Stability Of Solutions Research Articles

Weak set-equilibrium problems: existence and stability conditions

We formulate four types of set-equilibrium problems based on some well-known set-order relations. Then, we study the existence of solutions for these problems by imposing standard generalized convexity and Lipschitz conditions. A comparison with some widely studied vector equilibrium problems is performed. Finally, we are interested in an issue concerning the stability of solutions of some related approximate set-optimization problems.

Boundedness and asymptotic stability of solutions in an alopecia areata chemotaxis system with signal-dependent sensitivity

Boundedness and asymptotic stability of solutions in an alopecia areata chemotaxis system with signal-dependent sensitivity

Existence and Stability for Fractional Differential Equations with a ψ–Hilfer Fractional Derivative in the Caputo Sense

This article aims to explore the existence and stability of solutions to differential equations involving a ψ-Hilfer fractional derivative in the Caputo sense, which, compared to classical ψ-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the ψ-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the ψ-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies.

The efficiency of synchronization dynamics and the role of network syncreactivity

Synchronization of coupled oscillators is a fundamental process in both natural and artificial networks. While much work has investigated the asymptotic stability of the synchronous solution, the fundamental question of the transient behavior toward synchronization has received far less attention. In this work, we present the transverse reactivity as a metric to quantify the instantaneous rate of growth or decay of desynchronizing perturbations. We first use the transverse reactivity to design a coupling-efficient and energy-efficient synchronization strategy that involves varying the coupling strength dynamically according to the current state of the system. We find that our synchronization strategy is able to synchronize networks in both simulation and experiment over a significantly larger (often by orders of magnitude) range of coupling strengths than is possible when the coupling strength is constant. Then, we characterize the effects of network topology on the transient dynamics towards synchronization by introducing the concept of network syncreactivity: A network with a larger syncreactivity has a larger transverse reactivity at every point on the synchronization manifold, independent of the oscillator dynamics. We classify real-world examples of complex networks in terms of their syncreactivity.

Stability and boundedness of solutions to a certain third-order neutral integro-differential equation with constant delay

In this work, we examine a third-order nonlinear neutral integro-differential equation with constant delay. By building a Lyapunov functional, we obtain some sufficient criteria that ensure the asymptotic stability and boundedness of solutions for an analyzed equation. We present two examples to demonstrate the applicability of our conclusions, which extend and improve several well-known results in the literature.

Heat transfer inspection and entropy optimization in water/MoS2-SiO2 Casson hybrid nanofluid flow with quadratic radiation ray over an exponentially contracting permeable Riga surface: Duality and stability analysis

In this study, we have examined the primary behaviour of a hybridized electro-magneto-non-Newtonian MoS2-SiO2/H2O nanofluid over a porous Riga surface that is exponentially contracting, in particular its flow dynamics and heat transfer characteristics. The key factors of this research are the presence of quadratic thermal radiation over the Riga surface under velocity slip and convective boundary. We have solved the highly circuitous coupled partial differential equations system by utilizing the bvp4c numerical method with MATLAB programming. A dual solution is determined based on a certain amount of mass suction parameter and shrinking parameter under the critical value range; beyond that value, no solution exists. We investigated the stability of the dual solutions by determining the minimal eigenvalue. An evaluation of the temporal stability of these solutions reveals that only the first solution remains stable. This is supported by the positivity of the smallest eigenvalues (β1> 0), indicating stability. Conversely, for the second solution, the smallest eigenvalues (β1< 0) are negative, signifying its instability. It can be seen that the nanofluid temperature is substantially enhanced for both branches of solution for the quadratic thermal radiating parameter, Eckert number, and Biot number. It was found that, in the first solution branch, the velocity layout was boosted with the velocity slip factor while reduced by enhancing the Casson parameter, whereas, in the second solution branch, the opposite behavior was observed. As the suction parameter increases, the heat transfer coefficient for the first solution is decreased by about 0.5 % and increases by about 0.75 % for the second solution branch. Quadratic thermal radiation parameter Nr significantly impacts the Nusselt number, as it is reduced by approximately 27.44 % and 28.72 % for the first and second solution branches, respectively. The shear stress coefficient is decreased by about 29.32 % for the first solution branch with contracting parameter λ. Because the Riga plate, paired with Casson nanofluid flows, has industrial and civil engineering applications, this study's thermal relevance makes it applicable to a wide range of industrial and engineering disciplines.

