Boundedness Of Solutions Research Articles

Stability and Ultimate Boundedness of Solutions of Certain Third Order Nonlinear Rectangular Matrix Differential Equations

We present in this paper the qualitative study of solutions of certain third order nonlinear matrix differential equations where the unknown function X is matrix-valued. The properties of solutions were investigated using Lyapunov’s direct method by employing the use of suitable Lyapunov functionals obtained from the differential equations describing the system satisfying certain requirements for establishing the stability and boundedness of solutions of the system considered. An example is given to demonstrate the significance of the results obtained as well as analysis through geometric graphs describing the dynamics of the system’s solutions. The results obtained are novel and will significantly enhance and extend the results of those mentioned in the literature.

Modeling the Bifurcation Dynamics of Rumor Propagation in the Spatial Environment

The harm caused by rumors is immeasurable. Studying the dynamic characteristics of rumors can help control their spread. In this paper, we propose a nonsmooth rumor model with a nonlinear propagation rate. First, we utilize the positive invariant regions to prove the boundedness of solutions. Second, we analyze the conditions for the existence of equilibrium points in both the left and right systems. Additionally, we confirm the occurrence of saddle-node bifurcation in the left system. Next, by considering the influence of spatial diffusion, we establish the conditions for Turing instability. Then we discuss the conditions for spatial homogeneous and inhomogeneous Hopf bifurcations in the left and right systems, respectively. We differentiate between supercritical and subcritical bifurcations using the Lyapunov coefficient. Furthermore, we examine the conditions for the existence of discontinuous Hopf bifurcation at the demarcation point. Finally, in the numerical simulation section, we validate our theorems on Turing patterns. We also investigate the impact of parameter changes on rumor propagation and conclude that an increase in the psychological inhibitory factor significantly reduces the rate of rumor propagation, providing an effective strategy for curbing rumors. To that end, we fit actual data to our system and the results are excellent, confirming the validity of the system.

Boundedness of solutions to singular anisotropic elliptic equations

Boundedness of solutions to singular anisotropic elliptic equations

On behaviours for the solution to a certain third-order stochastic integro-differential equation with time delay

In the present paper, Lyapunov functional (LF) is employed to discuss the continuability and boundedness of solutions for a third-order non-autonomous stochastic integro-differential equation (SIDE) with time delay. The third-order differential equation is ablated to a system of first-order differential equations together with its equivalent quadratic function to derive a suitable downright LF and then we study the behaviour of the solutions. A numerical example is considered to support our results. Moreover, we use the Euler-Maruyama method to get an approximate numerical solution for the considered system. The obtained result complements some recent ones in the literature.

Some further progress for boundedness of solutions to a quasilinear higher–dimensional chemotaxis–haptotaxis model with nonlinear diffusion

Some further progress for boundedness of solutions to a quasilinear higher–dimensional chemotaxis–haptotaxis model with nonlinear diffusion

Ultimate Boundedness of Solutions of Some System of Third-Order Nonlinear Differential Equations

Ultimate Boundedness of Solutions of Some System of Third-Order Nonlinear Differential Equations

Further study on Hopf bifurcation and hybrid control strategy in BAM neural networks concerning time delay

<abstract><p>Delayed dynamical system plays a vital role in describing the dynamical phenomenon of neural networks. In this article, we proposed a class of new BAM neural networks involving time delay. The traits of solution and bifurcation behavior of the established BAM neural networks involving time delay were probed into. First, the existence and uniqueness is discussed using a fixed point theorem. Second, the boundedness of solution of the formulated BAM neural networks involving time delay was analyzed by applying an appropriate function and inequality techniques. Third, the stability peculiarity and bifurcation behavior of the addressed delayed BAM neural networks were investigated. Fourth, Hopf bifurcation control theme of the formulated delayed BAM neural networks was explored by virtue of a hybrid controller. By adjusting the parameters of the controller, we could control the stability domain and Hopf bifurcation onset, which was in favor of balancing the states of different neurons in engineering. To verify the correctness of gained major outcomes, computer simulations were performed. The acquired outcomes of this article were new and own enormous theoretical meaning in designing and dominating neural networks.</p></abstract>

Leveraging Transformers for Boundary Detection and Encroachment Identification in Land Surveys

Land management, urban planning, and legal compliance all depend on the prompt and accurate detection of encroachments and the establishment of land boundaries. Especially in vast or complicated land regions, traditional surveying methods are labor-intensive and prone to inaccuracy since they frequently rely on manual processes or GPS-based approaches. In this study, we offer a unique method to automate land boundary recognition and monitor encroachment in satellite and aerial photos using transformer-baseddeep learning models, specifically Vision Transformers (ViT). By utilizing transformers' capacity to identify local and global patterns in high-resolution images, the suggested approach offers scalability and accuracy improvements over current techniques.Our approach compares predicted boundaries with historical geospatial data to detect encroachments by integrating boundary detection with a change detection pipeline. Extensive experiments on publicly available and custom geospatial datasets demonstrate that the transformer model outperforms traditional convolutional neural networks (CNNs) in terms of precision, recall, and Intersection over Union (IoU) scores. This paper also explores the integration of the proposed system with Geographic Information Systems (GIS) for real-time monitoring and visualization. The results indicate that transformer-based models can revolutionize land surveying by providing a robust, scalable, and efficient solution for boundary and encroachment detection.

