Advances in Difference Equations

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

Deadline for manuscript submissions: 30 June 2024 | Viewed by 9121

Special Issue Editors


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Guest Editor
Department of Mathematics, Government College University, Lahore 54770, Pakistan
Interests: dynamical systems

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Guest Editor
Department of Mechanical Engineering, Sejong University, Seoul 05006, Republic of Korea
Interests: fluid dynamics (Newtonian and non-Newtonian fluids; heat and mass transfer; viscoelastic models with memory; fractional thermoelasticity)

Special Issue Information

Dear Colleagues,

As we are all aware, the extreme nature of difference equations’ representations of complex dynamical systems is widely recognized. The kernel of fractional order derivative operators has its own relevance as an empirical explanation of such complex phenomena, and the theory of fractional derivative operators has been successfully applied in recent years to study anomalous social and physical sciences. The difference counterpart of fractional calculus has been increasingly used to describe many occurrences that take place in the real world. Many academic fields, significantly the biological sciences, have started to place a substantial emphasis on systems of delay differential equations.

This Special Issue will accept top-notch papers with unique research findings and focuses on the theory and applications of differential and difference equations, particularly in science and engineering. Moreover, the goal of this Special Issue is to bring together mathematicians, physicists, and other scientists to a platform of differential and difference equations where they can present their beneficial research to the intellectual community.

Dr. Azhar Ali Zafar
Dr. Nehad Ali Shah
Guest Editors

Manuscript Submission Information

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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

  • difference equations
  • fractional difference equations
  • delay difference equations
  • weak solutions difference equations
  • numerical methods for difference equations
  • asymptotic behavior for difference equations

Published Papers (9 papers)

