FUNDING
4. DISCUSSION
Herein, the subordination procedure provides an equivalent description of the average dynamics of a single individual within a complex network, in terms of a linear fractional differential equation. The fractional rate equation is solved exactly, determining the Poisson statistics of the isolated individual becomes attenuated Mittag-Leffler statistics, owing to the interaction of that individual with the other members of a complex dynamic network.
1Adjusted goodness of fit,R2=1−SSSSregtot/(n−κ)/(n−1), is defined as the ratio of the sum of squared residues for the nonlinear fit with the MLF (SSreg) and for the fit to the average value of data points (SStot), wherenis the number of data points andκis the number of free parameters being estimated.
Consequently, an individual’s simple random behavior, when isolated, is replaced with behavior that might serve a more adaptive role in social networks. We conjecture that the behavior of the individual is generic, given that the DMM network dynamics belong to the Ising universality class.
Members of this universality class share the critical temporal behavior [28] that drives the subordination process. It is the renewal property of the event statistics, which, through the subordination process, gives rise to the linear fractional master equation for the typical individual’s dynamics. The solution to the tempered fractional rate equation manifests the subsequent robust behavior of the individual; it remains to be determined just how robust the behavior of the individual is relative to control signals that might be driving the network.
As pointed out by Liu et al. [29], the ultimate understanding of complex networks is reflected in the ability to control them.
Recent observations of the interconnectedness of infrastructure networks [30], facilitating the spread of failures [31] or the tight coupling between banking institutions, posing a danger to the stability of global financial system [32], demonstrate the importance of developing a systematic approach to influence and/or control the complex networks. The analysis presented here provides an alternative attempt to address this need directly. Subordination suggests a way to impose the conditions of traditional control theory [33] onto the complex network dynamics by, first, expressing the underlying nonlinear network dynamics in the form of a linear fractional equation of motion.
This approach at addressing control will be pursued in a future publication.
AUTHOR CONTRIBUTIONS
BW developed the theoretical formalism and performed the analytic calculations. MT performed numerical simulations.
Both authors discussed the results and contributed to the final manuscript.
ACKNOWLEDGMENTS
The authors acknowledge the US Army Research Laboratory for supporting this research.
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Conflict of Interest Statement: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Copyright © 2018 Turalska and West. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
ORIGINAL RESEARCH published: 24 October 2018 doi: 10.3389/fphy.2018.00118
Edited by:
Dumitru Baleanu, University of Craiova, Romania Reviewed by:
Xiao-Jun Yang, China University of Mining and Technology, China Jordan Yankov Hristov, University of Chemical Technology and Metallurgy, Bulgaria Abdullahi Yusuf, Federal University, Nigeria
*Correspondence:
Carlo Cattani [email protected]
Specialty section:
This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics Received:30 August 2018 Accepted:28 September 2018 Published:24 October 2018 Citation:
Cattani C (2018) Sinc-Fractional Operator on Shannon Wavelet Space.
Front. Phys. 6:118.
doi: 10.3389/fphy.2018.00118
Sinc-Fractional Operator on Shannon Wavelet Space
Carlo Cattani*
Engineering School, DEIM, Tuscia University, Viterbo, Italy
In this paper the sinc-fractional derivative is extended to the Hilbert space based on Shannon wavelets. Some new fractional operators based on wavelets are defined. One of the main task is to investigate the localization and compression properties of wavelets when dealing with the non-integer order of a differential operator.
Keywords: fractional calculus, shannon wavelet, sinc-function, operational matrix, connection coefficients
1. INTRODUCTION
In recent years, fractional calculus has been growing fast both in theory and applications to many different fields. Several classical and fundamental problems have been revised by using fractional methods, thus showing unexpected new results [1–3], while more and more new problems were shaped to fit the theoretical models of fractional calculus [4–7].
In fractional calculus is based on two universally accepted principles: the first one is that the definition of fractional derivative is not unique, thus giving raise to a neverending controversial debate on the best fractional operator. The second principle is that, although the missing uniqueness of the fractional operator, fractional calculus is an essential tool for a deeper and more comprehensive investigation of complex , non-linear , local, or non-local problems.
Therefore according to the suitable choice of the fractional differential operator, there follows a corresponding model of analysis so that the physical model and the corresponding physical interpretation of the results it strongly depends on the chosen fractional operator.
In some recent papers [8–15] the classical Lie symmetry analysis has been combined with the Riemman-Liouville fractional derivative to solve time fractional partial differential equations.
In these papers, Lie point symmetries have been used to convert a fractional partial differential equation into a non-linear ordinary differential equation, that can be solved by suitable methods.
Some fractional operators have been used also to study non-differentiable functions [see e.g., [16]
some of them are more suitable for the analysis of non-differentiable sets, or fractal sets like the Cantor fractal set [4–7] Some fractional operators have been specially defined to analyze complex functions [17–19]. For instance the chaotic decay to zero of the complexζ-Riemann function was easily shown by using a suitable fractional derivative [19].
Among the many interesting definitions of fractional operators, some Authors have recenlty proposed a fractional differential operator based on the sinc-function [20]. This function is very popular in the signal analysis, also because it is a localized function with slow decay. Moreover, it is the fundamental basic function for the definition of the so-called Shannon wavelet theory, i.e., the multiscale analysis on Shannon wavelets [21–26].
This paper will focus on the definition of a fractional derivative by the Shannon wavelets.
These functions belong to a special family of wavelets which have a sharp compact support in the
frequency space, so that their Fourier transform are box- functions in frequencies. This is a great advantage because, the frequency domain of a signal can be easily decomposed in terms of scaled box-functions.
Wavelet theory has been growing very fast so that there has been also a wide spreading of wavelets for the solution of theoretical and applied problems. However, alike the various definition of fractional operators there exist also many different families of wavelets and this missing uniqueness it might be considered as a drawback because of the arbitrary choice. Nevertheless all families of wavelets enjoy two fundamental properties their localization in time (or frequency) and the multiscale decomposition. Due to their localization they can be used to detect, and single out, localized singularities and/or peaks, while the multiscale property enable to decompose the approximation space into separate scales [27]. Thanks to these properties wavelets have been used to solve non-linear problems and moreover they are the most suitable tool for the analysis of multiscale problems.
The sinc-fractional operator will be generalized in order to compute the fractional derivative of the L2(R)-functions belonging to the Hilbert space defined by the Shannon wavelet.
In doing so, we will be able to compute the fractional derivative of these functions by knowing only their wavelet coefficients. Moreover, with this approach we will be able to decompose the fractional derivative at different scales, thus showing the influence of a given scale in multiscale physical problems.
The organization of this paper is as follows: Preliminary remarks on fractional operators are given in section 2. In section 3 the sinc fractional derivative, as given by Yang et al. [20] is described. Section 4 gives the basic properties on the multiscale approximation defined on Shannon wavelet. The differential properties of the functions belonging to the Hilbert space based on Shannon wavelet are given in section 5, together with the explicit form of the integer order derivatives (see also [24,26]).
Section 6 deals with the sinc-fractional derivative on the Hilbert space based on Shannon wavelets , i.e., sinc-fractional derivative of functions which can be represented as Shannon wavelet series.