Asymptotic Analysis

By means of piecewise continuous functions which are analogues of Lyapunov's functions sufficient conditions for the asymptotic stability of almost periodic systems of impulsive differential‐difference equations are obtained.

We present a mathematically rigorous derivation of a diffusion model previously introduced by the first author to model the diffusion of charged‐particles moving in the gap between two plane parallel plates. The particles are subject to crossed electric and magnetic fields and to collisions against the surface of the solid plates. The surface collisions are supposed to be elastic. Under appropriate scaling assumptions, the particle distribution function converges to a function of the energy and of the longitudinal position coordinates only, which evolves in time according to a diffusion equation. A rigorous convergence proof is given. The proof relies on precise estimates on the trace of the distribution function at the boundary.

In this work we introduce a new mathematical model for elastic beams with a cross‐section of constant width and longitudinally variable depth, obtained from the classical three‐dimensional linear elasticity problem by using an asymptotic expansion method. We characterize the first‐ and second‐order displacements and the first‐order stress field, giving results related to existence, uniqueness and convergence for the limit model solution. Finally, we present the computations for a classical example.