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Generalized Fractional Dynamics in Graphs and Complex Systems
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Generalized Fractional Dynamics in Graphs and Complex Systems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 9684

Special Issue Editors

Prof. Dr. Federico Polito

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Guest Editor
Department of Mathematics “Giuseppe Peano”, University of Torino, 10123 Torino, Italy
Interests: fractional stochastic processes; branching models
Dr. Thomas Michelitsch

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Guest Editor
Institut Jean le Rond ďAlembert, Sorbonne Université, CNRS UMR 7190 4 Place Jussieu, CEDEX 05, 75252 Paris, France
Interests: applied mathematics; mathematical modelling and analysis; dynamics and wave propagation; network science; random walks; stochastic processes; fractional calculus and its generalizations; applied probability; fractals
Dr. Alejandro P. Riascos

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Guest Editor
Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de México 01000, Mexico
Interests: complex systems; networks; random walks; fractional calculus; quantum mechanics

Special Issue Information

Dear Colleagues,

Random walks are among the most fundamental stochastic processes that occur ubiquitously in various interdisciplinary contexts, such as in biological networks, the foraging of animals, epidemic spreading, financial markets, human mobility patterns, friendship networks, communication in online networks, aging of living systems, anomalous transport and diffusion, and generally in the dynamics of “complex systems”.

It has become apparent that there are deep connections of several aspects of the “complexity” of a system and fractional dynamic behavior exhibiting long-range correlations and self-similarity (scaling laws) with non-Markovian long memory features.

The Special Issue “Generalized Fractional Dynamics in Graphs and Complex Systems” aims to gather research articles or reviews with recent advances in wider interdisciplinary areas such as continuous-time random walks, stochastic motions in networks with long-range jumps, turbulent diffusion and motions, random search, anomalous transport phenomena, Lévy flights, stochastic motions in directed graphs, epidemic spreading, quantitative models of aging phenomena in living systems, models on fractal aspects of stochastic motions and related topics.

The focus of this Special Issue also encourages contributions involving new approaches of recently emerging generalizations of fractional calculus; e.g., “Prabhakar general fractional calculus” and their applications.

Prof. Dr. Federico Polito
Dr. Thomas Michelitsch
Dr. Alejandro P. Riascos
Guest Editors

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Keywords

  • Random motions in networks
  • Long-range navigation in graphs
  • Anomalous transport
  • Biased random motion
  • General Fractional Calculus
  • Mittag-Leffler functions and generalizations
  • Fat-tailed distributions and scaling laws

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Published Papers (4 papers)

