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Fractal Fract
Special Issues
Anomalous and Non-ergodic Dynamics: Modelling, Analysis, and Simulation
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Anomalous and Non-ergodic Dynamics: Modelling, Analysis, and Simulation

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (20 November 2021) | Viewed by 6961

Special Issue Editors

Dr. Minghua Chen

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Interests: nonlocal/fractional PDEs; high-order algorithms; inverse problems; deep learning
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Hossein Jafari

E-Mail
Guest Editor
Department of Mathematical Sciences, University of South Africa, UNISA, Roodepoort 0003, South Africa
Interests: fractional differential equations; operational matrices; numerical methods; lie symmetry
Special Issues, Collections and Topics in MDPI journals
Dr. Can Li

E-Mail Website
Guest Editor
Department of Applied Mathematics, Xi'an University of Technology, Xi'an 713300, China
Interests: fractional differential equations; anomalous diffusion; nonlocal diffusion models; finite element methods; finite difference methods
Dr. Yajing Li

E-Mail Website
Guest Editor
College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China
Interests: fractional partial differential equation; stochastic partial differential equations; algorithm analysis; stochastic dynamics
Dr. Lijing Zhao

E-Mail Website
Guest Editor
Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, China
Interests: fractional operators; spectral method; finite difference method

Special Issue Information

Dear Colleagues,

Anomalous and non-ergodicity are important properties of some dynamical systems. These dynamical systems play important roles in many problems of physics, fluid dynamics, chemistry, biology, finance, hydrology and control theory. Recently, various kinds of efficient ways have been developed to describe these dynamical systems. Furthermore, many new mathematical models, includes fractional differential equations, nonlocal diffusion models, stochastic differential equation with non-Gaussian noises, are derived in the related research areas. Comparing with the classical mathematical physical models, the new models give rise to multiple dynamic behaviors. Many scholars have made great contributions in developing efficient numerical methods to solve the new mathematical models. But the research results in the above-mentioned areas are far from being mature. New mathematical-physical models and numerical methods are needed for simulating more potential complex dynamical systems.

The aim of this Special Issue is to collect original and high-quality contributions related to the recent advances in anomalous and non-ergodic dynamics as well as efficient numerical methods to simulate the related mathematical models. Topics that are invited for submission include (but are not limited to):

  • Anomalous diffusion process;
  • Non-ergodic fractional dynamical systems;
  • Nonlocal and Fractional mathematical models;
  • Applications of fractional differential models for biology;
  • Stochastic differential equations with non-Gaussian noises;
  • Fractional stochastic differential equations;
  • Integral equations with fractional calculus;
  • Numerical methods for fractional differential equations;
  • Numerical methods for nonlocal differential equations;
  • Nonlocal and fractional order mathematical models for COVID-19;
  • Analytical methods for fractional differential equations.

Dr. Minghua Chen
Prof. Dr. H Jafari
Dr. Can Li
Dr. Yajing Li
Dr. Lijing Zhao
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • anomalous diffusion process
  • non-ergodic fractional dynamical systems
  • nonlocal and fractional models
  • fractional biology model
  • Non-Gaussian noises
  • fractional stochastic differential equations
  • integral equations
  • finite difference methods
  • finite element methods
  • mathematical models for COVID-19
  • spectral methods
  • fractional lie symmetry
  • local fractional derivative

Published Papers (3 papers)

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Research

13 pages, 2426 KiB  
Article
Multi-Model Selection and Analysis for COVID-19
by Nuri Ma, Weiyuan Ma and Zhiming Li
Fractal Fract. 2021, 5(3), 120; https://doi.org/10.3390/fractalfract5030120 - 13 Sep 2021
Cited by 14 | Viewed by 2228
Abstract
In the face of an increasing number of COVID-19 infections, one of the most crucial and challenging problems is to pick out the most reasonable and reliable models. Based on the COVID-19 data of four typical cities/provinces in China, integer-order and fractional SIR, [...] Read more.
In the face of an increasing number of COVID-19 infections, one of the most crucial and challenging problems is to pick out the most reasonable and reliable models. Based on the COVID-19 data of four typical cities/provinces in China, integer-order and fractional SIR, SEIR, SEIR-Q, SEIR-QD, and SEIR-AHQ models are systematically analyzed by the AICc, BIC, RMSE, and R means. Through extensive simulation and comprehensive comparison, we show that the fractional models perform much better than the corresponding integer-order models in representing the epidemiological information contained in the real data. It is further revealed that the inflection point plays a vital role in the prediction. Moreover, the basic reproduction numbers R0 of all models are highly dependent on the contact rate. Full article
(This article belongs to the Special Issue Anomalous and Non-ergodic Dynamics: Modelling, Analysis, and Simulation)
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12 pages, 2371 KiB  
Article
Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator
by Igor V. Malyk, Mykola Gorbatenko, Arun Chaudhary, Shivani Sharma and Ravi Shanker Dubey
Fractal Fract. 2021, 5(3), 64; https://doi.org/10.3390/fractalfract5030064 - 2 Jul 2021
Cited by 12 | Viewed by 2000
Abstract
In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC [...] Read more.
In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator. Full article
(This article belongs to the Special Issue Anomalous and Non-ergodic Dynamics: Modelling, Analysis, and Simulation)
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16 pages, 910 KiB  
Article
Analytic Solution of the Langevin Differential Equations Dominated by a Multibrot Fractal Set
by Rabha W. Ibrahim and Dumitru Baleanu
Fractal Fract. 2021, 5(2), 50; https://doi.org/10.3390/fractalfract5020050 - 25 May 2021
Cited by 10 | Viewed by 1809
Abstract
We present an analytic solvability of a class of Langevin differential equations (LDEs) in the asset of geometric function theory. The analytic solutions of the LDEs are presented by utilizing a special kind of fractal function in a complex domain, linked with the [...] Read more.
We present an analytic solvability of a class of Langevin differential equations (LDEs) in the asset of geometric function theory. The analytic solutions of the LDEs are presented by utilizing a special kind of fractal function in a complex domain, linked with the subordination theory. The fractal functions are suggested for the multi-parametric coefficients type motorboat fractal set. We obtain different formulas of fractal analytic solutions of LDEs. Moreover, we determine the maximum value of the fractal coefficients to obtain the optimal solution. Through the subordination inequality, we determined the upper boundary determination of a class of fractal functions holding multibrot function ϑ(z)=1+3κz+z3. Full article
(This article belongs to the Special Issue Anomalous and Non-ergodic Dynamics: Modelling, Analysis, and Simulation)
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