Fractional Order Systems: Deterministic and Stochastic Analysis II

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

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 11146

Special Issue Editor

Special Issue Information

Dear Colleagues,

The field of fractional dynamic systems has become very popular and already attracted many scientists and research groups from around the world. Its main advantage is in the modeling of several complex phenomena, with the best results in numerous seemingly diverse and widespread areas of science and engineering. Since such developments are currently considered essential in applied sciences, it is very important to focus on possible novelties of the most promising new directions and open problems that have been formulated based on modern techniques and approaches indicated in the latest scientific achievements.  

Additionally, the theory of stochastic processes is considered to be an important contribution to probability theory, and continues to be an active topic of research for both theoretical reasons and applications. The word “stochastic” is used to describe many terms and objects in mathematics. Examples include the stochastic matrix, which describes a stochastic process known as a Markov process; and stochastic calculus, which involves differential equations and integrals based on stochastic processes such as the Wiener process, also called the Brownian motion process.

In this open call for papers we strictly invite strong and interesting contributions providing original results which have been obtained from modern computational techniques of theoretical, experimental, and applied aspects of both deterministic and stochastic fractional dynamic systems. We also strongly encourage young researchers/PhD students who have achieved exciting results while supervised and guided by their scientific advisors to submit their works to this Special Issue. It is necessary that papers have a high-level mathematical ground. Note that submitted papers should explicitly meet the Aims and Scope of the Fractal Fract journal.

Topics to be included are:

  • Appropriate fractional derivative senses in applied sciences;
  • Computational methods for fractional dynamical systems;
  • Fractional inverse problems: modeling and simulation;
  • Cancer dynamic fractional systems: optimality and modeling;
  • Optimal control for fractional models of HIV/AIDS infection;
  • Latest advancements on COVID-19 pandemic fractional systems;
  • Continuous and discrete fractional systems with randomness;
  • Uncertainty quantification for random fractional dynamic systems;
  • Stochastic analysis for fractional mathematical models;
  • Instantaneous impulsive fractional equations and inclusions;
  • Applications of fractional problems in science and engineering;
  • Implementation methods and simulations for fractional models;
  • Fractional reaction-diffusion and Navier–Stokes equations;
  • Automorphic and periodic solutions for fractional systems;
  • Approximation methods for fractional order systems;
  • Stochastic processes involving fractional PDEs;
  • Control and optimization for fractional systems;
  • Variable order differentiation and integration;
  • Heat transfer involving local fractional operators;
  • Waves, wavelets and fractals: fractional calculus approach.

Prof. Dr. Amar Debbouche
Guest Editor

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Published Papers (6 papers)

