This paper deals with the numerical solution of initial value problems for pantograph differential equations with variable delays. We investigate the stability of one leg θ-methods in the numerical solution of these ...This paper deals with the numerical solution of initial value problems for pantograph differential equations with variable delays. We investigate the stability of one leg θ-methods in the numerical solution of these problems. Sufficient conditions for the asymptotic stability of θ-methods are given by Fourier analysis and Ergodic theory.展开更多
This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented...This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small ε > 0. We will study the numerical solution defined by the linear θ-method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small ε > 0 if and only if θ = 1.展开更多
This paper deals with the stability analysis of numerical methods for the solution of delay differential equations. We focus on the behaviour of three θ-methodsin the solution of the linear test equation u'(t)-A(...This paper deals with the stability analysis of numerical methods for the solution of delay differential equations. We focus on the behaviour of three θ-methodsin the solution of the linear test equation u'(t)-A(t)u(t)+B(t)u( (t)) with (t)and A(t),B(t) continuous matrix functions. The stability regions for the threeθ-methods are determined.展开更多
This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solutio...This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solution of test equations u'(t) = a 11 u(t) + a12v(t) + b11 u(t - τ1) + b12v(t-τ2,v'(t) = a21 u(t) + a22 v(t) + b21 u(t -τ1,) + b22 v(t -τ2), t>0,with initial conditionsu(t)=u0(t),v(t) =v0(t), t≤0.where aij, bij∈C, τj >0, i,j = 1,2,, and u0(t), v0(t)are continuous and complex valued. Sufficient conditions for the asymptotic stability of test equation are derived. Furthermore, with respect to an appropriate definition of stability for the numerical method, it is proved that the linear θ-method is stable if and only if 1/2≤θ≤1 and the one-leg θ-method is stable if and only if θ= 1.展开更多
This paper deals with the asymptotic stability of theoretical solutions and numerical methods for the delay differential equations(DDEs)where a,b1,b2,. ..,bm and yo ∈ C, 0 < λm ≤ λm-1 ≤ ... ≤λl < 1. A suf...This paper deals with the asymptotic stability of theoretical solutions and numerical methods for the delay differential equations(DDEs)where a,b1,b2,. ..,bm and yo ∈ C, 0 < λm ≤ λm-1 ≤ ... ≤λl < 1. A sufficient condition such that the differential equations are asymptotically stable isderived.And it is shown that the linear θ-method is AGPm-stable if and only if1/2≤θ-≤ 1.展开更多
文摘This paper deals with the numerical solution of initial value problems for pantograph differential equations with variable delays. We investigate the stability of one leg θ-methods in the numerical solution of these problems. Sufficient conditions for the asymptotic stability of θ-methods are given by Fourier analysis and Ergodic theory.
基金This project is supported by NSF of China (No.10101012)Shanghai Rising Star Program (No.03QA14036) The Special Funds for Major Specialties of Shanghai Education Committee.
文摘This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small ε > 0. We will study the numerical solution defined by the linear θ-method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small ε > 0 if and only if θ = 1.
文摘This paper deals with the stability analysis of numerical methods for the solution of delay differential equations. We focus on the behaviour of three θ-methodsin the solution of the linear test equation u'(t)-A(t)u(t)+B(t)u( (t)) with (t)and A(t),B(t) continuous matrix functions. The stability regions for the threeθ-methods are determined.
文摘This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solution of test equations u'(t) = a 11 u(t) + a12v(t) + b11 u(t - τ1) + b12v(t-τ2,v'(t) = a21 u(t) + a22 v(t) + b21 u(t -τ1,) + b22 v(t -τ2), t>0,with initial conditionsu(t)=u0(t),v(t) =v0(t), t≤0.where aij, bij∈C, τj >0, i,j = 1,2,, and u0(t), v0(t)are continuous and complex valued. Sufficient conditions for the asymptotic stability of test equation are derived. Furthermore, with respect to an appropriate definition of stability for the numerical method, it is proved that the linear θ-method is stable if and only if 1/2≤θ≤1 and the one-leg θ-method is stable if and only if θ= 1.
文摘This paper deals with the asymptotic stability of theoretical solutions and numerical methods for the delay differential equations(DDEs)where a,b1,b2,. ..,bm and yo ∈ C, 0 < λm ≤ λm-1 ≤ ... ≤λl < 1. A sufficient condition such that the differential equations are asymptotically stable isderived.And it is shown that the linear θ-method is AGPm-stable if and only if1/2≤θ-≤ 1.
