Differential Equations in Applied Mathematics

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

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 12074

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Guest Editor
Department of Mathematics, Yunnan University, Kunming 650091, China
Interests: stochastic ordinary differential equations; fuzzy ordinary differential equations; fractional ordinary differential equations; functional ordinary; partial differential equations
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Special Issue Information

Dear Colleagues,

Numerous significant ideas in applied mathematics have been formulated within the framework of ordinary or partial differential equations, which provides a language for the illustration of these ideas. Through the years, mathematicians and scientists have developed extensions of these methodologies to almost all areas of science and technology.

This Special Issue provides an opportunity to showcase recent developments in the many branches of ordinary or partial differential equations in applied mathematics, which are related to stochastic, fuzzy, functional, and fractional differential or difference equations in sciences and engineering. Among the topics that this Special Issue will address, we will consider the following non-exhaustive list:

  • Qualitative analyses of ordinary or partial differential or difference equations;
  • Periodic, almost periodic, and almost automorphic oscillations, etc.;
  • Random attractors or blowup solutions;
  • Lattice dynamic systems;
  • Boundary valued problems;
  • Neural networks;
  • Mathematical models in biology, ecology, economy, etc.;
  • Variational methods and other nonlinear analysis techniques;
  • Control and optimization, e.g., synchronizations, adaptive controls, automatic control and stochastic optimizations, etc.;
  • Numerical analysis on ordinary or partial differential equations;
  • Differential equations with discontinuity, e.g., switch systems, Markovian jumping systems;
  • Uncertain systems, e.g., stochastic systems, fuzzy systems;
  • Discussions of fractional differential or difference equations.

In this Special Issue, we encourage you to submit papers on recent developments in the area of ordinary or partial differential or difference equations in applied areas.

Dr. Tianwei Zhang
Guest Editor

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Keywords

  • periodic oscillations
  • attractor
  • lattice
  • boundary valued problems
  • variational methods
  • fixed point theory
  • control and optimization
  • switch systems
  • markovian jump
  • discontinuous system
  • uncertain system
  • stochastic system
  • fuzzy

Published Papers (12 papers)

