Alfredo Bellen

Bio: Alfredo Bellen is an academic researcher from University of Trieste. The author has contributed to research in topics: Delay differential equation & Differential equation. The author has an hindex of 23, co-authored 42 publications receiving 2267 citations.

Numerical Methods for Delay Differential Equations

Abstract: 1. Introduction 2. Existence and regularity of solutions of DDEs 3. A review of DDE methods 4. The standard approach via continuous ODE methods 5. Continuous Runge-Kutta methods for ODEs 6. Runge-Kutta methods for DDEs 7. Local error estimation and variable stepsize 8. Stability analysis of Runge-Kutta methods for ODEs 9. Stability analysis of DDEs 10. Stability analysis of Runge-Kutta methods for DDEs

Methods for linear systems of circuit delay differential equations of neutral type

Abstract: 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.

Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems

Abstract: 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 $\tau >0$ , $L,M\in \mathbb{C}^{m\times m}$ and $\varphi \in C\left( \left[ -\tau ,0\right] ,\mathbb{C}^m\right) $ , 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).

Solving Ordinary Differential Equations

Abstract: Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Polychronization: Computation with Spikes

Abstract: We present a minimal spiking network that can polychronize, that is, exhibit reproducible time-locked but not synchronous firing patterns with millisecond precision, as in synfire braids. The network consists of cortical spiking neurons with axonal conduction delays and spike-timing-dependent plasticity (STDP); a ready-to-use MATLAB code is included. It exhibits sleeplike oscillations, gamma (40 Hz) rhythms, conversion of firing rates to spike timings, and other interesting regimes. Due to the interplay between the delays and STDP, the spiking neurons spontaneously self-organize into groups and generate patterns of stereotypical polychronous activity. To our surprise, the number of coexisting polychronous groups far exceeds the number of neurons in the network, resulting in an unprecedented memory capacity of the system. We speculate on the significance of polychrony to the theory of neuronal group selection (TNGS, neural Darwinism), cognitive neural computations, binding and gamma rhythm, mechanisms of attention, and consciousness as "attention to memories."

Analysis of the Parareal Time-Parallel Time-Integration Method

Abstract: The parareal algorithm is a method to solve time-dependent problems parallel in time: it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time. In this paper the relation of the parareal algorithm to space-time multigrid and multiple shooting methods is first briefly discussed. The focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals, and linear convergence for unbounded intervals.