Rossana Vermiglio

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

A Stable Numerical Approach for Implicit Non-Linear Neutral Delay Differential Equations

In this paper we consider implicit non-linear neutral delay differential equations to derive efficient numerical schemes with good stability properties. The basic idea is to reformulate the original problem eliminating the dependence on the derivative of the solution in the past values. Our hypothesis on the original equation allow us to study the boundedness and asymptotic stability of the true and numerical solutions by the theory of stability with respect to the forcing term.

Natural Continuous Extensions of Runge-Kutta Methods for Volterra Integral Equations of the Second-Kind and Their Applications

We consider a very general class of Runge-Kutta methods for the numerical solution of Volterra integral equations of the second kind, which includes as special cases all the more important methods which have been considered in the literature. The main purpose of this paper is to define and prove the existence of the Natural Continuous Extensions (NCEs) of Runge-Kutta methods, i.e., piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration. The particular properties required of the NCEs allow us to construct the tail approximations, which are quite efficient in terms of kernel evaluations.

Stability of solutions of delay functional integro-differential equations and their discretizations

AbstractIn this paper we study asymptotic stability and contractivity properties of solutions of a class of delay functional integro-differential equations. These results form the basis for obtaining insight into the analogous properties of numerical solutions generated by continuous Runge-Kutta or collocation methods, where these methods are applied to a suitable reformulation of the given initial-value problem.

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

This paper deals with the approximation of the eigenvalues of evolution operators for linear retarded functional differential equations through the reduction to finite dimensional operators by a pseudospectral collocation. Fundamental applications such as determination of asymptotic stability of equilibria and periodic solutions of nonlinear autonomous retarded functional differential equations follow at once. Numerical tests are provided.

On the formulation of epidemic models (an appraisal of Kermack and McKendrick)

The aim of this paper is to show that a large class of epidemic models, with both demography and non-permanent immunity incorporated in a rather general manner, can be mathematically formulated as a scalar renewal equation for the force of infection.

RUNGE-KUTTA METHODS FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS

We introduce Runge–Kutta (RK) methods for Retarded Functional Differential Equations (RFDEs). With respect to RK methods (A, b, c) for Ordinary Differential Equations the weights vector b ∈ ℝs and the coefficients matrix A ∈ ℝs×s are replaced by ℝs-valued and ℝs×s-valued polynomial functions b(·) and A(·) respectively. Such methods for RFDEs are different from Continuous RK (CRK) methods where only the weights vector is replaced by a polynomial function. We develop order conditions and construct explicit methods up to the convergence order four.

On the stability of Runge-Kutta methods for delay integral equations

SummaryWe present a class of Runge-Kutta methods for the numerical solution of a class of delay integral equations (DIEs) described by two different kernels and with a fixed delay τ. The stability properties of these methods are investigated with respect to a test equation with linear kernels depending on complex parameters. The results are then applied to collocation methods. In particular we obtain that any collocation method for DIEs, resulting from anA-stable collocation method for ODEs, with a stepsize which is submultiple of the delay τ, preserves the asymptotic stability properties of the analytic solutions.

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

1 Introduction.- 2 Local dynamics of a parabolic germ.- 3 Global theory.- 4 Numerical results.- 5 For dessert: several amusing examples.- Index.

Stability Analysis of the Gurtin-MacCamy Model

In this paper we propose a numerical scheme to investigate the stability of steady states of the nonlinear Gurtin-MacCamy system, which is a basic model in population dynamics. In fact the analysis of stability is usually performed by the study of transcendental characteristic equations that are too difficult to approach by analytical methods. The method is based on the discretization of the infinitesimal generator associated to the semigroup of the solution operator by using pseudospectral differencing techniques. The method computes the rightmost characteristic roots, and it is shown to converge with spectral accuracy behavior.