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Research

21 pages, 776 KiB  
Article
Boundary Controlling Synchronization and Passivity Analysis for Multi-Variable Discrete Stochastic Inertial Neural Networks
by Yongyan Yang, Tianwei Zhang and Zhouhong Li
Axioms 2023, 12(9), 820; https://doi.org/10.3390/axioms12090820 - 26 Aug 2023
Viewed by 636
Abstract
The current paper considers discrete stochastic inertial neural networks (SINNs) with reaction diffusions. Firstly, we give the difference form of SINNs with reaction diffusions. Secondly, stochastic synchronization and passivity-based control frames of discrete time and space SINNs are newly formulated. Thirdly, by designing [...] Read more.
The current paper considers discrete stochastic inertial neural networks (SINNs) with reaction diffusions. Firstly, we give the difference form of SINNs with reaction diffusions. Secondly, stochastic synchronization and passivity-based control frames of discrete time and space SINNs are newly formulated. Thirdly, by designing a boundary controller and constructing a Lyapunov-Krasovskii functional, we address decision theorems for stochastic synchronization and passivity-based control for the aforementioned discrete SINNs. Finally, to illustrate our main results, a numerical illustration is provided. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
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23 pages, 474 KiB  
Article
Global Dynamics of an Age-Structured Tuberculosis Model with Vaccine Failure and Nonlinear Infection Force
by Zhongkai Guo and Liang Zhang
Axioms 2023, 12(9), 805; https://doi.org/10.3390/axioms12090805 - 22 Aug 2023
Viewed by 799
Abstract
China bears a heavy burden due to tuberculosis (TB) with hundreds of thousands of people falling ill with the disease every year. Therefore, it is necessary to understand the effectiveness of current control measures in China. In this paper, we first present a [...] Read more.
China bears a heavy burden due to tuberculosis (TB) with hundreds of thousands of people falling ill with the disease every year. Therefore, it is necessary to understand the effectiveness of current control measures in China. In this paper, we first present a TB model that incorporates both vaccination and treatment. Additionally, the model considers TB transmission characteristics such as relapse and variable latency. We then define the basic reproduction number R0 of the proposed model and indicate that the disease-free equilibrium state is globally asymptotically stable if R0<1, and the endemic equilibrium state is globally asymptotically stable if R0>1. We then apply the Grey Wolf Optimizer algorithm to obtain the parameters and initial values of the model by combining TB data collected in China from 2007 to 2020. Through the partial rank correlation coefficient method, we identify the parameters that are most sensitive to R0. Based on the analysis results of the model, we propose some suggestions for TB control measures in the conclusion section. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
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18 pages, 772 KiB  
Article
Phase Portraits of Families VII and VIII of the Quadratic Systems
by Laurent Cairó and Jaume Llibre
Axioms 2023, 12(8), 756; https://doi.org/10.3390/axioms12080756 - 1 Aug 2023
Viewed by 726
Abstract
The quadratic polynomial differential systems in a plane are the easiest nonlinear differential systems. They have been studied intensively due to their nonlinearity and the large number of applications. These systems can be classified into ten classes. Here, we provide all topologically different [...] Read more.
The quadratic polynomial differential systems in a plane are the easiest nonlinear differential systems. They have been studied intensively due to their nonlinearity and the large number of applications. These systems can be classified into ten classes. Here, we provide all topologically different phase portraits in the Poincaré disc of two of these classes. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
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18 pages, 359 KiB  
Article
Asynchronous Switching Control of Discrete Time Delay Linear Switched Systems Based on MDADT
by Jimin Yu, Xiaoyu Qi and Yabin Shao
Axioms 2023, 12(8), 747; https://doi.org/10.3390/axioms12080747 - 29 Jul 2023
Viewed by 752
Abstract
Ideally, switching between subsystems and controllers occurs synchronously. In other words, whenever a subsystem requires switching, its corresponding sub-controller will be promptly activated. However, in reality, due to network delays, system detection, etc., the activation of candidate controllers frequently lags, which causes issues [...] Read more.
Ideally, switching between subsystems and controllers occurs synchronously. In other words, whenever a subsystem requires switching, its corresponding sub-controller will be promptly activated. However, in reality, due to network delays, system detection, etc., the activation of candidate controllers frequently lags, which causes issues with asynchronous switching between controllers and subsystems. This asynchronous switching problem may affect system performance and even make the system unstable because the state between the subsystem and the controller may be inconsistent, resulting in the controller not being able to control the subsystem correctly. To keep the system stable while using asynchronous switching, this work suggests an asynchronous control technique for a class of discrete linear switching systems with time delay based on the mode-dependent average dwell time (MDADT). First, we construct a state feedback controller and establish a closed-loop system. In the asynchronous and synchronous intervals of subsystems and controllers, different Lyapunov functions are selected, and sufficient conditions for exponential stability and the H performance of the closed-loop system under asynchronous switching are obtained. In addition, using the MDADT switching strategy, the relevant parameters of each subsystem are designed and the corresponding state–feedback controller gain matrix can be obtained. Finally, a switching system with three subsystems is shown. The approach is confirmed by simulating it using the average dwell time (ADT) switching strategy and the MDADT switching strategy separately. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
