Alfredo Bellen

Methods for linear systems of circuit delay differential equations of neutral type

Delay differential equations (DDEs) occur in many different fields including circuit theory. Circuits which include delayed elements have become more important due to the increase in performance of VLSI systems. The two types of circuits which include elements with delay are transmission lines and partial element equivalent circuits. The solution of systems which include these circuit elements are performed with solvers similar to conventional ODE circuits simulators. Since DDE solvers are more fragile with respect to stability, we investigate the conditions for contractivity and determine sufficient conditions for the asymptotic stability of the zero solution by utilizing a suitable reformulation of the system.

Strong contractivity properties of numerical methods for ordinary and delay differential equations

Abstract In the last 20 years various stability test problems for DDE solvers have been considered. These are mainly generalizations of those used for ODEs. For the simplest autonomous case y ′( t )= λy ( t )+ μy ( t − τ ), the concepts of P- and GP-stability were introduced and a significant number of results have already been found for both classes of linear multistep methods and one-step Runge-Kutta methods. More difficult is the situation for the non-autonomous linear test equation y ′( t )= λ ( t ) y ( t )+ μ ( t ) y ( t − τ ) and for the general nonlinear case y ′( t )= f ( t , y ( t ), y ( t − τ )), which gave rise to the concepts of PN-, GPN-, RN- and GRN-stability. In particular, in order to study PN- and GPN-stability notion for ODE solvers based on the test equation with forcing term y′(t) = λ(t)y(t)+ƒ(t) , which is called AN ƒ -stability, has recently been found. This paper pursues this further and introduces the concepts of A ƒ -stability and BN ƒ -stability, which are based on the test equations y′(t)=λy(t)+ƒ(t) and y′(t)=ƒ(t,y(t),u(t)) , respectively. Some relationships are established among all these concepts of stability for ODEs and DDEs. The situation for the class of Runge–Kutta methods up to order 2 is thoroughly examined.

Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems

Summary. In this paper we present an approach for the numerical solution of delay differential equations \begin{equation} \left\{ \begin{array}{l} y^{\prime }\left( t\right) =Ly\left( t\right) +My\left( t-\tau \right) \;\;t\geq 0 y\left( t\right) =\varphi \left( t\right) \;\;-\tau \leq t\leq 0, \end{array} \right. \end{equation} where

Asymptotic stability properties of T-methods for the pantographs equation

\tau >0

Numerical stability of nonlinear delay differential equations of neutral type

,

On the Contractivity and Asymptotic Stability of Systems of Delay Differential Equations of Neutral Type

L,M\in \mathbb{C}^{m\times m}

Numerical solution of delay differential equations by uniform corrections to an implicit Runge-Kutta method

and

Attraction between negatively charged surfaces mediated by spherical counterions with quadrupolar charge distribution

\varphi \in C\left( \left[ -\tau ,0\right] ,\mathbb{C}^m\right)

Contractivity of waveform relaxation Runge-Kutta iterations and related limit methods for dissipative systems in the maximum norm

, different from the classical step-by-step method. We restate (1) as an abstract Cauchy problem and then we discretize it in a system of ordinary differential equations. The scheme of discretization is proved to be convergent. Moreover the asymptotic stability is investigated for two significant classes of asymptotically stable problems (1).

Parallel algorithms for initial-value problems for difference and differential equations

In this paper we consider asymptotic stability properties of Q-methods for the following pantograph equation: 1 y’(t) = aY(t) + by(G) + cY’@)? Q E (O,lL Y(O) = 1, where a, b, c E @. In recent years stability properties of numerical methods for this kind of equation have been studied by numerous authors who have considered meshes with fixed stepsize. In general the developed techniques give rise to non-ordinary recurrence relations. In this work, instead, we study constrained variable stepsize schemes, suggested by theoretical and computational reasons, which lead to a non-stationary difference equation. For a first insight, we focus our attention on the class of @-methods and show that asymptotic stability is obtained for 0 > l/2. Finally, some preliminary considerations are devoted to the non-neutral and non-stationary pantograph equation. o 1997 Elsevier Science B.V.