the concept of boundary are discussed. The notion of boundaries like similarity boundary and dynamical boundary are extended to condensation IFSs. The relationships and measure theoretic properties of boundaries in dynamical systems are a…
Self-similarity and scaling of thermal shock fractures
2013
The problem of crack pattern formation due to thermal shock loading at the surface of half-space is solved numerically using two-dimensional boundary element method. The results of numerical simulations with 100-200 random simultaneously growing and interacting cracks are used to obtain scaling relations for crack length and spacing. The numerical results predict that such process of pattern formation with quasi-static crack growth is not stable and at some point the excess energy leads to unstable propagation of one of the longest crack. The onset of instability has also been determined from numerical results.
Closure to "Deducing a drain spacing formula by applying dimensional analysis and self-similarity theory" by Vito Ferro
2017
This paper is a closure of the discussions to the paper "Deducing a drain spacing formula by applying dimensional analysis and self-similarity theory"
Flow resistance of overland flow on a smooth bed under simulated rainfall
2020
In this paper a recently theoretically deduced flow resistance equation, based on a power-velocity profile, was tested using laboratory measurements by Yoon and Wenzel for an overland flow on a smooth bed under rainfall. These measurements of the Darcy-Weisbach friction factor, corresponding to a wide range of the flow Reynolds number (191–5700), were carried out for an overland flow under a simulated rainfall characterized by different intensity values ranging from 13 to 381 mm h−1. At first, the available measured values of flow velocity, water depth, cross sectional area, wetted perimeter and bed slope were used to calibrate the relationship between the velocity profile parameter Γ, the …
SELF SIMILARITY IN SWELLING SYSTEMS: FRACTAL PROPERTIES OF PEAT
1994
Sphagnum peat gives an example of a swelling system with a self-similar structure in sufficiently wide range of scales. The surface fractal dimension, dfs, has been calculated by means of thermodynamic method on the basis of water adsorption and capillary equilibrium measurements. This method makes possible the exploration of the self-similarity in the scale range over at least 4 decimal orders of magnitude from 1 nm to 10 μm. In a sample explored, two ranges of fractality have been observed: dfs ≈ 2.55 in the range 1.5–80 nm and dfs ≈ 2.42 in the range 0.25–9 µm.
Visualization of tonal content with self-organizing maps and self-similarity matrices
2005
This article presents a dynamic model of tonality perception based on a short-term memory model and a self-organizing map (SOM). The model can be used for dynamic visualization of perceived tonal content, making it possible to examine the clarity and locus of tonality at any given point of time. This article also presents a method for the visualization of tonal structure using self-similarity matrices. The methods are applied to compositions of J. S. Bach, S. Barber, and J. Pachelbel. Finally, a real-time application embracing the tonality perception model is presented.
Assessing flow resistance law in vegetated channels by dimensional analysis and self-similarity
2019
Abstract In this paper experimental data collected by Kouwen et al., Wilson and Horrit, Raffaelli et al. and Carollo et al., using straight flumes having a bed covered by grass-like vegetation with different stem concentrations, were used to analyze flow resistance for flexible submerged elements. At first, the dimensional analysis and the incomplete self-similarity hypothesis was applied to deduce the flow velocity distribution and the resulting theoretical expression of the Darcy-Weisbach friction factor. Then, a relationship between the Γ function of the velocity profile and the biomechanical characteristics of vegetation, the channel slope, the Reynolds number and the flow Froude number…
Closure to “New Theoretical Solution of Stage-Discharge Relationship for Slit Weirs” by Vito Ferro and Ismail Aydin
2018
In this paper, the flow-process of a slit weir was analyzed on the basis of a theorem of dimensional analysis and incomplete self-similarity theory. The theoretically deduced stage-discharge formula then was calibrated using experimental data obtained for a ratio between the weir and the channel width, ranging from 0.05 to 0.25. The deduced stage-discharge relationship allowed measuring discharge values characterized by errors that, for 98% of the measured values, were less than or equal to +/- 5%. The performance of the proposed theoretical stage-discharge formula also was improved by introducing the Reynolds number (for 98.5% of the measured values the error was less than or equal to +/- …