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13 pages, 353 KiB  
Article
Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with p-Laplacian
by Kaihong Zhao
Axioms 2023, 12(8), 733; https://doi.org/10.3390/axioms12080733 - 27 Jul 2023
Cited by 12 | Viewed by 829
Abstract
The fractional order p-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary [...] Read more.
The fractional order p-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary value problem (BVP) for a class of nonlinear Hadamard fractional differential equation with p-Laplacian operator. By virtue of an important fixed point theorem on a complete metric space with two distances, we study the solvability and approximation of this BVP. Based on nonlinear analysis methods, we further discuss the generalized Ulam-Hyers (GUH) stability of this problem. Eventually, we supply two example and simulations to verify the correctness and availability of our main results. Compared to many previous studies, our approach enables the solution of the system to exist in metric space rather than normed space. In summary, we obtain some sufficient conditions for the existence, uniqueness, and stability of solutions in the metric space. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
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26 pages, 1328 KiB  
Article
Weighted Pseudo-θ-Almost Periodic Sequence and Finite-Time Guaranteed Cost Control for Discrete-Space and Discrete-Time Stochastic Genetic Regulatory Networks with Time Delays
by Shumin Sun, Tianwei Zhang and Zhouhong Li
Axioms 2023, 12(7), 682; https://doi.org/10.3390/axioms12070682 - 11 Jul 2023
Cited by 1 | Viewed by 639
Abstract
This paper considers the dual hybrid effects of discrete-time stochastic genetic regulatory networks and discrete-space stochastic genetic regulatory networks in difference formats of exponential Euler difference and second-order central finite difference. The existence of a unique-weight pseudo-θ-almost periodic sequence solution for [...] Read more.
This paper considers the dual hybrid effects of discrete-time stochastic genetic regulatory networks and discrete-space stochastic genetic regulatory networks in difference formats of exponential Euler difference and second-order central finite difference. The existence of a unique-weight pseudo-θ-almost periodic sequence solution for discrete-time and discrete-space stochastic genetic regulatory networks on the basis of discrete constant variation formulation is discussed, as well as the theory of semi-flow and metric dynamical systems. Furthermore, a finite-time guaranteed cost controller is constructed to reach global exponential stability of these discrete networks via establishing a framework of drive, response, and error networks. The results indicate that spatial diffusions of non-negative dense coefficients have no influence on the global existence of the unique weighted pseudo-θ-almost periodic sequence solution of the networks. The present study is a basic work in the consideration of discrete spatial diffusion in stochastic genetic regulatory networks and serves as a foundation for further study. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
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25 pages, 714 KiB  
Article
A Proof of Chaos for a Seasonally Perturbed Version of Goodwin Growth Cycle Model: Linear and Nonlinear Formulations
by Marina Pireddu
Axioms 2023, 12(4), 344; https://doi.org/10.3390/axioms12040344 - 31 Mar 2023
Cited by 1 | Viewed by 923
Abstract
We show the existence of complex dynamics for a seasonally perturbed version of the Goodwin growth cycle model, both in its original formulation and for a modified formulation, encompassing nonlinear expressions of the real wage bargaining function and of the investment function. The [...] Read more.
We show the existence of complex dynamics for a seasonally perturbed version of the Goodwin growth cycle model, both in its original formulation and for a modified formulation, encompassing nonlinear expressions of the real wage bargaining function and of the investment function. The need to deal with a modified formulation of the Goodwin model is connected with the economically sensible position of orbits, which have to lie in the unit square, in contrast to what occurs in the model’s original formulation. In proving the existence of chaos, we follow the seminal idea by Goodwin of studying forced models in economics. Namely, the original and the modified formulations of Goodwin model are described by Hamiltonian systems, characterized by the presence of a nonisochronous center, and the seasonal variation of the parameter, representing the ratio between capital and output, which is common to both frameworks, is empirically grounded. Hence, exploiting the periodic dependence on time of that model parameter we enter the framework of Linked Twist Maps. The topological results valid in this context allow us to prove that the Poincaré map, associated with the considered systems, is chaotic, focusing on sets that lie in the unit square, and also when dealing with the original version of the Goodwin model. Accordingly, the trademark features of chaos follow, such as sensitive dependence on initial conditions and positive topological entropy. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
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14 pages, 355 KiB  
Article
Existence and Stability of a Nonlinear Distributed Delayed Periodic AG-Ecosystem with Competition on Time Scales
by Kaihong Zhao
