Stability Analysis of Fractional Systems-II

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 June 2023) | Viewed by 8197

Special Issue Editors


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Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria
Interests: fractional differential equations; functional-differential equations; impulsive differential equations; differential equations in Banach spaces; integral equations; integral inequalities; stability analysis; real and functional analysis; applied mathematics
Special Issues, Collections and Topics in MDPI journals

E-Mail Website
Guest Editor
Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria
Interests: fractional differential equations; impulsive differential equations; functional-differential equations; differential equations in Banach spaces
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nowadays, fractional dynamical systems (FDSs) based on different kinds of fractional derivatives (Riemann–Liouville, Caputo, Gruenwald–Letnikov, and other fractional derivatives with a good mathematical and physical background) have become powerful tools for modelling real-world phenomena.

This Special Issue, “Stability Analysis of Fractional Systems-II”, invites papers that focus on the recent and novel developments in the stability theory of fractional dynamical systems of different types (with or without delays, impulsive or not, and so on). We expect high-quality articles concerning various types of stabilities, namely: Lyapunov’s type, finite time stability, Mittag-Leffler stability, robust stability, Hyers–Ulam–Rassias stability, and so on.

Works exploring the possibilities for using FDSs as relevant models in the applied sciences, for example, neural networks modeled with FDS, are welcome too.

Prof. Dr. Andrey Zahariev
Prof. Dr. Hristo Kiskinov
Guest Editors

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Keywords

  • fractional dynamical systems 
  • fractional differential systems 
  • stability 
  • fractional derivatives and integrals

Published Papers (6 papers)

