Asymptotic Behavior Of Solutions Research Articles

Analysis, Symmetry and Asymptotic Behaviour of Solutions for the Belousov-Zhabotinsky Model

This research provides analytical insights in connection with the solutions of the Oregonator model, a refined iteration of the iconic Belousov-Zhabotinsky (BZ) reaction problem. This chemical process, initially observed by B. P. Belousov while replicating the Krebs cycle in vitro and later modified by Zhabotinsky using Fe-phenanthroline (ferroin), has become a hallmark example of non-linear dynamics, chaos theory, and has parallels in various biological systems. Our study systematically delves into the boundedness, regularity, and possible symmetries of weak solutions. We explore traveling waves using the Tanh-method, alongside examining asymptotic solutions entrenched in self-similar forms and exponential scaling leading to a Hamilton-Jacobi equation. This research emphasizes on mathematical arguments along with the dynamics of the involved chemical concentrations. We provide new forms of analytical solutions showing them in a comprehensive manner that connects with the interpretation of the Oregonator model and its broader implications in chemical systems.

The influence of damping on the asymptotic behavior of solution for laminated beam

<p>This paper dealt with a laminated beam system along with structural damping, past history, distributed delay, and in the presence of both temperatures and micro-temperatures effects. The damping terms left the system dissipative. Employing the semigroup approach, we established the existence and uniqueness of the solution. Additionally, with the help of convenient assumptions on the kernel, we demonstrated a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagation. The main aim was to address how specific behaviors of the system were related to memory and delays. We aimed to investigate the joint impact of an infinite memory, distributed delay and micro-temperature effects on the system. We found a new relationship between the decay rate of solution and the growth of g at infinity. The objective was to find studies that use no- trivial results and their applications to relevant problems from mathematical physics.</p>

Comparative Analysis of Fractional Derivative Operators: Stability, Asymptotic Behavior, and Computational Challenges

This research paper explores novel theorems related to fractional differential operators, including Grunwald-Letnikov, Riemann-Liouville, Caputo, and Weyl. Each operator is rigorously defined, and their mathematical properties are investigated. The paper presents a detailed analysis of the asymptotic behavior of solutions to fractional differential equations governed by these operators. The advantages and disadvantages of each operator in capturing non-local behaviors, power-law decay, and handling initial conditions are discussed. Special emphasis is given to the stability characteristics of solutions, shedding light on the suitability of these operators for different types of problems. Through a comparative study, we highlight the unique features and computational challenges associated with each fractional derivative. Theoretical results are complemented by numerical simulations, providing insights into the practical implications of choosing a particular fractional operator in real-world applications.

Stability analysis for a Rao-Nakra sandwich beam equation with time-varying weights and frictional dampings

<abstract><p>In this paper, we studied the asymptotic behavior of solutions for a Rao-Nakra sandwich beam equation with time-varying weights and frictional damping terms acting complementarily in the domain. We studied the effect of the three damping on the asymptotic behavior of the energy function. Under nonrestrictive on the growth assumption on the frictional damping terms, we established exponential and general energy decay rates for this system by using the multiplier approach. The results generalized some earlier decay results on the Rao-Nakra sandwich beam equation.</p></abstract>

Asymptotic Behavior of Solutions for the Three-dimensional Generalized Incompressible MHD Equations with Nonlinear Damping Terms

Asymptotic Behavior of Solutions for the Three-dimensional Generalized Incompressible MHD Equations with Nonlinear Damping Terms

On extensibility and qualitative properties of solutions to Riccati's equation

We consider Riccati's equation on the real axis with continuous coefficients and non-negative discriminant of the right-hand side. We study the extensibility of its solutions to unbounded intervals. We obtain asymptotic formulae for its solutions in their dependence on the initial values and the properties of the functions representing roots of the right-hand side of the equation. We obtain results on the asymptotical behaviour of solutions defined near $\pm\infty$. We study the structure of the set of bounded solutions in the case when the roots of the right-hand side of the equation are $C^1$-functions which are different on the whole of their domain and tend monotonically to some limits as $x\to\pm\infty$. We extend, improve, or refine some well-known results. Bibliography: 47 titles.

On asymptotic behavior of solutions of multivariate linear stochastic differential equations with multiplicative noise

On asymptotic behavior of solutions of multivariate linear stochastic differential equations with multiplicative noise

On the boundedness of solutions of some fuzzy dynamical control systems

<abstract> <p>The asymptotic behavior of solutions of fuzzy control systems is a component of the study of fuzzy control theory. The study of stability for T-S (Takagi-Sugeno) fuzzy systems, which process qualitative data through linguistic expressions, is the subject of this paper. Asymptotic stability is conservative in many real-world applications due to measurement noise and other disruptions. The ultimate limit, which indicates that the mistakes stay in a specific area close to the origin after a long enough amount of time, is a crucial characteristic that is frequently defined for such systems. We are interested with the problem of the state feedback controller for T-S fuzzy models with uncertainties where the global exponential ultimate boundedness of solutions is studied for certain fuzzy control systems. We use common quadratic Lyapunov function and parallel distributed compensation controller techniques to study the asymptotic behavior of the solutions of fuzzy control system in presence of perturbations. An example demonstrating the validity of the main result is discussed.</p> </abstract>

The existence and asymptotic behavior of solutions to 3D viscous primitive equations with Caputo fractional time derivatives

On the one hand, the primitive three-dimensional viscous equations for large-scale ocean and atmosphere dynamics are commonly used in weather and climate predictions. On the other hand, ever since the middle of the last century, it has been widely recognized that the climate variability exhibits long-time memory. In this paper, we first prove the global existence of weak solutions to the primitive equations of large-scale ocean and atmosphere dynamics with Caputo fractional time derivatives. Then we establish the existence of an absorbing set, which is positively invariant. Finally, an attractor (strictly speaking, the minimal attracting set containing all the limiting dynamics) is constructed for the time fractional primitive equations, which means that the present state of a system may have long-time influences on the states in far future. However, there was no work on the long-time behavior of the time fractional primitive equations and we fill this gap in this paper.