Axioms 2023, 12(3), 315; https://doi.org/10.3390/axioms12030315 - 22 Mar 2023
Cited by 15 | Viewed by 1042
Abstract
The Ayala-Gilpin (AG) kinetics system is one of the famous mathematical models of ecosystem. This model has been widely concerned and studied since it was proposed. This paper stresses on a nonlinear distributed delayed periodic AG-ecosystem with competition on time scales. In the [...] Read more.
The Ayala-Gilpin (AG) kinetics system is one of the famous mathematical models of ecosystem. This model has been widely concerned and studied since it was proposed. This paper stresses on a nonlinear distributed delayed periodic AG-ecosystem with competition on time scales. In the sense of time scale, our model unifies and generalizes the discrete and continuous cases. Firstly, with the aid of the auxiliary function having only two zeros in the real number field, we apply inequality technique and coincidence degree theory to obtain some sufficient criteria which ensure that this model has periodic solutions on time scales. Meanwhile, the global asymptotic stability of the periodic solution is founded by employing stability theory in the sense of Lyapunov. Eventually, we provide an illustrative example and conduct numerical simulation by means of MATLAB tools. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
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16 pages, 295 KiB  
Article
Monotonically Iterative Method for the Cantilever Beam Equations
by Yujun Cui, Huiling Chen and Yumei Zou
Axioms 2023, 12(2), 178; https://doi.org/10.3390/axioms12020178 - 8 Feb 2023
Viewed by 787
Abstract
In this paper, we consider the existence of extremal solutions for the nonlinear fourth-order differential equation. By use of a new comparison result, some sufficient conditions for the existence of extremal solutions are established by combining the monotone iterative technique and the methods [...] Read more.
In this paper, we consider the existence of extremal solutions for the nonlinear fourth-order differential equation. By use of a new comparison result, some sufficient conditions for the existence of extremal solutions are established by combining the monotone iterative technique and the methods of lower and upper solutions. Finally, an example is given to illustrate the validity of our main results. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
14 pages, 341 KiB  
Article
On the Crossing Bridge between Two Kirchhoff–Love Plates
by Alexander Khludnev
Axioms 2023, 12(2), 120; https://doi.org/10.3390/axioms12020120 - 26 Jan 2023
Cited by 3 | Viewed by 886
Abstract
The paper is concerned with equilibrium problems for two elastic plates connected by a crossing elastic bridge. It is assumed that an inequality-type condition is imposed, providing a mutual non-penetration between the plates and the bridge. The existence of solutions is proved, and [...] Read more.
The paper is concerned with equilibrium problems for two elastic plates connected by a crossing elastic bridge. It is assumed that an inequality-type condition is imposed, providing a mutual non-penetration between the plates and the bridge. The existence of solutions is proved, and passages to limits are justified as the rigidity parameter of the bridge tends to infinity and to zero. Limit models are analyzed. The inverse problem is investigated when both the displacement field and the elasticity tensor of the plate are unknown. In this case, additional information concerning a displacement of a given point of the plate is assumed be given. A solution existence of the inverse problem is proved. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
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12 pages, 849 KiB  
Article
Coincidence Theory of a Nonlinear Periodic Sturm–Liouville System and Its Applications
by Kaihong Zhao
Axioms 2022, 11(12), 726; https://doi.org/10.3390/axioms11120726 - 13 Dec 2022
Cited by 9 | Viewed by 1057
Abstract
Based on the second derivative, this paper directly establishes the coincidence degree theory of a nonlinear periodic Sturm–Liouville (SL) system. As applications, we study the existence of periodic solutions to the S–L system with some special nonlinear functions by applying Mawhin’s continuation theorem. [...] Read more.
Based on the second derivative, this paper directly establishes the coincidence degree theory of a nonlinear periodic Sturm–Liouville (SL) system. As applications, we study the existence of periodic solutions to the S–L system with some special nonlinear functions by applying Mawhin’s continuation theorem. Some examples and simulations are furnished to inspect the correctness and availability of the chief findings. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
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12 pages, 414 KiB  
Article
Probing the Oscillatory Behavior of Internet Game Addiction via Diffusion PDE Model
by Kaihong Zhao
Axioms 2022, 11(11), 649; https://doi.org/10.3390/axioms11110649 - 16 Nov 2022
Cited by 11 | Viewed by 1276
Abstract
We establish a non-linear diffusion partial differential equation (PDE) model to depict the dynamic mechanism of Internet gaming disorder (IGD). By constructing appropriate super- and sub-solutions and applying Schauder’s fixed point theorem and continuation method, we study the existence and asymptotic stability of [...] Read more.
We establish a non-linear diffusion partial differential equation (PDE) model to depict the dynamic mechanism of Internet gaming disorder (IGD). By constructing appropriate super- and sub-solutions and applying Schauder’s fixed point theorem and continuation method, we study the existence and asymptotic stability of traveling wave solutions to probe into the oscillating behavior of IGD. An example is numerically simulated to examine the correctness of our outcomes. Full article
(This article belongs to the Special Issue Differential Equations in Applied Mathematics)
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