Asymptotic Analysis - Volume 102, issue 1-2

Authors: Fikioris, George | Andrianesis, Panagiotis

Article Type: Research Article

Abstract: This paper deals with two trigonometric sums that are pervasive in the literature dealing with the Gibbs phenomenon. In particular, the two sums often serve as test cases for methods – such as the method of Fejér averaging – that aim to overcome the Gibbs phenomenon. Each of the two is the partial sum of a convergent infinite series with a discontinuous limit function. Our starting points are some recently-published results, both exact and asymptotic, for the two sums. For the case of a large number of terms, we proceed from those results to develop simple and revealing asymptotic formulas …for the two sums and, also, for their Fejér averages. These formulas break down as we approach the point of discontinuity, so we further develop similar formulas that are appropriate near the discontinuity point. We repeat all these tasks for a third sum whose corresponding infinite series exhibits a logarithmic singularity. Our asymptotic results, which view the three sums from a new perspective, illuminate many aspects of the Gibbs phenomenon and the Fejér averaging method. As an illustrative example of the applicability of our formulas, we exploit their properties to develop a convergence acceleration method. We then use this method to accelerate some (more complicated) sums that exhibit logarithmic singularities, including a sum that arises in several physics applications. Show more

Keywords: Fourier series, Gibbs phenomenon, Fejér averaging, trigonometric sums

DOI: 10.3233/ASY-171408

Citation: Asymptotic Analysis, vol. 102, no. 1-2, pp. 1-19, 2017

Authors: Kozlov, Vladimir | Radosavljevic, Sonja | Wennergren, Uno

Article Type: Research Article

Abstract: Population growth is governed by many external and internal factors. In order to study their effects on population dynamics, we develop an age-structured time-dependent population model with logistic-type nonlinearity. We prove existence of a unique nonnegative bounded solution. Our main concern is to study asymptotic behavior of a solution in the general case, and especially for a periodic environment. We use the method of lower and upper solutions known in the theory of integral equations to formulate lower and upper boundaries of population density. In the periodic case, we discover a connection between the period of oscillation and its effect …on population growth. Show more

Keywords: Age-structure, time-variability, density-dependency, asymptotic behavior of solution

DOI: 10.3233/ASY-171409

Citation: Asymptotic Analysis, vol. 102, no. 1-2, pp. 21-54, 2017

Authors: Kang, Junjun | Tang, Yanbin

Article Type: Research Article

Abstract: In this paper we consider the asymptotical behavior of the density function of one dimensional nonsymmetric layered stable processes via Lévy–Khinchin exponent. For the corresponding parabolic partial integro-differential equation u t − L [ u ] = 0 generated by the Lévy operator L , the infinitesimal generator of the layered stable process, we first develop a new analytical method to instead of the usual probability method, then we give the long-time and the short-time asymptotical behavior of the solutions to the corresponding Cauchy problem of the equation u t − …L [ u ] = 0 respectively, and these imply the asymptotical behavior of transition density. Finally we localize the parabolic partial integro-differential equation to the bounded domains and give the error estimates due to the localization. Show more

Keywords: Transition density, Lévy–Khinchin exponent, layered stable process, asymptotical behavior, Gårding’s inequality

DOI: 10.3233/ASY-171410

Citation: Asymptotic Analysis, vol. 102, no. 1-2, pp. 55-70, 2017

Authors: Yan, Guan | Miara, Bernadette

Article Type: Research Article

Abstract: In this paper we properly justify the modeling of a thin piezoelectric shallow shell in unilateral contact with a rigid plane. Starting from the three-dimensional nonlinear Signorini problem, we establish the convergence of the displacement field and of the electric potential as the thickness of the shell goes to zero. More precisely we obtain that the transverse mechanical displacement field coupled with the in-plane components solve an obstacle problem described new piezoelectric characteristics. We also investigate the very popular case of cubic crystals and show that, for two-dimensional shallow shells, the coupling piezoelectric effect disappears.

Keywords: Signorini problem, obstacle problem, asymptotic analysis, shallow shell

DOI: 10.3233/ASY-171411

Citation: Asymptotic Analysis, vol. 102, no. 1-2, pp. 71-97, 2017

Authors: Carles, Rémi | Nouri, Anne

Article Type: Research Article

Abstract: Solutions to a singular one-dimensional Vlasov equation are obtained as the semiclassical limit of the Wigner transform associated to a logarithmic Schrödinger equation. Two frameworks are considered, regarding in particular the initial position density: Gaussian initial density, or smooth initial density away from vacuum. For Gaussian initial densities, the analysis also yields global solutions to the isothermal Euler system that do not enter the frame of regular solutions to hyperbolic systems by P.D. Lax.

Keywords: Plasmas, logarithmic Schrödinger equation, Vlasov equation, isothermal Euler system, semi-classical limit

DOI: 10.3233/ASY-171412

Citation: Asymptotic Analysis, vol. 102, no. 1-2, pp. 99-117, 2017