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Axioms
Special Issues
Infinite Dynamical System and Differential Equations
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Infinite Dynamical System and Differential Equations

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 27 February 2025 | Viewed by 6081

Special Issue Editors

Dr. Cheng-Cheng Zhu

E-Mail Website
Guest Editor
School of Science, Jiangnan University, Wuxi, China
Interests: infinite dynamical system; stability analysis in nonlinear systems; mathematical biology; infectious diseases
Prof. Dr. Hari Mohan Srivastava

grade E-Mail Website
Guest Editor
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Interests: real and complex analysis; fractional calculus and its applications; integral equations and transforms; higher transcendental functions and their applications; q-series and q-polynomials; analytic number theory; analytic and geometric Inequalities; probability and statistics; inventory modelling and optimization
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Infinite dimensional dynamical systems and differential equations are important branches of mathematical research that have wide applications in many disciplines. Interdisciplinary research has become an inevitable trend in modern scientific research. Infinite dimensional dynamical systems and differential equations have become an indispensable bridge and link for interdisciplinary research, providing strong mathematical theoretical support and analytical methods for many fields such as economics, medicine, ecology, epidemiology, machine learning, artificial intelligence, etc.

The purpose of this Special Issue is to allow scientists and researchers to showcase their latest theories, research methods, and application achievements in infinite dynamical systems and differential equations, as well as in interdisciplinary research, for example, revolving around the existence, uniqueness, stability of solutions to various differential equations, optimal control theory, and the application of infinite dynamical systems in disciplines such as biomedicine, epidemiology, and population dynamics.

Dr. Cheng-Cheng Zhu
Prof. Dr. Hari Mohan Srivastava
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. Axioms 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 2400 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

  • infinite dynamical systems
  • global attractor
  • ODEs
  • PDEs
  • DDEs
  • operator semigroup theory
  • mathematical biology
  • population dynamics
  • mathematical epidemiology
  • nonlinear dynamical systems
  • non-autonomous systems
  • stability analysis
  • threshold dynamics
  • persistence theory
  • bifurcations and chaos
  • infectious diseases
  • optimal control theory
  • Stieltjes differential equations

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Published Papers (7 papers)