Analysis of multi-term arbitrary order implicit differential equations with variable type delay

Due to their capacity to simulate intricate dynamic systems containing memory effects and non-local interactions, fractional differential equations have attracted a great deal of attention lately. This study examines multi-term fractional differential equations with variable type delay with the goal of illuminating their complex dynamics and analytical characteristics. The introduction to fractional calculus and the justification for its use in many scientific and technical domains sets the stage for the remainder of the essay. It describes the importance of including variable type delay in differential equations and then applying it to model more sophisticated and realistic behaviours of real-world phenomena. The research study then presents the mathematical formulation of variable type delay and multi-term fractional differential equations. The system’s novelty stems from its unique combination of variable delay, generalized multi terms fractional differential operators (n and m), and integral implicit parameters, and studying the stability of the the newly formulated system as compared to the work in the existing literature. While the variable type delay is introduced as a function of time to describe instances where the delay is not constant, the fractional order derivatives are generated using the Caputo approach. The existence, uniqueness, and stability of solutions are the main topics of the theoretical analysis of the suggested differential equations. In order to establish important mathematical features, the inquiry makes use of spectral techniques, and fixed-point theorems. The study finishes by summarizing the major discoveries and outlining potential future research avenues in this developing field. It highlights the potential contribution of multi-term fractional differential equations with variable type delay to improving the control and design of complex systems. Overall, this study adds to the growing body of knowledge in the field of fractional calculus and provides insightful information about the investigation of multi-term fractional differential equations with variable type delay, making it pertinent for academics and practitioners from a variety of fields.

Non-commutative Schwarzschild black hole surrounded by Perfect fluid dark matter: Plasma lensing and thermodynamics analysis

The focus of this paper is to examine the properties of thermodynamics and weak gravitational lensing about the geometry of black holes within the context of a non-commutative Schwarzschild black hole surrounded by Perfect fluid dark matter. We examine the geometric mass and thermal temperature in this context to discuss the stability of the black hole solution. We examine the phase transition and stability while calculating the specific heat. We also research the black hole's energy emission process. We deduce that our researched black hole solution is thermally stable based on its thermodynamic features. Furthermore, we analyze uniform and non-uniform plasma by calculating the deflection angle, and we examine gravitational lensing in the weak plasma field. It is observed that in uniform plasma, the deflection angle is larger than in non-uniform plasma. We also looked at the image magnification caused by source brightness and found that the source image is enlarged more in uniform plasma than in non-uniform plasma.

The Reduced-Dimension Method for Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors of the Extended Fisher–Kolmogorov Equation

A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank–Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear extended Fisher–Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the dimension of the unknown CNMFE solution coefficient vectors. For this purpose, the CNMFE scheme of the above EFK equation is established, and the uniqueness, stability and convergence of the CNMFE solutions are discussed. Subsequently, the matrix-based RDCNMFE scheme is derived by applying the POD method. Furthermore, the uniqueness, stability and error estimates of the linearized RDCNMFE solution are proved. Finally, numerical experiments are carried out to validate the theoretical findings. In addition, we contrast the RDCNMFE method with the CNMFE method, highlighting the advantages of the dimensionality reduction method.

Synchronization of Chaotic Systems with Huygens-Like Coupling

One of the earliest reports on synchronization of inert systems dates back to the time of the Dutch scientist Christiaan Huygens, who discovered that a pair of pendulum clocks coupled through a wooden bar oscillate in harmony. A remarkable feature in Huygens’ experiment is that different synchronous behaviors may be observed by just changing a parameter in the coupling. Motivated by this, in this paper, we propose a novel synchronization scheme for chaotic oscillators, in which the design of the coupling is inspired in Huygens’ experiment. It is demonstrated that the coupled oscillators may exhibit not only complete synchronization, but also mixed synchronization—some states synchronize in anti-phase whereas other states synchronize in-phase—depending on a single parameter of the coupling. Additionally, the stability of the synchronous solution is investigated by using the master stability function approach and the largest transverse Lyapunov exponent. The Lorenz system is considered as particular application example, and the performance of the proposed synchronization scheme is illustrated with computer simulations and validated by means of experiments using electronic circuits.

A class of explicit second derivative general linear methods for non-stiff ODEs

In this paper, we construct explicit second derivative general linear methods (SGLMs) with quadratic stability and a large region of absolute stability for the numerical solution of non-stiff ODEs. The methods are constructed in two different cases: SGLMs with p = q = r = s and SGLMs with p = q and r = s = 2 in which p, q, r and s are respectively the order, stage order, the number of external stages and the number of internal stages. Examples of the methods up to order five are given. The efficiency of the constructed methods is illustrated by applying them to some well-known non-stiff problems and comparing the obtained results with those of general linear methods of the same order and stage order.

Periodic and Quasi-Periodic Orbits in the Collinear Four-Body Problem: A Variational Analysis

This paper investigated the periodic and quasi-periodic orbits in the symmetric collinear four-body problem through a variational approach. We analyze the conditions under which homographic solutions minimize the action functional. We compute the minimal value of the action functional for these solutions and show that, for four equal masses organized in a linear configuration, these solutions are the minimizers of the action functional. Additionally, we employ numerical experiments using Poincaré sections to explore the existence and stability of periodic and quasi-periodic solutions within this dynamical system. Our results provide a deeper understanding of the variational principles in celestial mechanics and reveal complex dynamical behaviors, crucial for further studies in multi-body problems.

