Mathematical Models and Methods in Applied Sciences

We consider a system of Markov processes of finitely-many particles which exchange their energies in pairs at random times. A law of large numbers for this system means that the empirical measures of the processes may be approximated (as the number of particles increases) by the solution of a nonlinear evolution equation (the so-called McKean-Vlasov limit). This work presents two results of this type. The first one concerns the empirical processes and gives a probabilistic method for solving the nonlinear equation. The second is stated in the path scheme and extends classical results of chaos propagation by Kac (1956) and McKean (1967).

In this paper we apply the modified Chapman-Enskog expansion procedure to find the asymptotic solution of the Fokker-Planck equation related to Brownian motion. We prove that the asymptotic solution is defined by the diffusion equation and show that the difference between the exact and asymptotic solutions is of order ε² where 1/ε is related to the magnitude of the collision operator.

A BGK-type kinetic model is derived for describing the interaction of chemical substances. The ensuing equation is then solved asymptotically on certain space-time scales on which an appreciable interplay between kinetic and chemical effects, or the prevailing of one on the other, can be observed. The description of the interaction at the macroscopic level consists of a hierarchy of reaction-diffusion equations satisfied by the densities. Comparison is made with similar results previously obtained from certain phenomenological models, and illustrative examples are given.

This paper deals with a linear Schrödinger type equation in a rectangular domain with mixed Dirichlet-Neumann boundary conditions. The well-posedness of the continuous problem is proved, then a discrete problem is defined by combining a Legendre type spectral method in the first direction and a leap-frog scheme in the other one. The numerical analysis of the discretization is performed and error estimates are given. Numerical tests are presented.

When a flexible rectangular homogeneous beam, which is horizontal in equilibrium state, has an end rigidly fixed while a load is attached to the other end, the transverse vibrations of the beam are described by the elastic beam equation together with, amongst others, a dynamic boundary condition at that end of the beam to which the load is attached. In this paper we introduce various types of energy dissipation for this problem, viz. damping of Kelvin-Voigt type as well as structural damping. The resulting boundary-value problems are studied within the framework of the abstract theories of B-evolutions and fractional powers of a closed pair of operators by formulating an abstract evolution problem in the paired space X₁×X, with X a Hilbert space and X₁ continuously imbedded in X. This approach yields a unique solution in the strong sense which exhibits exponential decay as time tends to infinity for any initial displacement in X₁ and any initial velocity in X.

In this paper we prove some results about the existence of positive periodic solutions of nonlinear delay integral equations which formulate a model to explain the evolution of some infectious diseases and which have also been used as a growth equation for single species population when the birth rate varies seasonally. We use monotone methods (upper and lower solutions) to obtain our results and we compare them with other results obtained by different authors which have used topological methods to study the mentioned equation.

We prove a uniqueness result for the drift-diffusion model of semiconductor devices under weak regularity assumptions. Our proof rests on the convexity of the free energy functional and uses a new concavity argument.

We consider a mathematical model for solidification of semicrystalline polymers, describing the evolution of temperature, crystalline volume fraction, number and average size of crystals. In turn, the model couples a suitable kinetics of nonisothermal crystallization, taking into account both formation and growth of nuclei, with the thermal energy balance equation. We also present a model of secondary crystallization. The numerical approximation is performed by semiexplicit finite differences in time and finite elements in space. The fully discrete scheme amounts to solve, at any time step, a symmetric positive definite linear system preceded by an elementwise explicit computation. The computed numerical crystal structures match qualitatively the experimental ones.