Asymptotic Analysis - Volume 4, issue 3

Authors: Degond, P. | Raviart, P.A.

Article Type: Research Article

Abstract: We perform the asymptotic analysis of the one-dimensional Vlasov–Poisson system when singular boundary data are prescribed. Such a singular perturbation problem arises in the modelling of vacuum diodes under very large applied bias, and gives rise to the well-known “Child-Langmuir law”. In this paper, we provide a mathematical framework to this physical theory, by successively investigating the reduced problem (when the perturbation parameter ε is set equal to zero) and the boundary layer problem, which gives a sharp qualitative information.

DOI: 10.3233/ASY-1991-4301

Citation: Asymptotic Analysis, vol. 4, no. 3, pp. 187-214, 1991

Authors: Tovbis, A.I.

Article Type: Research Article

Abstract: Our purpose is to investigate a lateral connection problem for matrix differential equations with an irregular singularity in case of a multiple Stokes ray. We reduce this problem to constructing the so-called Stokes operators in certain functional spaces using the results of [7,9]. We also extend the recently obtained [5,10] analytical expression for Stokes matrix in case of a multiple Stokes ray.

DOI: 10.3233/ASY-1991-4302

Citation: Asymptotic Analysis, vol. 4, no. 3, pp. 215-233, 1991

Authors: Mohamed, Abderemane

Article Type: Research Article

Abstract: We give a BKW method in the semiclassical analysis of a Schrödinger operator in a magnetic field, with an electrical potential having singularities of Coulombian type. As applications, we obtain sharp estimates for the splitting in the double wells problem and for the first band in the periodic problem.

DOI: 10.3233/ASY-1991-4303

Citation: Asymptotic Analysis, vol. 4, no. 3, pp. 235-255, 1991

Authors: Figueiredo, I.M.N.

Article Type: Research Article

Abstract: We apply here the asymptotic expansion method to the nonlinear three-dimensional equations for the equilibrium of a thin elastic shell, following the work of Destuynder (1980) for the linear case. We consider two small parameters: ε that is the half-thickness of the shell, and ρ that is the ratio of ε to the lower bound of the radius of curvature of the middle surface of the shell. We show that the leading term of the asymptotic expansion is the solution of the Donnell–Mushtari–Vlasov model (cf. Sanders [1963]), if ρ is of the order ε2 . We also study the cases …ρ=O(ε2+r ), r > 0 and ρ= constant as ε→0. Show more

DOI: 10.3233/ASY-1991-4304

Citation: Asymptotic Analysis, vol. 4, no. 3, pp. 257-269, 1991

Authors: Barles, G. | Souganidis, P.E.

Article Type: Research Article

Abstract: We present a simple, purely analytic method for proving the convergence of a wide class of approximation schemes to the solution of fully non linear second-order elliptic or parabolic PDE. Roughly speaking, we prove that any monotone, stable and consistent scheme converges to the correct solution provided that there exists a comparison principle for the limiting equation. This method is based on the notion of viscosity solution of Crandall and Lions and it gives completely new results concerning the convergence of numerical schemes for stochastic differential games.

DOI: 10.3233/ASY-1991-4305

Citation: Asymptotic Analysis, vol. 4, no. 3, pp. 271-283, 1991