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Research

16 pages, 298 KiB  
Article
Investigation of the Oscillatory Behavior of the Solutions of a Class of Third-Order Delay Differential Equations with Several Terms
by Asma Al-Jaser, Insaf F. Ben Saoud, Higinio Ramos and Belgees Qaraad
Axioms 2024, 13(10), 703; https://doi.org/10.3390/axioms13100703 - 11 Oct 2024
Viewed by 432
Abstract
In this paper, we address the study of the oscillatory properties of the solutions of a class of third-order delay differential equations. The primary objective of this study is to provide new relationships that can be employed to obtain criteria for excluding increasing [...] Read more.
In this paper, we address the study of the oscillatory properties of the solutions of a class of third-order delay differential equations. The primary objective of this study is to provide new relationships that can be employed to obtain criteria for excluding increasing positive solutions and decreasing positive solutions so that the resulting criteria are easier to apply than other criteria that have appeared in the literature. We have obtained new oscillation criteria that hold up more robustly upon application. Some examples are presented to illustrate the significance of our main findings. Full article
(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)
14 pages, 249 KiB  
Article
Qualitative Outcomes on Monotone Iterative Technique and Quasilinearization Method on Time Scale
by Şahap Çetin, Yalçın Yılmaz and Coşkun Yakar
Axioms 2024, 13(9), 640; https://doi.org/10.3390/axioms13090640 - 19 Sep 2024
Viewed by 451
Abstract
In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. [...] Read more.
In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. Unlike other studies, we consider the LS and US pair of the seventh type instead of the natural type. It was determined that the solutions of the dynamic equation converge uniformly and monotonically to the unique solution of the IVP, and the convergence is quadratic. Moreover, we will use the delta derivative (Δγ) instead of the classical derivative (dγ) in the proof because it studies a time scale. In the second part of the paper, we applied the monotone iterative technique (MIT) coupled with the LS and US, which is an effective method, proving a clear analytical representation for the solution of the equation when the relevant functions are monotonically non-decreasing and non-increasing. Then an example is given to illustrate the results obtained. Full article
(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)
19 pages, 300 KiB  
Article
On Some New Dynamic Hilbert-Type Inequalities across Time Scales
by Mohammed Zakarya, Ahmed I. Saied, Amirah Ayidh I Al-Thaqfan, Maha Ali and Haytham M. Rezk
Axioms 2024, 13(7), 475; https://doi.org/10.3390/axioms13070475 - 14 Jul 2024
Viewed by 762
Abstract
In this article, we present some novel dynamic Hilbert-type inequalities within the framework of time scales T. We achieve this by utilizing Hölder’s inequality, the chain rule, and the mean inequality. As specific instances of our findings (when T=N and [...] Read more.
In this article, we present some novel dynamic Hilbert-type inequalities within the framework of time scales T. We achieve this by utilizing Hölder’s inequality, the chain rule, and the mean inequality. As specific instances of our findings (when T=N and T=R), we obtain the discrete and continuous analogues of previously established inequalities. Additionally, we derive other inequalities for different time scales, such as T=qN0 for q>1, which, to the best of the authors’ knowledge, is a largely novel conclusion. Full article
(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)
27 pages, 364 KiB  
Article
Fractional-Order Sequential Linear Differential Equations with Nabla Derivatives on Time Scales
by Cheng-Cheng Zhu and Jiang Zhu
Axioms 2024, 13(7), 447; https://doi.org/10.3390/axioms13070447 - 1 Jul 2024
Viewed by 623
Abstract
In this paper, we present a general theory for fractional-order sequential differential equations with Riemann–Liouville nabla derivatives and Caputo nabla derivatives on time scales. The explicit solution, in the case of constant coefficients, for both the homogeneous and the non-homogeneous problems, are given [...] Read more.
In this paper, we present a general theory for fractional-order sequential differential equations with Riemann–Liouville nabla derivatives and Caputo nabla derivatives on time scales. The explicit solution, in the case of constant coefficients, for both the homogeneous and the non-homogeneous problems, are given using the ∇-Mittag-Leffler function, Laplace transform method, operational method and operational decomposition method. In addition, we also provide some results about a solution to a new class of fractional-order sequential differential equations with convolutional-type variable coefficients using the Laplace transform method. Full article
(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)
28 pages, 769 KiB  
Article
C0–Semigroups Approach to the Reliability Model Based on Robot-Safety System
by Ehmet Kasim and Aihemaitijiang Yumaier
Axioms 2024, 13(7), 423; https://doi.org/10.3390/axioms13070423 - 24 Jun 2024
Viewed by 661
Abstract
This paper considers a system with one robot and n safety units (one of which works while the others remain on standby), which is described by an integro-deferential equation. The system can fail in the following three ways: fails with an incident, fails [...] Read more.
This paper considers a system with one robot and n safety units (one of which works while the others remain on standby), which is described by an integro-deferential equation. The system can fail in the following three ways: fails with an incident, fails safely and fails due to the malfunction of the robot. Using the C0–semigroups theory of linear operators, we first show that the system has a unique non-negative, time-dependent solution. Then, we obtain the exponential convergence of the time-dependent solution to its steady-state solution. In addition, we study the asymptotic behavior of some time-dependent reliability indices and present a numerical example demonstrating the effects of different parameters on the system. Full article
(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)
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16 pages, 312 KiB  
Article
On Using Relative Information to Estimate Traits in a Darwinian Evolution Population Dynamics
by Eddy Kwessi
Axioms 2024, 13(6), 406; https://doi.org/10.3390/axioms13060406 - 16 Jun 2024
Viewed by 632
Abstract
Since its inception, evolution theory has garnered much attention from the scientific community for a good reason: it theorizes how various living organisms came to be and what changes are to be expected in a certain environment. While many models of evolution have [...] Read more.
Since its inception, evolution theory has garnered much attention from the scientific community for a good reason: it theorizes how various living organisms came to be and what changes are to be expected in a certain environment. While many models of evolution have been proposed to track changes in species’ traits, not much has been said about how to calculate or estimate these traits. In this paper, using information theory, we propose an estimation method for trait parameters in a Darwinian evolution model for species with one or multiple traits. We propose estimating parameters by minimizing the relative information in a Darwinian evolution population model using either a classical gradient ascent or a stochastic gradient ascent. The proposed procedure is shown to be possible in a supervised or unsupervised learning environment, similarly to what occurs with Boltzmann machines. Simulations are provided to illustrate the method. Full article
(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)
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15 pages, 581 KiB  
Article
Approximate and Parametric Solutions to SIR Epidemic Model
by Lazhar Bougoffa, Smail Bougouffa and Ammar Khanfer
Axioms 2024, 13(3), 201; https://doi.org/10.3390/axioms13030201 - 16 Mar 2024
Cited by 2 | Viewed by 1829
Abstract
This article provides a detailed exploration of the SIR epidemic model, starting with its meticulous formulation. The study employs a novel approach called the upper and lower bounds technique to approximate the solution to the SIR model, providing insights into the dynamic interplay [...] Read more.
This article provides a detailed exploration of the SIR epidemic model, starting with its meticulous formulation. The study employs a novel approach called the upper and lower bounds technique to approximate the solution to the SIR model, providing insights into the dynamic interplay between susceptible S, infected I, and recovered R populations. A new parametric solution to this model has been presented. Applying the Adomian decomposition method (ADM) allows for the attaining of highly accurate approximate solutions in the context of the SIR epidemic model. To validate the accuracy and robustness of the proposed approach, a numerical exploration is conducted, considering a diverse range of experimental parameters. This numerical analysis provides valuable insights into the sensitivity and responsiveness of the SIR epidemic model under varying conditions, contributing to the broader understanding of infectious disease dynamics. The interplay between theoretical formulation and numerical exploration establishes a comprehensive framework for studying the SIR model, with implications for refining our ability to predict and manage the spread of infectious diseases. Full article
(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)
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