Nicola Guglielmi

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.

Implementing radau IIA methods for stiff delay differential equations

Abstract This article discusses the numerical solution of a general class of delay differential equations, including stiff problems, differential-algebraic delay equations, and neutral problems. The delays can be state dependent, and they are allowed to become small and vanish during the integration. Difficulties encountered in the implementation of implicit Runge–Kutta methods are explained, and it is shown how they can be overcome. The performance of the resulting code – RADAR5 – is illustrated on several examples, and it is compared to existing programs.

Asymptotic stability properties of T-methods for the pantographs equation

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.

Computing breaking points in implicit delay differential equations

Systems of implicit delay differential equations, including state-dependent problems, neutral and differential-algebraic equations, singularly perturbed problems, and small or vanishing delays are considered. The numerical integration of such problems is very sensitive to jump discontinuities in the solution or in its derivatives (so-called breaking points). In this article we discuss a new strategy – peculiar to implicit schemes – that allows codes to detect automatically and then to compute very accurately those breaking points which have to be inserted into the mesh to guarantee the required accuracy. In particular for state-dependent delays, where breaking points are not known in advance, this treatment leads to a significant improvement in accuracy. As a theoretical result we obtain a general convergence theorem which was missing in the literature (see Bellen and Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003). Furthermore, as a useful by-product, we design strategies that are able to detect points of non-uniqueness or non-existence of the solution so that the code can terminate when such a situation occurs. A new version of the code RADAR5 together with drivers for some real-life problems is available on the homepages of the authors.

On the asymptotic properties of a family of matrices

In this paper we consider bounded families F of complex n n-matrices. After introducing the concept of asymptotic order, we investigate how the norm of products of matrices behaves as the number of factors goes to infinity. In the case of defective families F ,u sing the asymptotic order allows us to get a deeper knowledge of the asymptotic behaviour than just considering the so-called generalized spectral radius. With reference to the well-known finiteness conjecture for finite families, we also introduce the concepts of spectrum-maximizing product and limit spectrum-maximizing product , showing that, for finite families of 2 2matrices, defectivity is equivalent to the existence of defective such limit products.

Fast Algorithms for the Approximation of the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

The

Numerical stability of nonlinear delay differential equations of neutral type

\varepsilon

On the zero-stability of variable stepsize multistep methods: the spectral radius approach

-pseudospectral abscissa and radius of an

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

n\times n

Differential Equations for Roaming Pseudospectra: Paths to Extremal Points and Boundary Tracking

matrix are, respectively, the maximal real part and the maximal modulus of points in its