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Research

18 pages, 309 KiB  
Article
New Nonlinear Retarded Integral Inequalities and Their Applications to Nonlinear Retarded Integro-Differential Equations
by Mahvish Samar, Xinzhong Zhu, Abdul Shakoor and Mawia Osman
Axioms 2024, 13(6), 356; https://doi.org/10.3390/axioms13060356 - 27 May 2024
Viewed by 303
Abstract
The purpose of this article is to present some new nonlinear retarded integral inequalities which can be utilized to study the existence, stability, boundedness, uniqueness, and asymptotic behavior of solutions of nonlinear retarded integro-differential equations, and these inequalities can be used in the [...] Read more.
The purpose of this article is to present some new nonlinear retarded integral inequalities which can be utilized to study the existence, stability, boundedness, uniqueness, and asymptotic behavior of solutions of nonlinear retarded integro-differential equations, and these inequalities can be used in the symmetrical properties of functions. These inequalities also generalize some former famous inequalities in the literature. Two examples as applications will be provided to demonstrate the strength of our inequalities in estimating the boundedness and global existence of the solution to initial value problems for nonlinear integro-differential equations and differential equations which can be seen in graphs. This research work will ensure opening new opportunities for studying nonlinear dynamic inequalities on a time-scale structure of a varying nature. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
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23 pages, 357 KiB  
Article
Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times
by Huifu Xia, Yunfei Peng and Peng Zhang
Axioms 2024, 13(2), 126; https://doi.org/10.3390/axioms13020126 - 18 Feb 2024
Cited by 1 | Viewed by 867
Abstract
In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on [...] Read more.
In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on the continuous dependence and Gâteaux differentiable of the solution relative to the initial value are also presented in a new topology sense. For the autonomous impulsive system, the periodicity of the solution is given. As an application, the properties of the solution for a type of controlled nonlinear ordinary differential equation with impulses at variable times is obtained. These results are a foundation to study optimal control problems of systems governed by differential equations with impulses at variable times. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
12 pages, 270 KiB  
Article
New Conditions for Testing the Oscillation of Solutions of Second-Order Nonlinear Differential Equations with Damped Term
by Asma Al-Jaser, Belgees Qaraad, Higinio Ramos and Stefano Serra-Capizzano
Axioms 2024, 13(2), 105; https://doi.org/10.3390/axioms13020105 - 4 Feb 2024
Cited by 1 | Viewed by 949
Abstract
This paper deals with the oscillatory behavior of solutions of a new class of second-order nonlinear differential equations. In contrast to most of the previous results in the literature, we establish some new criteria that guarantee the oscillation of all solutions of the [...] Read more.
This paper deals with the oscillatory behavior of solutions of a new class of second-order nonlinear differential equations. In contrast to most of the previous results in the literature, we establish some new criteria that guarantee the oscillation of all solutions of the studied equation without additional restrictions. Our approach improves the standard integral averaging technique to obtain simpler oscillation theorems for new classes of nonlinear differential equations. Two examples are presented to illustrate the importance of our findings. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
19 pages, 456 KiB  
Article
The FitzHugh–Nagumo Model Described by Fractional Difference Equations: Stability and Numerical Simulation
by Tareq Hamadneh, Amel Hioual, Omar Alsayyed, Yazan Alaya Al-Khassawneh, Abdallah Al-Husban and Adel Ouannas
Axioms 2023, 12(9), 806; https://doi.org/10.3390/axioms12090806 - 22 Aug 2023
Cited by 2 | Viewed by 1198
Abstract
The aim of this work is to describe the dynamics of a discrete fractional-order reaction–diffusion FitzHugh–Nagumo model. We established acceptable requirements for the local asymptotic stability of the system’s unique equilibrium. Moreover, we employed a Lyapunov functional to show that the constant equilibrium [...] Read more.
The aim of this work is to describe the dynamics of a discrete fractional-order reaction–diffusion FitzHugh–Nagumo model. We established acceptable requirements for the local asymptotic stability of the system’s unique equilibrium. Moreover, we employed a Lyapunov functional to show that the constant equilibrium solution is globally asymptotically stable. Furthermore, numerical simulations are shown to clarify and exemplify the theoretical results. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
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14 pages, 431 KiB  
Article
Synchronization of Fractional Partial Difference Equations via Linear Methods
by Ibraheem Abu Falahah, Amel Hioual, Mowafaq Omar Al-Qadri, Yazan Alaya AL-Khassawneh, Abdallah Al-Husban, Tareq Hamadneh and Adel Ouannas
Axioms 2023, 12(8), 728; https://doi.org/10.3390/axioms12080728 - 27 Jul 2023
Cited by 2 | Viewed by 790
Abstract
Discrete fractional models with reaction-diffusion have gained significance in the scientific field in recent years, not only due to the need for numerical simulation but also due to the stated biological processes. In this paper, we investigate the problem of synchronization-control in a [...] Read more.
