In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the behavior of solutions and to discuss the stability and the accuracy of our approximation.1991 Mathematics Subject Classification. 65M06, 35L50, 92B05, 92C17, 92C42.. IntroductionThe movement of bacteria, cells or other microorganisms under the effect of a chemical stimulus, represented by a chemoattractant, has been widely studied in mathematics in the last two decades, see [21,23,26], and numerous models involving partial differential equations have been proposed. The basic unknowns in these chemotactic models are the density of individuals and the concentrations of some chemical attractants. One of the most considered models is the Patlak-Keller-Segel system [19], where the evolution of the density of cells is described by a parabolic equation, and the concentration of a chemoattractant is generally given by a parabolic or elliptic equation, depending on the different regimes to be described and on authors' choices. The behavior of this system is quite well known now: in the one-dimensional case, the solution is always global in time, while in two and more dimensions the solutions exist globally in time or blow up according to the size of the initial data. However, a drawback of this model is that the diffusion leads to a fast dissipation or an explosive behavior, and prevents us to observe intermediate organized structures, like aggregation patterns.By contrast, models based on hyperbolic/kinetic equations for the evolution of the density of individuals, are characterized by a finite speed of propagation and have registered a growing consideration in the last few years [5-7, 15, 26]. In such models, the population is divided in compartments depending on the velocity of propagation of individuals, giving raise to kinetic type equations, either with continuous or discrete velocities.

Abstract. In this paper we propose a Godunov-based discretization of a hyperbolic system of conservation laws with discontinuous flux, modeling vehicular flow on a network. Each equation describes the density evolution of vehicles having a common path along the network. We show that the algorithm selects automatically an admissible solution at junctions, hence ad hoc external procedures (e.g., maximization of the flux via a linear programming method) usually employed in classical approaches are no needed. Since users have not to deal explicitly with vehicle dynamics at junction, the numerical code can be implemented in minutes. We perform a detailed numerical comparison with a Godunov-based scheme coming from the classical theory of traffic flow on networks which maximizes the flux at junctions.

Abstract. In this paper we introduce a computation algorithm to trace car paths on road networks, whose load evolution is modeled by conservation laws. This algorithm is composed by two parts: computation of solutions to conservation equations on each road and localization of car position resulting by interactions with waves produced on roads. Some applications and examples to describe the behavior of a driver traveling in a road network are showed. Moreover a convergence result for wave front tracking approximate solutions, with BV initial data on a single road is established.

We consider a mathematical model for fluid-dynamic flows on networks which is based on conservation laws. Road networks are considered as graphs composed by arcs that meet at some junctions. The crucial point is represented by junctions, where interactions occurr and the problem is underdetermined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and the more recent discrete velocities kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm which proceeds processing each junction. We present the algorithm and its application to some simple test cases and to portions of urban network.

Multidimensional extensions of the Bernoulli and Appell polynomials are defined generalizing the corresponding generating functions, and using the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials. Furthermore the differential equations satisfied by the corresponding 2D polynomials are derived exploiting the factorization method, introduced in [15].

We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag-Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials. The main properties of these polynomial sets are shown. In particular, the differential equations can be constructed by means of the factorization method.

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This paper is focused on continuum-discrete models for supply\ud chains. In particular, we consider the model introduced in [10], where a system\ud of conservation laws describe the evolution of the supply chain status on\ud sub-chains, while at some nodes solutions are determined by Riemann solvers.\ud Fixing the rule of flux maximization, two new Riemann Solvers are defined. We\ud study the equilibria of the resulting dynamics, moreover some numerical experiments\ud on sample supply chains are reported. We provide also a comparison,\ud both of equilibria and experiments, with the model of [15]