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20 pages, 566 KiB  
Article
Semi-Markovian Discrete-Time Telegraph Process with Generalized Sibuya Waiting Times
by Thomas M. Michelitsch, Federico Polito and Alejandro P. Riascos
Mathematics 2023, 11(2), 471; https://doi.org/10.3390/math11020471 - 16 Jan 2023
Cited by 1 | Viewed by 1546
Abstract
In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred to this random walk as the ‘squirrel random walk’ (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is [...] Read more.
In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred to this random walk as the ‘squirrel random walk’ (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unchanged at uneventful time instants. We first recall general notions of the SRW. The main subject of the paper is the study of the SRW where the step direction switches at the arrival times of a generalization of the Sibuya discrete-time renewal process (GSP) which only recently appeared in the literature. The waiting time density of the GSP, the ‘generalized Sibuya distribution’ (GSD), is such that the moments are finite up to a certain order r≤m−1 (m≥1) and diverging for orders r≥m capturing all behaviors from broad to narrow and containing the standard Sibuya distribution as a special case (m=1). We also derive some new representations for the generating functions related to the GSD. We show that the generalized Sibuya SRW exhibits several regimes of anomalous diffusion depending on the lowest order m of diverging GSD moment. The generalized Sibuya SRW opens various new directions in anomalous physics. Full article
(This article belongs to the Special Issue Generalized Fractional Dynamics in Graphs and Complex Systems)
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14 pages, 1467 KiB  
Article
An Explicit Wavelet Method for Solution of Nonlinear Fractional Wave Equations
by Jiong Weng, Xiaojing Liu, Youhe Zhou and Jizeng Wang
Mathematics 2022, 10(21), 4011; https://doi.org/10.3390/math10214011 - 28 Oct 2022
Cited by 1 | Viewed by 1323
Abstract
An explicit method for solving time fractional wave equations with various nonlinearity is proposed using techniques of Laplace transform and wavelet approximation of functions and their integrals. To construct this method, a generalized Coiflet with N vanishing moments is adopted as the basis [...] Read more.
An explicit method for solving time fractional wave equations with various nonlinearity is proposed using techniques of Laplace transform and wavelet approximation of functions and their integrals. To construct this method, a generalized Coiflet with N vanishing moments is adopted as the basis function, where N can be any positive even number. As has been shown, convergence order of these approximations can be N. The original fractional wave equation is transformed into a time Volterra-type integro-differential equation associated with a smooth time kernel and spatial derivatives of unknown function by using the technique of Laplace transform. Then, an explicit solution procedure based on the collocation method and the proposed algorithm on integral approximation is established to solve the transformed nonlinear integro-differential equation. Eventually the nonlinear fractional wave equation can be readily and accurately solved. As examples, this method is applied to solve several fractional wave equations with various nonlinearities. Results show that the proposed method can successfully avoid difficulties in the treatment of singularity associated with fractional derivatives. Compared with other existing methods, this method not only has the advantage of high-order accuracy, but it also does not even need to solve the nonlinear spatial system after time discretization to obtain the numerical solution, which significantly reduces the storage and computation cost. Full article
(This article belongs to the Special Issue Generalized Fractional Dynamics in Graphs and Complex Systems)
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13 pages, 405 KiB  
Article
Certain New Models of the Multi-Space Fractal-Fractional Kuramoto-Sivashinsky and Korteweg-de Vries Equations
by Hari M. Srivastava, Khaled Mohammed Saad and Walid M. Hamanah
Mathematics 2022, 10(7), 1089; https://doi.org/10.3390/math10071089 - 28 Mar 2022
Cited by 32 | Viewed by 1916
Abstract
The main objective of this paper is to introduce and study the numerical solutions of the multi-space fractal-fractional Kuramoto-Sivashinsky equation (MSFFKS) and the multi-space fractal-fractional Korteweg-de Vries equation (MSFFKDV). These models are obtained by replacing the classical derivative by the fractal-fractional derivative based [...] Read more.
The main objective of this paper is to introduce and study the numerical solutions of the multi-space fractal-fractional Kuramoto-Sivashinsky equation (MSFFKS) and the multi-space fractal-fractional Korteweg-de Vries equation (MSFFKDV). These models are obtained by replacing the classical derivative by the fractal-fractional derivative based upon the generalized Mittag-Leffler kernel. In our investigation, we use the spectral collocation method (SCM) involving the shifted Legendre polynomials (SLPs) in order to reduce the new models to a system of algebraic equations. We then use one of the known numerical methods, the Newton-Raphson method (NRM), for solving the resulting system of the nonlinear algebraic equations. The efficiency and accuracy of the numerical results are validated by calculating the absolute error as well as the residual error. We also present several illustrative examples and graphical representations for the various results which we have derived in this paper. Full article
(This article belongs to the Special Issue Generalized Fractional Dynamics in Graphs and Complex Systems)
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15 pages, 1084 KiB  
Article
ARDL as an Elixir Approach to Cure for Spurious Regression in Nonstationary Time Series
by Ghulam Ghouse, Saud Ahmad Khan, Atiq Ur Rehman and Muhammad Ishaq Bhatti
Mathematics 2021, 9(22), 2839; https://doi.org/10.3390/math9222839 - 9 Nov 2021
Cited by 12 | Viewed by 3665
Abstract
In conventional Econometrics, the unit root and cointegration analysis are the only ways to circumvent the spurious regression which may arise from missing variable (lag values) rather than the nonstationarity process in time series data. We propose the Ghouse equation solution of autoregressive [...] Read more.
In conventional Econometrics, the unit root and cointegration analysis are the only ways to circumvent the spurious regression which may arise from missing variable (lag values) rather than the nonstationarity process in time series data. We propose the Ghouse equation solution of autoregressive distributed lag mechanism which does not require additional work in unit root testing and bound testing. This advantage makes the proposed methodology more efficient compared to the existing cointegration procedures. The earlier tests weaken their position in comparison to it, as they had numerous linked testing procedures which further increase the size of the test and/or reduce the test power. The simplification of the Ghouse equation does not attain any such type of error, which makes it a more powerful test as compared to widely cited exiting testing methods in econometrics and statistics literature. Full article
(This article belongs to the Special Issue Generalized Fractional Dynamics in Graphs and Complex Systems)
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