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
In this paper two classes of equivalence transform methods for solving ordinary differential equations are proposed. One class of method is the equivalence integral transform method for special differential algebraic ...In this paper two classes of equivalence transform methods for solving ordinary differential equations are proposed. One class of method is the equivalence integral transform method for special differential algebraic problems. The advantage of this class of method is such that the amount of work calculating one integration with parameters becomes that of two interpolations, when the system of nonlinear equations is solved on the right hand side function. The other class of method is the equivalence substitution method for avoiding calculating derivative on the right hand side function. In order to avoid calculation derivatives, two equivalence substitution methods are proposed here. The application instances of some special effect of the equivalence substitution methods are given.展开更多
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.
基金The project was supported by the National Natural Science Faundation of China
文摘In this paper two classes of equivalence transform methods for solving ordinary differential equations are proposed. One class of method is the equivalence integral transform method for special differential algebraic problems. The advantage of this class of method is such that the amount of work calculating one integration with parameters becomes that of two interpolations, when the system of nonlinear equations is solved on the right hand side function. The other class of method is the equivalence substitution method for avoiding calculating derivative on the right hand side function. In order to avoid calculation derivatives, two equivalence substitution methods are proposed here. The application instances of some special effect of the equivalence substitution methods are given.
