Recently a new parallel ODE solver implementing a “parallelism across the steps” has been proposed (Amodio and Brugnano, 1997; Brugnano and Trigiante, 1998). In the mentioned references, the attention was devoted to some essential features of the parallel method, which are already present in the case where it is used to approximate linear continuous problems. In this paper, the previous analysi...

A problem of frequent interest in analyzing nonlinear ODE models of ecological systems is the location of equilibrium states and bifurcations. Interval-Newton techniques are explored for identifying, with certainty, all equilibrium states and all codimension-1 and codimension-2 bifurcations of interest within specified model parameter intervals. The methodology is applied to a tritrophic food c...

In [36], we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of β. Using this general result, the case β = 6 is further considered here. This is the smallest even β, when the corresponding Lax pair and its relation to Painlevé II (PII) have not been known before, unlike cases β = 2 and 4. It turns out that again everything can be expressed in terms of the Ha...

This paper proposes Runge-Kutta neural networks (RKNNs) for identification of unknown dynamical systems described by ordinary differential equations (i.e., ordinary differential equation or ODE systems) with high accuracy. These networks are constructed according to the Runge-Kutta approximation method. The main attraction of the RKNNs is that they precisely estimate the changing rates of syste...

A tool for the order reduction of differential algebraic equations (DAEs) is outlined in this report. Through the use of an equation dependency analysis and nonlinear function approximation, the algebraic equations can be divided into sets that require implicit or explicit solutions. If all of the algebraic variables can be solved or approximated explicitly, the DAE becomes a set of ordinary di...

The 2-soliton solutions of the Korteweg-de Vries equation satisfy a fourth-order nonlinear ordinary differential equation which depends on two parameters λ, μ. As is well known, for each fixed choice of (λ, μ) in R, the ODE is actually a completely integrable Hamiltonian system with two degrees of freedom. Here we address the question of whether the soliton and 2-soliton solutions of KdV are th...

This paper proposes a methodology for the synthesis of nonlinear finite-dimensional time-varying output feedback controllers for systems of quasi-linear parabolic Ž . partial differential equations PDEs with time-dependent spatial domains, whose dynamics can be partitioned into slow and fast ones. Initially, a nonlinear model reduction scheme, similar to the one introduced in Christofides and D...

The goal of input-output modeling is to apply a test input to a system, analyze the results, and learn something useful from the causeeffect pair. Any automated modeling tool that takes this approach must be able to reason effectively about sensors and actuators and their interactions with the target system. Distilling qualitative information from sensor data is fairly easy, but a variety of di...

A systematic algorithm for building integrating factors of the form μ(x, y′) or μ(y, y′) for nonlinear second order ODEs is presented. When such an integrating factor exists, the scheme returns the integrating factor itself, without solving any differential equations. The scheme was implemented in Maple, in the framework of the ODEtools package and its ODE-solver. A comparison between this impl...

The paper discusses the modeling of a central solar heating plant with seasonal storage in the ground using the new object-oriented physical systems modeling language Modelica. Main emphasis is put on the hierarchical decomposition of the system model and on the re-engineering of an existing Fortran code for the ground store model. The object-oriented physical system model is compiled to a math...