Luciano Galeone

Fractional Adams-Moulton methods

In the simulation of dynamical systems exhibiting an ultraslow decay, differential equations of fractional order have been successfully proposed. In this paper we consider the problem of numerically solving fractional differential equations by means of a generalization of k-step Adams-Moulton multistep methods. Our investigation is focused on stability properties and we determine intervals for the fractional order for which methods are at least A(@p/2)-stable. Moreover we prove the A-stable character of k-step methods for k=0 and k=1.

Convergence analysis of time-point relaxation iterates for linear systems of differential equations

We study convergence properties of time-point relaxation (TR) Runge-Kutta methods for linear systems of ordinary differential equations. TR methods are implemented by decoupling systems in Gauss-Jacobi, Gauss-Seidel and successive overrelaxation modes (continuous-time iterations) and then solving the resulting subsystems by means of continuous extensions of Runge-Kutta (CRK) methods (discretized iterations). By iterating to convergence, these methods tend to the same limit called diagonally split Runge-Kutta (DSRK) method. We prove that TR methods are equivalent to decouple in the same modes the linear algebraic system obtained by applying DSRK limit method. This issue allows us to study the convergence of TR methods by using standard principles of convergence of iterative methods for linear algebraic systems. For a particular problem regions of convergence are plotted.

Stability of the numerical solution of the periodic reaction- diffusion systems

Abstract In this paper we study the behavior of the numerical solution of nonlinear reaction-diffusion systems, with periodic in time nonlinear term, obtained via the known θ-method. In particular we are interested to the existence and asymptotic stability of the numerical periodic solutions in order to simulate the behaviour of the theoretical solution. To this end, by imposing the positivity of the numerical scheme, we can use some results about M-matrices. So, by means of particular over and upper solutions, we study some conditions for the stability and instability of the trivial solution. Finally we show when a positive numerical periodic solution exists and when it is unique and asymptotically stable.

Decay to spatially homogeneous states for the numerical solution of reaction-diffusion systems

The aim of this paper is to discuss the behaviour of the numerical solution of systems of nonlinear reaction-diffusion equations with homogeneous Neumann boundary conditions. We show that, under certain conditions, the solution obtained using known finite difference methods reproduces the behaviour of the exact solution. In particular we prove that the numerical solution decays as time increases to a spatially homogeneous vector, which is a suitably «weighted» mean value of the numerical solution itself.

Generalizzazione del metodo di Laguerre

The following paper concerns the construction of a class of methods of every high odd order for the evaluation of the roots of a polynomial. Such methods are a generalization of the Laguerre“s method and for them, as for the Laguerre“s method, we proof the convergence for real roots. The method has been proved for polynomials with complex roots and, as the Laguerre“s method, it results usually convergent.SommarioNella seguente nota costruiamo una classe di metodi di ordine dispari comunque elevato per il calcolo delle radici di un polinomioP(x) generalizzando il metodo di Laguerre e, come per questo, dimostriamo la globale convergenza per radici reali. Il metodo è stato provato per polinomi con radici complesse e, al pari di Laguerre, risulta usualmente convergente.

A GALERKIN NUMERICAL METHOD FOR A CLASS OF NONLINEAR REACTION-DIFFUSION SYSTEMS

Publisher Summary This chapter focuses on Galerkin numerical method for a class of nonlinear reaction–diffusion systems. It presents the approximate solution of a special class of nonlinear weakly coupled parabolic systems, which occur in models in chemical, ecological, epidemic, and many other biological processes. The chapter explores systems of nonlinear initial-boundary value problems. Numerical experiments have been mainly devoted to epidemic models with diffusion and oro-faecal epidemic models. The analytical methods used to study reaction–diffusion systems are not capable of describing completely the qualitative behavior of these systems. Hence, numerical simulation appears to be very helpful in giving further insight into such behavior. The chapter presents Galerkin procedures in the space variables and uses finite difference schemes for the resulting systems of nonlinear ordinary differential equations.

Un metodo iterativo non stazionario per la risoluzione di equazioni algebriche

In this paper, by means of a class of methods that generalise the Laguerre’s method, we describe a nonstationary iterative method to solve polynomial equations.We then apply this method to the matrix eigen-value problem. We have numerical appreciable results expecially for complex roots and ill-conditioned eigenvalues.SommarioIn questa nota, mediante una successione di metodi che generalizzano quello di Laguerre ([4]), costruiamo un metodo non stazionario per la risoluzione di equazioni algebriche.Applichiamo poi il metodo al calcolo degli autovalori di matrici in forma di Hessemberg.Si ottengono risultati rilevanti, in particolare per zeri complessi e per autovalori mal condizionati.