Discrete fractional models with reaction-diffusion have gained significance in the scientific field in recent years, not only due to the need for numerical simulation but also due to the stated biological processes. In this paper, we investigate the problem of synchronization-control in a fractional discrete nonlinear bacterial culture reaction-diffusion model using the Caputo h-difference operator and a second-order central difference scheme and an L1 finite difference scheme after deriving the discrete fractional version of the well-known Degn–Harrison system and Lengyel–Epstein system. Using appropriate techniques and the direct Lyapunov method, the conditions for full synchronization are determined.Furthermore, this research shows that the L1 finite difference scheme and the second-order central difference scheme may successfully retain the properties of the related continuous system. The conclusions are proven throughout the paper using two major biological models, and numerical simulations are carried out to demonstrate the practical use of the recommended technique. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
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16 pages, 330 KiB  
Article
Solvability of a Boundary Value Problem Involving Fractional Difference Equations
by Zhiwei Lv, Chun Wu, Donal O’Regan and Jiafa Xu
Axioms 2023, 12(7), 650; https://doi.org/10.3390/axioms12070650 - 29 Jun 2023
Viewed by 719
Abstract
In this current work, we apply the topological degree and fixed point theorems to investigate the existence, uniqueness, and multiplicity of solutions for a boundary value problem associated with a fractional-order difference equation. Moreover, we provide some appropriate examples to verify our main [...] Read more.
In this current work, we apply the topological degree and fixed point theorems to investigate the existence, uniqueness, and multiplicity of solutions for a boundary value problem associated with a fractional-order difference equation. Moreover, we provide some appropriate examples to verify our main conclusions. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
18 pages, 1109 KiB  
Article
The Dynamics of a General Model of the Nonlinear Difference Equation and Its Applications
by Osama Moaaz and Aseel A. Altuwaijri
Axioms 2023, 12(6), 598; https://doi.org/10.3390/axioms12060598 - 16 Jun 2023
Cited by 1 | Viewed by 951
Abstract
This article investigates the qualitative properties of solutions to a general difference equation. Studying the properties of solutions to general difference equations greatly contributes to the development of theoretical methods and provides many pieces of information that may help to understand the behavior [...] Read more.
This article investigates the qualitative properties of solutions to a general difference equation. Studying the properties of solutions to general difference equations greatly contributes to the development of theoretical methods and provides many pieces of information that may help to understand the behavior of solutions of some special models. We present the sufficient and necessary conditions for the existence of prime period-two and -three solutions. We also obtain a complete perception of the local stability of the studied equation. Then, we investigate the boundedness and global stability of the solutions. Finally, we support the validity of the results by applying them to some special cases, as well as numerically simulating the solutions. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
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16 pages, 2606 KiB  
Article
Study of Burgers–Huxley Equation Using Neural Network Method
by Ying Wen and Temuer Chaolu
Axioms 2023, 12(5), 429; https://doi.org/10.3390/axioms12050429 - 26 Apr 2023
Cited by 3 | Viewed by 1323
Abstract
The study of non-linear partial differential equations is a complex task requiring sophisticated methods and techniques. In this context, we propose a neural network approach based on Lie series in Lie groups of differential equations (symmetry) for solving Burgers–Huxley nonlinear partial differential equations, [...] Read more.
The study of non-linear partial differential equations is a complex task requiring sophisticated methods and techniques. In this context, we propose a neural network approach based on Lie series in Lie groups of differential equations (symmetry) for solving Burgers–Huxley nonlinear partial differential equations, considering initial or boundary value terms in the loss functions. The proposed technique yields closed analytic solutions that possess excellent generalization properties. Our approach differs from existing deep neural networks in that it employs only shallow neural networks. This choice significantly reduces the parameter cost while retaining the dynamic behavior and accuracy of the solution. A thorough comparison with its exact solution was carried out to validate the practicality and effectiveness of our proposed method, using vivid graphics and detailed analysis to present the results. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
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14 pages, 290 KiB  
Article
Higher-Order Nabla Difference Equations of Arbitrary Order with Forcing, Positive and Negative Terms: Non-Oscillatory Solutions
by Jehad Alzabut, Said R. Grace, Jagan Mohan Jonnalagadda, Shyam Sundar Santra and Bahaaeldin Abdalla
Axioms 2023, 12(4), 325; https://doi.org/10.3390/axioms12040325 - 27 Mar 2023
Cited by 5 | Viewed by 873
Abstract
This work provides new adequate conditions for difference equations with forcing, positive and negative terms to have non-oscillatory solutions. A few mathematical inequalities and the properties of discrete fractional calculus serve as the fundamental foundation to our approach. To help establish the main [...] Read more.
This work provides new adequate conditions for difference equations with forcing, positive and negative terms to have non-oscillatory solutions. A few mathematical inequalities and the properties of discrete fractional calculus serve as the fundamental foundation to our approach. To help establish the main results, an analogous representation for the main equation, called a Volterra-type summation equation, is constructed. Two numerical examples are provided to demonstrate the validity of the theoretical findings; no earlier publications have been able to comment on their solutions’ non-oscillatory behavior. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
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