Asymptotic Behavior of Solutions to Difference Equations of Neutral Type

We present sufficient conditions for the existence of a solution x to an equation Δm(xn-unxn-k)=anf(xn-τ)+bn,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Delta ^m(x_n-u_nx_{n-k})=a_nf(x_{n-\ au })+b_n, \\end{aligned}$$\\end{document}which is “close” to a given solution y to the linear homogeneous equation of neutral type Δm(yn-λyn-k)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta ^m(y_n-\\lambda y_{n-k})=0$$\\end{document}, where λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document} is the limit of the sequence u. Closeness of solutions to above equations is understood as xn-yn=o(ωn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_n-y_n=\ extrm{o}(\\omega _n)$$\\end{document}, where ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} is a given nonincreasing sequence with positive values. Moreover, we establish under which conditions for a given solution x to Δm(xn-unxn-k)=anf(xn-τ)+bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta ^m(x_n-u_nx_{n-k})=a_nf(x_{n-\ au })+b_n$$\\end{document} and a given nonincreasing sequence with positive values ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} there exists a polynomial sequence φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} of degree less than m such that xn=φ(n)+o(ωn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_n=\\varphi (n)+\ extrm{o}(\\omega _n)$$\\end{document}. Presented conditions strongly depend on λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}.

Retracted: Asymptotic Behavior of Solution for Functional Evolution Equations with Stepanov Forcing Terms

Retracted: Asymptotic Behavior of Solution for Functional Evolution Equations with Stepanov Forcing Terms

ASYMPTOTIC REPRESENTATION OF SOLUTIONS CLOSE TO LINEAR DIFFERENTIAL EQUATIONS OF THE THIRD ORDER

In the work, using the a priori properties of the Pω(λ0) class of solutions, the conditions for the existence of one class of solutions in the binomial non-autonomous of a differential equation of the third order with a nonlinearity close in some sense to linear, in the critical case, namely when λ0 = ±∞. Asymptotic at t↑ω (ω ≤ +∞) images for such solutions and their derivatives of the first and second order in the case λ0 = ±∞. Proven for of the nonlinear equation, the lemma and the theorem are transferred to linear differential equations of the third order with asymptotically small coefficients. The transferred results do not contradict, and to some extent, complement known results regarding the asymptotic behavior of solutions of linear differential equations of the third order.

Splitting of Solutions of Weakly Nonlinear Singularly Perturbed Equations Under Regular Degeneration

We consider a weakly nonlinear singularly perturbed equation in complex domains. The problem is posed about the possibility of splitting the equation into several components. By introducing new unknown functions, a system of two equations is obtained. Next, the asymptotic behavior of solutions of the resulting equations in complex domains is studied. It has been proven that the solution to each of these equations is dominant in certain parts of the areas under consideration. The solution of one of these equations determines the boundary lines and regions, and the solution of the other system determines the regular region.

Existence and Asymptotic Behavior of Solutions for Generalized Choquard Systems

Existence and Asymptotic Behavior of Solutions for Generalized Choquard Systems

Asymptotic Behavior of Solutions of the Nonstationary Schrödinger Equation with Potential that Slowly Depends on Time

Asymptotic Behavior of Solutions of the Nonstationary Schrödinger Equation with Potential that Slowly Depends on Time

The Quenching Solutions of a Singular Parabolic Equation

This article is dedicated to the study of the self-similar solutions of a nonlinear parabolic equation. More precisely, we consider the following uni-dimensional equation: (E) : ut(x, t) = (u m)_xx(x, t) − |x|^q u^−p (x, t), x ∈ R, t > 0, where m > 1, q > 1 and p > 0. Initially, we employed a fixed point theorem and an associated energy function to establish the existence of solutions. Subsequently, we derived some important results on the asymptotic behavior of solutions near the origin.

Asymptotic behavior of solutions for nonlinear parabolic problems with Marcinkiewicz data

Asymptotic behavior of solutions for nonlinear parabolic problems with Marcinkiewicz data

Existence and asymptotic behavior of solutions for the Schrödinger–Born–Infeld system with steep potential well

Existence and asymptotic behavior of solutions for the Schrödinger–Born–Infeld system with steep potential well

On asymptotic behavior of solutions of stochastic differential equations in multidimensional space

Consider the multidimensional SDE dX(t) = a(X(t)) dt + b(X(t)) dW(t). We study the asymptotic behavior of its solution X(t) as t → ∞, namely, we study sufficient conditions of transience of its solution X(t), stabilization of its multidimensional angle X(t)/|X(t)|, and asymptotic equivalence of solutions of the given SDE and the following ODE without noise: dx(t) = a(x(t)) dt.

Asymptotic behavior of solutions and spectrum of states in the quantum scalar field theory in the Schwarzschild spacetime

Asymptotic behavior of solutions and spectrum of states in the quantum scalar field theory in the Schwarzschild spacetime