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Research

26 pages, 1514 KiB  
Article
Deterministic and Fractional-Order Co-Infection Model of Omicron and Delta Variants of Asymptomatic SARS-CoV-2 Carriers
by Waqas Ali Faridi, Muhammad Imran Asjad, Shabir Ahmad, Adrian Iftene, Magda Abd El-Rahman and Mohammed Sallah
Fractal Fract. 2023, 7(2), 192; https://doi.org/10.3390/fractalfract7020192 - 14 Feb 2023
Viewed by 1910
Abstract
The Delta and Omicron variants’ system was used in this research study to replicate the complex process of the SARS-CoV-2 outbreak. The generalised fractional system was designed and rigorously analysed in order to gain a comprehensive understanding of the transmission dynamics of both [...] Read more.
The Delta and Omicron variants’ system was used in this research study to replicate the complex process of the SARS-CoV-2 outbreak. The generalised fractional system was designed and rigorously analysed in order to gain a comprehensive understanding of the transmission dynamics of both variants. The proposed dynamical system has heredity and memory effects, which greatly improved our ability to perceive the disease propagation dynamics. The non-singular Atangana–Baleanu fractional operator was used to forecast the current pandemic in order to meet this challenge. The Picard recursions approach can be used to ensure that the designed fractional system has at least one solution occupying the growth condition and memory function regardless of the initial conditions. The Hyers–Ulam–Rassias stability criteria were used to carry out the stability analysis of the fractional governing system of equations, and the fixed-point theory ensured the uniqueness of the solution. Additionally, the model exhibited global asymptotically stable behaviour in some conditions. The approximate behaviour of the fatal virus was investigated using an efficient and reliable fractional numerical Adams–Bashforth approach. The outcome demonstrated that there will be a significant decline in the population of those infected with the Omicron and Delta SARS-CoV-2 variants if the vaccination rate is increased (in both the symptomatic and symptomatic stages). Full article
(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis II)
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16 pages, 349 KiB  
Article
Approximate Controllability of Delayed Fractional Stochastic Differential Systems with Mixed Noise and Impulsive Effects
by Naima Hakkar, Rajesh Dhayal, Amar Debbouche and Delfim F. M. Torres
Fractal Fract. 2023, 7(2), 104; https://doi.org/10.3390/fractalfract7020104 - 18 Jan 2023
Cited by 20 | Viewed by 1873
Abstract
We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter H^(1/2,1). Using fixed point techniques, a q-resolvent family, [...] Read more.
We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter H^(1/2,1). Using fixed point techniques, a q-resolvent family, and fractional calculus, we discuss the existence of a piecewise continuous mild solution for the proposed system. Moreover, under appropriate conditions, we investigate the approximate controllability of the considered system. Finally, the main results are demonstrated with an illustrative example. Full article
(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis II)
22 pages, 370 KiB  
Article
Fractional Stochastic Integro-Differential Equations with Nonintantaneous Impulses: Existence, Approximate Controllability and Stochastic Iterative Learning Control
by Kinda Abuasbeh, Nazim I. Mahmudov and Muath Awadalla
Fractal Fract. 2023, 7(1), 87; https://doi.org/10.3390/fractalfract7010087 - 12 Jan 2023
Cited by 1 | Viewed by 1282
Abstract
In this paper, existence/uniqueness of solutions and approximate controllability concept for Caputo type stochastic fractional integro-differential equations (SFIDE) in a Hilbert space with a noninstantaneous impulsive effect are studied. In addition, we study different types of stochastic iterative learning control for SFIDEs with [...] Read more.
In this paper, existence/uniqueness of solutions and approximate controllability concept for Caputo type stochastic fractional integro-differential equations (SFIDE) in a Hilbert space with a noninstantaneous impulsive effect are studied. In addition, we study different types of stochastic iterative learning control for SFIDEs with noninstantaneous impulses in Hilbert spaces. Finally, examples are given to support the obtained results. Full article
(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis II)
9 pages, 286 KiB  
Article
On Averaging Principle for Caputo–Hadamard Fractional Stochastic Differential Pantograph Equation
by Mounia Mouy, Hamid Boulares, Saleh Alshammari, Mohammad Alshammari, Yamina Laskri and Wael W. Mohammed
Fractal Fract. 2023, 7(1), 31; https://doi.org/10.3390/fractalfract7010031 - 28 Dec 2022
Cited by 16 | Viewed by 1563
Abstract
In this paper, we studied an averaging principle for Caputo–Hadamard fractional stochastic differential pantograph equation (FSDPEs) driven by Brownian motion. In light of some suggestions, the solutions to FSDPEs can be approximated by solutions to averaged stochastic systems in the sense of mean [...] Read more.
In this paper, we studied an averaging principle for Caputo–Hadamard fractional stochastic differential pantograph equation (FSDPEs) driven by Brownian motion. In light of some suggestions, the solutions to FSDPEs can be approximated by solutions to averaged stochastic systems in the sense of mean square. We expand the classical Khasminskii approach to Caputo–Hadamard fractional stochastic equations by analyzing systems solutions before and after applying averaging principle. We provided an applied example that explains the desired results to us. Full article
(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis II)
18 pages, 350 KiB  
Article
Controllability of Fractional Stochastic Delay Systems Driven by the Rosenblatt Process
by Barakah Almarri and Ahmed M. Elshenhab
Fractal Fract. 2022, 6(11), 664; https://doi.org/10.3390/fractalfract6110664 - 10 Nov 2022
Cited by 3 | Viewed by 1365
Abstract
In this work, we consider linear and nonlinear fractional stochastic delay systems driven by the Rosenblatt process. With the aid of the delayed Mittag-Leffler matrix functions and the representation of solutions of these systems, we derive the controllability results as an application. By [...] Read more.
In this work, we consider linear and nonlinear fractional stochastic delay systems driven by the Rosenblatt process. With the aid of the delayed Mittag-Leffler matrix functions and the representation of solutions of these systems, we derive the controllability results as an application. By introducing a fractional delayed Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear fractional stochastic delay systems. Furthermore, by employing Krasnoselskii’s fixed point theorem, we establish sufficient conditions for the controllability of nonlinear fractional stochastic delay systems. Finally, an example is given to illustrate the main results. Full article
(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis II)
13 pages, 1246 KiB  
Article
Synchronization of Fractional Stochastic Chaotic Systems via Mittag-Leffler Function
by T. Sathiyaraj, Michal Fečkan and JinRong Wang
Fractal Fract. 2022, 6(4), 192; https://doi.org/10.3390/fractalfract6040192 - 30 Mar 2022
Cited by 7 | Viewed by 1950
Abstract
This paper is involved with synchronization of fractional order stochastic systems in finite dimensional space, and we have tested its time response and stochastic chaotic behaviors. Firstly, we give a representation of solution for a stochastic fractional order chaotic system. Secondly, some useful [...] Read more.
This paper is involved with synchronization of fractional order stochastic systems in finite dimensional space, and we have tested its time response and stochastic chaotic behaviors. Firstly, we give a representation of solution for a stochastic fractional order chaotic system. Secondly, some useful sufficient conditions are investigated by using matrix type Mittag-Leffler function, Jacobian matrix via stochastic process, stability analysis and feedback control technique to assure the synchronization of stochastic error system. Thereafter, numerical illustrations are provided to verify the theoretical parts. Full article
(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis II)
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