Stefano Maset

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

TRACE-DDE: a Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations

\tau >0

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

,

Stability of Runge-Kutta methods for linear delay differential equations

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

Attraction between like-charged surfaces induced by orientational ordering of divalent rigid rod-like counterions : theory and simulations

and

Recent trends in the numerical solution of retarded functional differential equations

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

Approximation of Eigenvalues of Evolution Operators for Linear Retarded Functional Differential Equations

, 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).

RUNGE-KUTTA METHODS FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS

In the recent years the authors developed numerical schemes to detect the stability properties of different classes of systems involving delayed terms. The base of all methods is the use of pseudospectral differentiation techniques in order to get numerical approximations of the relevant characteristic eigenvalues. This chapter is aimed to present the freely available Matlab package TRACE-DDE devoted to the computation of characteristic roots and stability charts of linear autonomous systems of delay differential equations with discrete and distributed delays and to resume the main features of the underlying pseudospectral approach.

Stability of linear delay differential equations: a numerical approach with MATLAB

We observed monoclonal antibody mediated coalescence of negatively charged giant unilamellar phospholipid vesicles upon close approach of the vesicles. This feature is described, using a mean field density functional theory and Monte Carlo simulations, as that of two interacting flat electrical double layers. Antibodies are considered as spherical counterions of finite dimensions with two equal effective charges spatially separated by a fixed distance l inside it. We calculate the equilibrium configuration of the system by minimizing the free energy. The results obtained by solving the integrodifferential equation and by performing the Monte Carlo simulation are in excellent agreement. For high enough charge densities of the interacting surfaces and large enough l, we obtain within a mean field approach an attractive interaction between like-charged surfaces originating from orientational ordering of quadrupolar counterions. As expected, the interaction between surfaces turns repulsive as the distance between charges is reduced.

Unconditional stability of explicit exponential Runge-Kutta methods for semi-linear ordinary differential equations

Summary. This paper investigates the stability of Runge-Kutta methods when they are applied to the complex linear scalar delay differential equation