S De Lillo

Collective learning modeling based on the kinetic theory of active particles.

This paper proposes a systems approach to the theory of perception and learning in populations composed of many living entities. Starting from a phenomenological description of these processes, a mathematical structure is derived which is deemed to incorporate their complexity features. The modeling is based on a generalization of kinetic theory methods where interactions are described by theoretical tools of game theory. As an application, the proposed approach is used to model the learning processes that take place in a classroom.

MODELLING EPIDEMICS AND VIRUS MUTATIONS BY METHODS OF THE MATHEMATICAL KINETIC THEORY FOR ACTIVE PARTICLES

The present study is devoted to modelling the onset and the spread of epidemics. The mathematical approach is based on the generalized kinetic theory for active particles. The modelling includes virus mutations and the role of the immune system. Moreover, the heterogeneous distribution of patients is also taken into account. The structure allows the derivation of specific models and of numerical simulations related to real systems.

The Burgers equation on the semi-infinite and finite intervals

The initial/boundary value problem on the semiline and on a finite interval, for the Burgers equations ut=uxx+2uxu, is solved, i.e. reduced, by quadratures, to a linear integral equation of Volterra type in one independent variable, which can itself be solved by quadratures if the boundary data are time independent.

Mathematical tools of the kinetic theory of active particles with some reasoning on the modelling progression and heterogeneity

The mathematical approach proposed in this paper refers to the modelling, and related mathematical problems, of large systems of interacting entities whose microscopic state includes not only geometrical and mechanical variables (typically position and velocity), but also peculiar functions or specific activities. The number of the above entities is sufficiently large for describing the overall state of the system using a suitable probability distribution over the microscopic state. The first part of the paper is devoted to the derivation of suitable mathematical structures which can be properly used to model a variety of models in different fields of applied sciences. Then some research perspectives are analyzed, focussed on applications to biological systems.

The Burgers equation on the semiline with general boundary conditions at the origin

A technique is given to solve the initial/boundary value problem for the Burgers equation ut(x,t)=uxx(x,t)+2 ux(x,t) u(x,t) on the semiline 0≤x<∞, with the general boundary condition at the origin H[u(0,t),ux(0,t);t]=0. Here ‘‘to solve’’ means ‘‘to reduce to an equation in one variable only.’’ This equation is generally nonlinear and integrodifferent ial; it comes in several (equivalent) avatars, which contain nontrivially a free parameter, whose value can be assigned arbitrarily since the solution of the equation is independent of it. In the special case when H(y,z;t)=a(t)y+b(t)(z+y2) −F(t), which is the case relevant for most applications, the equations reduce to linear integral equations of Volterra type, which can in fact be solved by quadratures if a(t)/F(t)=c1 and b(t)/F(t)=c2 are time‐independent.

On the modeling of collective learning dynamics

This paper deals with the modeling of the collective learning dynamics of two systems of a heterogeneously distributed population. The first one evolves autonomously towards higher levels of knowledge, while the second system learns from the first one. The approach is based on the mathematical kinetic theory for active particles. The modeling focuses on applications to life sciences.

The Dirichlet-to-Neumann map for the heat equation on a moving boundary

We construct the Dirichlet-to-Neumann map for a moving initial/boundary value problem for the linear heat equation. The unknown Neumann boundary value is expressed in terms of the Dirichlet boundary value and of the initial condition through the solution of a linear Volterra integral equation of the second type. This equation involves an exponentially decaying kernel, and this leads to efficient numerical integration, as illustrated by some concrete examples.

Cauchy problems on the semiline and on a finite interval for the Eckhaus equation

The Cauchy problems on the semiline and on a finite interval, for the Eckhaus equation i psi t+ psi xx+2( mod psi mod 2)x psi + mod psi mod 4 psi =0, are solved, i.e. to a linear integrodifferential equation in one independent variable.

Forced and semiline solutions of the Burgers equation

Abstract The Burgers equation with forcing of the type F(t)δ(x) is considered. By relating this problem to semiline solutions one can solve the problem via a linear Volterra integral equation.

The Burgers equation under deterministic and stochastic forcing

Abstract The forced Burgers equation is linearized and investigated in the case when the forcing is the product of a distribution (a derivative of a dirac delta function) multiplied by an arbitrary function of time: G ( x , t ) = δ ′( x ) F ( t ). In the case when F ( t ) is a deterministic function of time, explicit solutions are obtained and the asymptotic behaviour is analyzed for different choices of F ( t ). The case when F ( t ) is random Gaussian noise and weakly correlated, is also analyzed. Explicit expressions are obtained for the statistical average of the solution and for some relevant correlation functions. In the large time and long wavelength limit, the two-time correlation function of the system, exhibits a scaling behaviour of diffusive type.