Asymptotic behavior of the solutions of an integrodifferential system

Summary We consider the system(L): , t ⩾ p, y(t)=f(t), t⩽0, where y is an n-vector and each A_i, B(t) are n × n matrices. System(L) generates a semigroup by means of T_tf(s)=y (t+s, f), f(s) ∈ BC_l(− ∞, 0]. Under some hypotheses concerning the roots ofdet where is the Laplace transform of B(t), the asymptotic behavior of y(t) is discussed. Two typical results are: Theorem 3.1: suppose ∥B(t)∥ ɛ L₁[0, ∞), thendet forRe λ>0 iff for every ɛ>0 there is an M_ɛ>0 such that ∥T_tf∥_l ⩽ ⩽ M_ɛ exp [ɛt]∥f∥_l for t ⩾ 0. Corollary 3.1.1: suppose exp [at]B(t) ∈ ∈ L₁[0, ∞) for some a>0 anddet forRe λ>−a. Then the solution of(L) is exponentially asymptotically stable. Entrata in Redazione il 21 marzo 1975. The author is grateful to ProfessorC. Corduneanu for suggesting this problem and for many helpful discussions during the preparation of the paper.     

Asymptotic solution of a system of linear integrodifferential equations of rational rank

Ukrainian Mathematical Journal -     

Asymptotic stability of impulsive stochastic partial integrodifferential equations with delays

In this paper, we study the existence and asymptotic stability in the p-th moment of mild solutions of nonlinear impulsive stochastic partial functional integrodifferential equations with delays. We suppose that the linear part possesses a resolvent operator in the sense given in Grimmer [R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc. 273(1) (1982), 333–349] and the nonlinear terms are assumed to be Lipschitz continuous. A fixed point approach is employed for achieving the required result. An example is provided to illustrate the results of this work.     

Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability

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The existence, uniqueness and stability of positive periodic solution for periodic reaction-diffusion system

1. IntroductionLet fi e RN be a bounded open domain with smooth boundary afl, and for eachi E {1, 2,', m}, let Li be a second order differelltial operator defined byand Bi be a boundary operator given bywhere % denotes the outward normal derivative of m on an.We consider the following boundary value problem of the reaction-diffusion system withtime delaywhere i = 1, 2,'. I m) mr = "i(x, t -- r), r 2 0 is a constant. If r ~ 0, it means that system(I) does not include the terms of time lag…     

A numerical solution of singular integrodifferential equations with variable coefficients

A method for the numerical solution of singular integrodifferential equations is presented where the integrals are discretized by using a convenient quadrature rule. Then the problem is reduced to a system of linear algebraic equations by applying the discretized functional equation to appropriately selected collocation points. This technique constitutes an extension of an analogous method convenient for solving singular integral equations which was proposed by the authors.     

Asymptotic behavior of a reaction-diffusion system in bacteriology

This paper is concerned with the asymptotic behavior of the solution for a coupled system of reaction-diffusion equations which describes the bacteria growth and the diffusion of histidine and buffer concentrations. Under the basic boundary condition of Neumann type or mixed type the coupled system can have infinitely many steady-state solutions. The present paper gives some explicit information on the asymptotic limit of the time-dependent solution in relation to these steady states. This information exhibits some rather distinct properties of the solutions between the Neumann boundary problem and the Dirichlet or mixed boundary problem.     

Asymptotic stability of an N-dimensional impulsive competitive system

In this paper, we first consider an $N$-dimensional Lotka–Volterra system with fixed moments of impulsive perturbations. By means of piecewise continuous functions which are modifications of classical Lyapunov's functions we give sufficient conditions for asymptotic stability of the solutions.     

Asymptotic stability of reaction-diffusion systems in chemical reactor and combustion theory

Some coupled reaction-diffusion systems arising from chemical diffusion processes and combustion theory are analyzed. This analysis includes the existence and uniqueness of positive time-dependent solutions, upper and lower bounds of the solution, asymptotic behavior and invariant sets, and the stability of steady-state solutions, including an estimate of the stability region. Explicit conditions for the asymptotic behavior and the stability of a steady-state solution are given. These conditions establish some interrelationship among the physical parameters of the diffusion medium, the reaction mechanism, the initial function and the type of boundary condition. Under the same set of physical parameters and reaction function, a comparison between the Neumann type and Dirichlet or third type boundary condition exhibits quite different asymptotic behavior of the solution. For the general nonhomogeneous system, multiple steady-state solutions may exist and only local stability results are obtained. However, for certain models it is possible to obtain global stability of a steady-state solution by either increasing the diffusion coefficients or decreasing the size of the diffusion medium. This fact is demonstrated by a one-dimensional tubular reactor model commonly discussed in the literature.     

A fast numerical solution of the 3D nonlinear tempered fractional integrodifferential equation

In this paper, we investigate the numerical solution of the three-dimensional (3D) nonlinear tempered fractional integrodifferential equation which is subject to the initial and boundary conditions. The backward Euler (BE) method in association with the first-order convolution quadrature rule is employed to discretize this equation for time, and the Galerkin finite element method is applied for space, which is combined with an alternating direction implicit (ADI) algorithm, in order to reduce the computational cost for solving the three-dimensional nonlocal problem. Then a fully discrete BE ADI Galerkin finite element scheme can be obtained by linearizing the non-linear term. Thereafter we prove a positive-type lemma, from which the stability and convergence of the proposed numerical scheme are derived based on the energy method. Numerical experiments are performed to verify the effectiveness of the proposed approach.     

Numerical solution of two-dimensional reaction-diffusion Brusselator system

In this paper, polynomial based differential quadrature method (DQM) is applied for the numerical solution of a class of two-dimensional initial-boundary value problems governed by a non-linear system of partial differential equations. The system is known as the reaction-diffusion Brusselator system. The system arises in the modeling of certain chemical reaction-diffusion processes. In Brusselator system the reaction terms arise from the mathematical modeling of chemical systems such as in enzymatic reactions, and in plasma and laser physics in multiple coupling between modes. The numerical results reported for three specific problems. Convergence and stability of the method is also examined numerically.     

Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution

A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ?₊, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.