Small scale model for predicting transportation-induced particle formation in biotherapeutics

Understanding protein adsorption and aggregation at the air-liquid interfaces of protein solutions is an important open challenge in biopharmaceutical, medical, and biotechnological applications, among others. Proteins, being amphiphilic, adsorb at the surface, partially unfold, and form a viscoelastic film through non-covalent interactions. Mechanical agitation of the surface can break this film up, releasing insoluble protein particles into the solution. These aggregates are usually highly undesirable and even toxic in cases, such as for biopharmaceutical application. Therefore, it is imperative to be able to predict the behavior of such solutions undergoing surface agitation during handling, usually transport or mixing. We apply the findings on the viscoelastic protein film, formed at the air-liquid interface, to the prediction of surface mediated aggregation in selected protein solutions of direct biopharmaceutical relevance. Our broad study of Brewster angle microscopy and aggregation monitoring across multiple size ranges by micro-flow imaging, light scattering, and size exclusion chromatography shows that formation of protein particles is driven by the adsorption rate as compared to the rate of surface turnover and that surface film dynamics in the quiescent phase directly affect aggregation. We demonstrate how these learnings can be directly applied to the design of a novel small scale biopharmaceutical stability study, simulating relevant transport conditions. More generally, we show the impact of adsorption dynamics at the air-liquid interface on the stability of a distinct protein solution, as a general contribution to understanding different colloidal and biological interfacial systems.

Multiplicity and Stability of Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function

Multiplicity and Stability of Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function

Preparation of Near-Spherical α-Al2O3 Particles by Enhanced Precipitation of Gibbsite from Sodium Aluminate Solution Using Highly Dispersed Al2O3 as Seeds.

The near-spherical α-Al2O3 particles (∼200 nm) were prepared for the first time through the precipitation of gibbsite from a supersaturated sodium aluminate solution, with the addition of an acidic solution containing highly dispersed nanosized Al2O3 seeds. The near-spherical α-Al2O3 were achieved by employing seeds at 1100 °C, with the initial nucleation temperature being only 900 °C, due to the highly homogeneous dispersion of the seeds into gibbsite formation by in situ growth that can effectively enhance the role of the seeds in the phase transition. Moreover, the precipitation rate of sodium aluminate solution was significantly accelerated by 17-fold. This acceleration was due to the slightly acidic solution breaking the stability of sodium aluminate solution through a neutralization reaction on the seed surface, leading to the rapid formation of gibbsite nuclei on the seed surface. Our findings provide a feasible approach for the fabrication of well-dispersed α-Al2O3 particles and make the industrial production of well-dispersed α-Al2O3 from sodium aluminate solution a commercial reality.

Single transition layer in mass-conserving reaction-diffusion systems with bistable nonlinearity

Abstract Mass-conserving reaction-diffusion systems with bistable nonlinearity are useful models for studying cell polarity formation, which is a key process in cell division and differentiation. We rigorously show the existence and stability of stationary solutions with a single internal transition layer in such reaction-diffusion systems under general assumptions by the singular perturbation theory. Moreover, we present a meaningful model for understanding the existence of an unstable transition layer solution; our numerical simulations show that the unstable solution is a separatrix of the dynamics of the model.

Existence, Uniqueness, and Stability of Solutions for Nabla Fractional Difference Equations

In this paper, we study a class of nabla fractional difference equations with multipoint summation boundary conditions. We obtain the exact expression of the corresponding Green’s function and deduce some of its properties. Then, we impose some sufficient conditions in order to ensure existence and uniqueness results. Also, we establish some conditions under which the solution to the considered problem is generalized Ulam–Hyers–Rassias stable. In the end, some examples are included in order to illustrate our main results.

Analysis of Caputo fractional variable order multi-point initial value problems: existence, uniqueness, and stability

In this paper, we examine the existence, uniqueness, and stability of solutions for a Caputo variable order ϑ-initial value problem (ϑ-IVP) with multi-point initial conditions. The proofs for uniqueness and existence leverage Sadovski’s and Banach’s fixed point theorems, along with the Kuratowski measure of noncompactness. Furthermore, we explore the Ulam–Hyers–Rassias (UHR) stability of the solution. To validate our findings, we present a numerical example.

Existence and stability of mild solutions to non autonomous impulsive stochastic neutral integrodifferential equations

Existence and stability of mild solutions to non autonomous impulsive stochastic neutral integrodifferential equations

Time-delay feedback control on vertical vibration of maglev train under unsteady aerodynamic force

The stability and safety of maglev trains are significantly affected by unsteady aerodynamic forces during high-speed operation. In this paper, we employ a time-delay feedback control strategy to suppress the nonlinear vertical vibrations of maglev train under periodic unsteady aerodynamic force. Firstly, the amplitude-frequency response equation of the maglev vehicle is derived using the multiple scales method, and the stability of the steady-state solutions is determined. Then, the influences of velocity feedback gain coefficient and velocity time delay on the nonlinear vibrations of the maglev vehicle are investigated. The optimal value range of the time delay is identified. The results demonstrate that the time delay feedback control can effectively suppress the nonlinear vibration of maglev vehicles and enhance their operational stability under the premise of reasonable selection of time delay parameters. The theoretical research presented in this paper can serve as a theoretical reference for the design, improvement, and optimization of control systems for high-speed maglev trains.