Luisa Fermo

TOWARDS THE MODELING OF VEHICULAR TRAFFIC AS A COMPLEX SYSTEM: A KINETIC THEORY APPROACH

Kinetic theory methods are applied in this paper to model the dynamics of vehicular traffic. The basic idea is to consider each vehicular-driver system as a single part, or micro-system, of a large complex system, in order to capture the heterogeneous behavior of all the micro-systems that compose the overall system. The evolution of the system is ruled by nonlinearly additive interactions described by stochastic games. A qualitative analysis for the proposed model with discrete states is developed, showing well-posedness of the related Cauchy problem for the spatially homogeneous case and for the spatially nonhomogeneous case, the latter with periodic boundary conditions. Numerical simulations are also performed, with the aim to show how the model proposed is able to reproduce empirical data and to show emerging behavior as the formation of clusters.

A Fully-Discrete-State Kinetic Theory Approach to Modeling Vehicular Traffic

This paper presents a new mathematical model of vehicular traffic, based on the methods of the generalized kinetic theory, in which the space of microscopic states (position and speed) of the vehicles is genuinely discrete. While in the recent literature discrete-velocity kinetic models of car traffic have already been successfully proposed, this is, to our knowledge, the first attempt to account for all aspects of the physical granularity of car flow within the formalism of the aforesaid mathematical theory. Thanks to a rich but handy structure, the resulting model allows one to easily implement and simulate various realistic scenarios giving rise to characteristic traffic phenomena of practical interest (e.g., queue formation due to roadwork or to a traffic light). Moreover, it is analytically tractable under quite general assumptions, whereby fundamental properties of the solutions can be rigorously proved.

On a class of integro-differential equations modeling complex systems with nonlinear interactions

Abstract This work deals with the qualitative analysis of the initial value problem for a class of large systems of interacting entities in the framework of the mathematical kinetic theory for active particles. The contents are specifically focused on the case where the system interacts with the outer environment and the entities are subject to nonlinearly additive interactions.

Heterogeneous distribution of mechanical stress in human lung: A mathematical approach to evaluate abnormal remodeling in IPF ☆

Idiopathic pulmonary fibrosis (IPF) is the most common and severe form of idiopathic interstitial pneumonia. A recently proposed pathogenic model suggests that the concurrent action of cell senescence, exposure to cigarette smoke and mechanical stress due to respiratory lung movements lead to a localized exhaustion of tissue renewal capacity with eventual alveolar loss and abnormal lung remodeling. In this study we have compared the distribution of IPF lesions, as shown by TC radiological images, with the hypothetical distribution of maximal mechanical stress obtained by a simplified mathematical model. The geometry and distribution of stress as determined by our simulation are closely similar to those demonstrated in vivo in the lungs of patients with idiopathic pulmonary fibrosis using high resolution CT scan radiological imaging. These data argue in favor of the recently proposed contribution of mechanical stress to progressive damage and remodeling of lung parenchyma in IPF. The parameters of the model can be tuned on the age of the patients.

On the mathematical theory of living systems II: The interplay between mathematics and system biology

This paper aims at showing how the so-called mathematical kinetic theory for active particles can be properly developed to propose a new system biology approach. The investigation begins with an analysis of complexity in biological systems, continues with reviewing a general methodology to reduce complexity and furnishes the mathematical tools to describe the time evolution of such systems by capturing all their features.

Bifurcation diagrams for the moments of a kinetic type model of keloid-immune system competition

The mathematical modelling of the keloid disease triggered by a virus has been recently investigated by one of the authors, Bianca (2011) [5], where it was shown that the model is able to depict the emerging behaviours which occur during the keloid formation. This paper deals with further numerical investigations of that model related to the bifurcation analysis of the measurable macroscopic variables associated to each functional subsystem. It is shown that there exists a critical value of a bifurcation parameter separating situations where the immune system controls the keloid formation from those where malignant effects are not contrasted.

A Nyström method for a boundary integral equation related to the Dirichlet problem on domains with corners

The authors consider the interior Dirichlet problem for Laplace’s equation on planar domains with corners. They provide a complete analysis of a natural method of Nyström type based on the global Gauss–Lobatto quadrature rule, in order to approximate the solution of the corresponding double layer boundary integral equation. Mellin-type integral operators are involved and, as usual, a modification of the method close to the corners is needed. A new modification is proposed and the convergence and stability of the “modified” quadrature method are proved. Some numerical tests are also included.

Emerging problems in approximation theory for the numerical solution of the nonlinear Schrödinger equation

We present some open problems pertaining to the approximation theory involved in the solution of the Nonlinear Schrodinger (NLS) equation. For this important equation, any Initial Value Problem (IVP) can be theo- retically solved by the Inverse Scattering Transform (IST) technique whose main steps involve the solution of Volterra equations with structured kernels on unbounded domains, the solution of Fredholm integral equations and the identification of coefficients and parameters of monomial-exponential sums. The aim of the paper is twofold: propose a method for solving the above mentioned problems under particular hypothesis; arise interest in the issues illustrated to achieve an effective method for solving the problem under more general assumptions

Numerical methods for Fredholm integral equations with singular right-hand sides

Fredholm integral equations with the right-hand side having singularities at the endpoints are considered. The singularities are moved into the kernel that is subsequently regularized by a suitable one-to-one map. The Nyström method is applied to the regularized equation. The convergence, stability and well conditioning of the method is proved in spaces of weighted continuous functions. The special case of the weakly singular and symmetric kernel is also investigated. Several numerical tests are included.

Parameter estimation of monomial-exponential sums in one and two variables

We propose a matrix-pencil method to identify monomial-exponential sums.In the univariate case, it is based on the QR factorization of Hankel matrices.In the bivariate case, it assures the same accuracy of the univariate case. In this paper we propose a matrix-pencil method for the numerical identification of the parameters of monomial-exponential sums in one and two variables. While in the univariate case the proposed method is a variant of that developed by the authors in a preceding paper, the bivariate case is treated for the first time here. In the bivariate case, the method we propose, easily extendible to more variables, reduces the problem to a pair of univariate problems and subsequently to the solution of a linear system. As a result, the relative errors in the univariate and in the bivariate case are almost of the same order.