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Research

15 pages, 316 KiB  
Article
Formulation of Impulsive Ecological Systems Using the Conformable Calculus Approach: Qualitative Analysis
by Anatoliy Martynyuk, Gani Stamov, Ivanka Stamova and Ekaterina Gospodinova
Mathematics 2023, 11(10), 2221; https://doi.org/10.3390/math11102221 - 9 May 2023
Cited by 1 | Viewed by 1049
Abstract
In this paper, an impulsive conformable fractional Lotka–Volterra model with dispersion is introduced. Since the concept of conformable derivatives avoids some limitations of the classical fractional-order derivatives, it is more suitable for applied problems. The impulsive control approach which is common for population [...] Read more.
In this paper, an impulsive conformable fractional Lotka–Volterra model with dispersion is introduced. Since the concept of conformable derivatives avoids some limitations of the classical fractional-order derivatives, it is more suitable for applied problems. The impulsive control approach which is common for population dynamics’ models is applied and fixed moments impulsive perturbations are considered. The combined concept of practical stability with respect to manifolds is adapted to the introduced model. Sufficient conditions for boundedness and generalized practical stability of the solutions are obtained by using an analogue of the Lyapunov function method. The uncertain case is also studied. Examples are given to demonstrate the effectiveness of the established results. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems-II)
10 pages, 396 KiB  
Article
On Solutions of Fractional Integrodifferential Systems Involving Ψ-Caputo Derivative and Ψ-Riemann–Liouville Fractional Integral
by Hamid Boulares, Abdelkader Moumen, Khaireddine Fernane, Jehad Alzabut, Hicham Saber, Tariq Alraqad and Mhamed Benaissa
Mathematics 2023, 11(6), 1465; https://doi.org/10.3390/math11061465 - 17 Mar 2023
Viewed by 1126
Abstract
In this paper, we investigate a new class of nonlinear fractional integrodifferential systems that includes the Ψ-Riemann–Liouville fractional integral term. Using the technique of upper and lower solutions, the solvability of the system is examined. We add two examples to demonstrate and [...] Read more.
In this paper, we investigate a new class of nonlinear fractional integrodifferential systems that includes the Ψ-Riemann–Liouville fractional integral term. Using the technique of upper and lower solutions, the solvability of the system is examined. We add two examples to demonstrate and validate the main result. The main results highlight crucial contributions to the general theory of fractional differential equations. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems-II)
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24 pages, 720 KiB  
Article
High-Order Approximation to Generalized Caputo Derivatives and Generalized Fractional Advection–Diffusion Equations
by Sarita Kumari, Rajesh K. Pandey and Ravi P. Agarwal
Mathematics 2023, 11(5), 1200; https://doi.org/10.3390/math11051200 - 28 Feb 2023
Cited by 1 | Viewed by 1209
Abstract
In this article, a high-order time-stepping scheme based on the cubic interpolation formula is considered to approximate the generalized Caputo fractional derivative (GCFD). Convergence order for this scheme is (4α), where [...] Read more.
In this article, a high-order time-stepping scheme based on the cubic interpolation formula is considered to approximate the generalized Caputo fractional derivative (GCFD). Convergence order for this scheme is (4α), where α(0<α<1) is the order of the GCFD. The local truncation error is also provided. Then, we adopt the developed scheme to establish a difference scheme for the solution of the generalized fractional advection–diffusion equation with Dirichlet boundary conditions. Furthermore, we discuss the stability and convergence of the difference scheme. Numerical examples are presented to examine the theoretical claims. The convergence order of the difference scheme is analyzed numerically, which is (4α) in time and second-order in space. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems-II)
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21 pages, 382 KiB  
Article
On Stability Criteria Induced by the Resolvent Kernel for a Fractional Neutral Linear System with Distributed Delays
by Ekaterina Madamlieva, Marian Milev and Tsvetana Stoyanova
Mathematics 2023, 11(3), 626; https://doi.org/10.3390/math11030626 - 26 Jan 2023
Cited by 1 | Viewed by 1105
Abstract
We consider an initial problem (IP) for a linear neutral system with distributed delays and derivatives in Caputo’s sense of incommensurate order, with different kinds of initial functions. In the case when the initial functions are with bounded variation, it is proven that [...] Read more.
We consider an initial problem (IP) for a linear neutral system with distributed delays and derivatives in Caputo’s sense of incommensurate order, with different kinds of initial functions. In the case when the initial functions are with bounded variation, it is proven that this IP has a unique solution. The Krasnoselskii’s fixed point theorem, a very appropriate tool, is used to prove the existence of solutions in the case of the neutral systems. As a corollary of this result, we obtain the existence and uniqueness of a fundamental matrix for the homogeneous system. In the general case, without additional assumptions of boundedness type, it is established that the existence and uniqueness of a fundamental matrix lead existence and uniqueness of a resolvent kernel and vice versa. Furthermore, an explicit formula describing the relationship between the fundamental matrix and the resolvent kernel is proven in the general case too. On the base of the existence and uniqueness of a resolvent kernel, necessary and sufficient conditions for the stability of the zero solution of the homogeneous system are established. Finally, it is considered a well-known economics model to describe the dynamics of the wealth of nations and comment on the possibilities of application of the obtained results for the considered systems, which include as partial case the considered model. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems-II)
20 pages, 361 KiB  
Article
On the Preservation with Respect to Nonlinear Perturbations of the Stability Property for Nonautonomous Linear Neutral Fractional Systems with Distributed Delays
by Ekaterina Madamlieva, Hristo Kiskinov, Milena Petkova and Andrey Zahariev
Mathematics 2022, 10(15), 2642; https://doi.org/10.3390/math10152642 - 28 Jul 2022
Cited by 3 | Viewed by 1007
Abstract
In the present paper, sufficient conditions are obtained under which the Cauchy problem for a nonlinearly perturbed nonautonomous neutral fractional system with distributed delays and Caputo type derivatives has a unique solution in the case of initial functions with first-kind discontinuities. For this [...] Read more.
In the present paper, sufficient conditions are obtained under which the Cauchy problem for a nonlinearly perturbed nonautonomous neutral fractional system with distributed delays and Caputo type derivatives has a unique solution in the case of initial functions with first-kind discontinuities. For this system, by applying a formula for the integral presentation of the solution of the nonhomogeneous linear neutral fractional system, we found some additional natural conditions to ensure that from the global asymptotically stability of the zero solution of the linear part of the nonlinearly perturbed system, global asymptotic stability of the zero solution of the whole nonlinearly perturbed system follows. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems-II)
33 pages, 22629 KiB  
Article
Dynamical Analysis of Nutrient-Phytoplankton-Zooplankton Model with Viral Disease in Phytoplankton Species under Atangana-Baleanu-Caputo Derivative
by Songkran Pleumpreedaporn, Chanidaporn Pleumpreedaporn, Jutarat Kongson, Chatthai Thaiprayoon, Jehad Alzabut and Weerawat Sudsutad
Mathematics 2022, 10(9), 1578; https://doi.org/10.3390/math10091578 - 7 May 2022
Cited by 5 | Viewed by 1774
Abstract
A mathematical model of the nutrient-phytoplankton-zooplankton associated with viral infection in phytoplankton under the Atangana-Baleanu derivative in Caputo sense is investigated in this study. We prove the theoretical results for the existence and uniqueness of the solutions by using Banach’s and Sadovskii’s fixed [...] Read more.
A mathematical model of the nutrient-phytoplankton-zooplankton associated with viral infection in phytoplankton under the Atangana-Baleanu derivative in Caputo sense is investigated in this study. We prove the theoretical results for the existence and uniqueness of the solutions by using Banach’s and Sadovskii’s fixed point theorems. The notion of various Ulam’s stability is used to guarantee the context of the stability analysis. Furthermore, the equilibrium points and the basic reproduction numbers for the proposed model are provided. The Adams type predictor-corrector algorithm has been applied for the theoretical confirmation to establish the approximate solutions. A variety of numerical plots corresponding to various fractional orders between zero and one are presented to describe the dynamical behavior of the fractional model under consideration. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems-II)
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