Mathematical Analysis of COVID-19 model with Vaccination and Partial Immunity to Reinfection

COVID-19 is an infectious respiratory disease caused by a new virus, called SARS-CoV-2. Since itsinception, it has been a major cause of deaths and illnesses in the general population across the globe. Inthis paper, we have formulated and theoretically analyzed a non-linear deterministic model for COVID-19transmission dynamics by incorporating vaccination of the susceptible population. The system properties,such as the boundedness of solutions, the basic reproduction number R0, the local stability of disease-freeequilibrium(DFE), and endemic equilibrium (EE) points, are explored. Besides, the Lyapunov function isutilized to prove the global stability of both DFE and EE. The bifurcation analysis was carried out by utilizingthe center manifold theory. Then, the model is fitted with real COVID-19 cumulative data of infected casesin Kenya as from March 30, 2020, to March 30, 2022. Furthermore, sensitivity analysis was performed forthe proposed model to ascertain the relative significance of model parameters to COVID-19 transmissiondynamics. The simulations revealed that the spread of COVID-19 can be curtailed not only via vaccinationof susceptible populations but also increased administration of COVID-19 booster vaccine to the vaccinatedpersons and early detection and treatment of asymptomatic individuals.

Stability and Hopf bifurcation for age-structured SVIR epidemic model with different compartment ages and two delays effects

An age-structured SVIR epidemic model with different compartment ages and two delays is investigated. The model is transformed into a non-densely defined abstract Cauchy problem. The basic properties, including the positivity and boundedness of solutions, basic reproduction number R0 and the existence of equilibria, are first derived. The linearized system and characteristic equation at an equilibrium from the corresponding abstract Cauchy problem are also obtained. When R0<1, the local stability of the disease-free equilibrium is proved, and when R0>1 and the two delays vanish, the local stability of the endemic equilibrium is proved. When R0>1, and Assumptions 1 and 2 are satisfied, the existence of Hopf bifurcation are established respectively in the general case with two delays as bifurcation parameters and in three special cases with only one delay as the bifurcation parameter. The results show that disease with the incubation period and the infection period of infection age has very complex dynamical behavior at the endemic equilibrium. Finally, numerical examples validate the theoretical results.

Experimental Investigation on Salt Precipitation Behavior during Carbon Geological Sequestration: Considering the Influence of Formation Boundary Solutions

Experimental Investigation on Salt Precipitation Behavior during Carbon Geological Sequestration: Considering the Influence of Formation Boundary Solutions

Analysis of a delayed spatiotemporal model of HBV infection with logistic growth

In this paper, following previous works of ours, we deal with a mathematical model of hepatitis B virus (HBV) infection. We assume spatial diffusion of free HBV particles, logistic growth for both healthy and infected hepatocytes, and use the standard incidence function for viral infection. Moreover, one time delay is introduced to account for actual virus production. Another time delay is used to account for virus maturation. The existence, uniqueness, positivity and boundedness of solutions are established. Analyzing the model qualitatively and using a Lyapunov functional, we establish the existence of a threshold T 0 such that, if the basic reproduction number R 0 verifies R 0 ≤ T 0 < 1 , the infection-free equilibrium is globally asymptotically stable. When R 0 is greater than one, we discuss the local asymptotic stability of the unique endemic equilibrium and the occurrence of a Hopf bifurcation. Also, when R 0 is greater than one, the system is uniformly persistent, which means that the HBV infection is endemic. Finally, we carry out some relevant numerical simulations to clarify and interpret the theoretical results.

Boundedness of solutions for parabolic-elliptic predator-prey chemotaxis-fluid system with logistic source term

This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics{(n1)t+u⋅∇n1=Δn1−χ∇⋅(n1∇w)+n1(λ1−μ1n1+an2),x∈Ω,t>0,(n2)t+u⋅∇n2=Δn2+ξ∇⋅(n2∇z)+n2(λ2−μ2n2−bn1),x∈Ω,t>0,wt+u⋅∇w=Δw−w+n2,x∈Ω,t>0,u⋅∇z=Δz−z+n1,x∈Ω,t>0,ut+∇P=Δu+(n1+n2)∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 under no-flux boundary conditions for n1, n2, w and z in three-dimensional bounded domains and no-slip boundary conditions for u, this is∂n1∂ν=∂n2∂ν=∂w∂ν=∂z∂ν=0,u=0,x∈∂Ω,t>0, where χ>0, ξ>0, μ1≥0, μ2≥0, λ1≥0, λ2≥0, a≥0, b≥0 and ϕ∈W2,∞(Ω). This system is a coupled system of the chemotaxis equations and viscous incompressible fluid equations. Under appropriate assumptions, this problem exhibits a global classical bounded solution.

