This paper deals with the asymptotic stability of theoretical solutions and numerical methods for systems of neutral differential equations x' = Ax'(t -r) + Bx(t) + Cx(t -~-), where A, B, and C are constant complex N x N matrices, and r > 0. A necessary and sufficient condition such that the differential equations are asymptotically stable is derived. We also focus on the numerical stability properties of adaptations of one-parameter methods. Further, we investigate carefully the characterization of the stability region.
This paper deals with the stability analysis of implicit RungeKutta methods for the numerical solutions of the systems of neutral delay differential equations. We focus on the behavior of such methods with respect to the linear test equationswhere τ > 0, L, M and N are d × d complex matrices. We show that an implicit Runge-Kutta method is NGP-stable if and only if it is A-stable. Classification (1991): 65L20
Mathematics Subject
This paper is concerned with delay-independent asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods. We first give some sufficient conditions for the delay-independent asymptotic stability of these equations. Then we study and derive a sufficient and necessary condition for the delay-independent asymptotic stability of numerical solutions obtained by linear multistep methods combined with Lagrange interpolation. Finally, one numerical example is performed to confirm our theoretical result.
We are concerned with the asymptotic stability of a system of linear neutral differential equations with many delays in the formwhere L , M i , N i ∈ C N ×N (i = 1, 2, . . . , d) are constant complex matrices, τ i > 0 (i = 1, 2, . . . , d) are constant delays and y(t) = (y 1 (t), y 2 (t) . . . y N (t)) T is an unknown vector-valued function for t > 0. We first establish a new result for the distribution of the roots of its characteristic function, next we obtain a sufficient condition for its asymptotic stability and then we investigate the corresponding numerical stability of linear multistep methods applied to such systems. One numerical example is given to testify our numerical analysis.
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