No abstract

Many problems of growing interest in science, engineering, biology, and medicine are modeled with systems of differential equations involving delay terms. In general, the presence of the delay in a model increases its reliability in describing the relevant real phenomena and predicting its behavior. Besides, the introduction of history in the evolution law of a system also augments its complexity since, opposite to Ordinary Differential Equations (ODEs), Delay Differential Equations (DDEs) represent infinite dimensional dynamical systems. Thus their time integration and the study of their stability properties require much more effort, together with efficient numerical methods. Since the introduction of the delay terms in the differential equations may drastically change the system dynamics, inducing dangerous instability and loss of performance as well as improving stability, analyzing the asymptotic stability of either an equilibrium or a periodic solution of nonlinear DDEs is a crucial requirement. Several monographs have been written on this subject and the theory is well developed. By the Principle of Linearized Stability, the stability questions can be reduced to the analysis of linear(ized) DDEs. In the literature, a great number of analytical, geometrical, and numerical techniques have been proposed to answer such questions. Part of these techniques aim at analyzing the distribution in the complex plane of the eigenvalues of certain infinite dimensional linear operators, in particular the solution operators associated to the linear(ized) problem and their infinitesimal generator. This monograph does not aim to be a survey, but presents the authors' recent work on the numerical methods for the stability analysis of the zero solution of linear DDEs, which consist in applying pseudospectral techniques to discretize either the solution operator or the infinitesimal generator. The eigenvalues of the resulting matrices are then used to approximate the exact spectra. The purpose of the book is to provide a complete and self-contained treatment, which includes the basic underlying mathematics and numerics, examples from applications and, above all, MATLAB programs implementing the proposed algorithms. MATLAB is a high-level language and interactive environment, which is nowadays well developed and widely used for a variety of mathematical problems arising from vii

We consider the numerical solution of boundary value problems for general neutral functional differential equations. The problems are restated in an abstract form and, then, a general discretization of the abstract form is introduced and a convergence analysis of this discretization is developed

No abstract

Like-charged macroions in aqueous electrolyte solution can attract each other because of the presence of inter- and/or intramolecular correlations. Poisson-Boltzmann theory is able to predict attractive interactions if the spatially extended structure (which reflects the presence of intramolecular correlations) of the mobile ions in the electrolyte is accounted for. We demonstrate this for the case of divalent, mobile ions where each ion consists of two individual charges separated by a fixed distance. Variational theory applied to this symmetric 2:2 electrolyte of rodlike ions leads to an integro-differential equation, valid for arbitrary rod length. Numerical solutions reveal the existence of a critical rod length above which electrostatic attraction starts to emerge. This electrostatic attraction is distinct from nonelectrostatic depletion forces. Analysis of the orientational distribution functions suggests a bridging mechanism of the rodlike ions to hold the two macroions together. For sufficiently large rod length, we also observe "overcharging", that is, an over-compensation of the macroion charges by the diffuse layer of mobile rodlike ions. Our results emphasize the importance of the often rodlike internal structure that condensing agents such as polyamines, peptides, or polymer segments exhibit. The results were compared with Monte Carlo simulations.

No abstract

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.

No abstract