Stability and Hopf Bifurcation Analysis of a Multi-Delay Vector-Borne Disease Model with Presence Awareness and Media Effect

Vector-borne diseases, being one of the most difficult infectious diseases to understand, model, and control, account for a large proportion of human infectious diseases. In the current transmission process of infectious diseases, the latent period of pathogens in vivo, the influence of media coverage, and the presence of awareness on the spread and control of diseases are important factors that cannot be ignored. Based on this, a novel vector-borne disease model with latent delay and media coverage delay is proposed to discuss the impact of these factors. First, the global existence and ultimate boundedness of solutions for this model are obtained. Further, the exact expressions for the basic reproduction number are given, from which the existence and local stability of the disease-free and endemic equilibria are analyzed. Moreover, using the delay as a bifurcation parameter, we also discuss the existence, direction, and stability of the Hopf bifurcation. Finally, some numerical examples are carried out to explain the main theoretical results and discuss the impacts of the main parameters of this model on the transmission of vector-borne disease.

Sparse series solutions of random boundary and initial value problems

The study investigates the utilization of sparse representations through dictionary elements to address the classical Dirichlet and Cauchy types of stochastic boundary value problems (BVPs) and initial value problems (IVPs). A novel approach is introduced based on the recently developed stochastic pre-orthogonal adaptive Fourier decomposition (SPOAFD) technique. By employing SPOAFD, both analytic and numerical solutions for the stochastic BVPs and IVPs are formulated. Furthermore, the scope of the study is extended to include BVPs and IVPs associated with a specific class of fractional heat equations and fractional Poisson equations. In addition to establishing the theoretical framework, important computational aspects are thoroughly discussed to enable the implementation of practical algorithms. The proposed methodology is validated through numerical examples, demonstrating its effectiveness and computational efficiency.

Boundedness of solutions for semilinear Duffing equations

We are concerned with the semilinear Duffing equationx″+ω2x+g(x)+ϕ(x)=p(t), where ω∈R+﹨Q, g(x), ϕ(x), p(t) are real analytic, ϕ(x+T)=ϕ(x), p(t+2π)=p(t). Under the assumptions that limx→±∞⁡g(x)=g(±∞) exist and g(+∞)≠g(−∞), and without the polynomial-like growth condition on g(x), we prove the boundedness of all solutions and the existence of quasi-periodic solutions. The result strengthens the existing results in the literature where the polynomial-like growth condition on g(x) is needed.

Fractional-order SIR epidemic model with treatment cure rate

In this paper, we will study a fractional-order SIR epidemic model with treatment cure rate. The paper starts with the investigation of some needed preliminaries related to fractional calculus. Next, we investigate the well-posedness of the fractional-order model in terms of positivity and boundedness of solution. Then, we give the basic reproduction number via the next generation matrix method. The model has two steady states, namely, the disease-free equilibrium and the endemic equilibrium. It was proven that when the basic reproduction number is less than unity, then, the disease-free-equilibrium is globally asymptotically stable. While, when the basic reproduction number is greater than unity, then the endemic-equilibrium is globally asymptotically stable. Numerical simulations are carried out in order to confirm the theoretical results. To this end, we will apply a numerical approach based on the fundamental theorem of fractional calculus and a three-step Lagrange polynomial interpolation technique. It is shown that any increase of the fractional order derivative leads to a decrease in the convergence speed towards the studied equilibrium.

Nonlinear transmission exponent for boundedness of solutions to a chemotaxis system with indirect signal production

This paper deals with the Neumann initial–boundary problem for a fully parabolic chemotaxis system with indirect signal production, and the size of phenotypic transmission exponent is identified to guarantee the global existence and boundedness of classical solutions to the problem with suitably regular initial data.

Global existence of solutions in two-species chemotaxis system with two chemicals with sub-logistic sources in 2d

This paper investigates the global existence and boundedness of solutions in a two-species chemotaxis system with two chemicals and sub-logistic sources. The presence of a sub-logistic source in only one cell density equation effectively prevents the occurrence of blow-up solutions, even in fully parabolic chemotaxis systems.

Stability and boundedness of regular solutions for a Sisko flow in an infinite annular porous space

The present article is intended to study the boundedness of solutions for an unsteady non-Newtonian flow, whose strain–stress relationship is provided by the Sisko fluid model. Such kind of flow can appear in different scenarios, but we make it particular to the field of magnetohydrodynamics, as it is a remarkable area of research in its own right. The ideas exposed in this work can be extended, in a similar manner, to a wide range of applications. To make our fluid further general, we use the Darcy’s law to characterize the porous space in which the fluid is flowing. We develop the boundedness criteria provided that the velocity w and the function g=−∂w/∂r satisfy w,g,∂g/∂r,∂2g/∂r2∈L2(0,T;BMO), where BMO means bounded mean oscillation space. For this purpose, we will consider energy estimates in Sobolev spaces and develop the boundedness criteria for the resulting unsteady parabolic nonlinear equation.