Topic
Asymptotology
About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.
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TL;DR: In this article, a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called "asymptotic function", was presented.
Abstract: We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper "asymptotic function," similar to but different from J. F Colombeau's algebras ofnew generalized functions.
52 citations
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TL;DR: In this paper, it was shown that three identically-constructed pairs of expansions exist for solving the Mathieu equation: one pair in terms of trigonometrical functions, one pair on terms of Hermite functions of a real variable, and another pair on the Hermite function of an imaginary variable.
Abstract: Straight forward and systematic procedures are developed for finding asymptotic expansions of Mathieu functions and their characteristic numbers. In particular, it is shown that three identically-constructed pairs of expansions exist for Solutions to the Mathieu equation: one pair in terms of trigonometrical functions, one pair in terms of Hermite functions of a real variable, and one pair in terms of Hermite functions of an imaginary variable. By linking these different expansions together in regions where they overlap, the behaviour of Mathieu functions can be investigated over the entire ränge of the independent variable.
52 citations
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TL;DR: In this article, an asymptotic theory for a large class of Boltzmann type equations suitable to model the evolution of multicellular systems in biology with special attention to the onset of nonlinear phenomena was developed.
Abstract: This paper develops an asymptotic theory for a large class of Boltzmann type equations suitable to model the evolution of multicellular systems in biology with special attention to the onset of nonlinear phenomena. The mathematical method shows how various levels of diffusion phenomena, linear and non-linear, can be obtained by suitable asymptotic limits. The time scaling corresponding to different speeds related to cell movement and biological evolution plays a crucial role and different macroscopic equations corresponds to different scaling.
52 citations
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TL;DR: In this paper, the solitary-wave solutions of Benjamin's model were investigated for a class of equations that include Benjamin's equation, which feature conflicting contributions to dispersion from dynamic effects on the interface and surface tension.
Abstract: Benjamin recently put forward a model equation for the evolution of waves on the interface of a two-layer system of fluids in which surface tension effects are not negligible. In this case, the fluid motion η on the interface of these two fluids can be approximately described by an equation ηt + ηx + ηηx − αLηx ± βηxxx = 0, where η depends on saptial variable x and time variable t, and L = H∂x is the composition of the Hilbert transform and the spatial derivative in the direction of primary propagation, or, equivalently, L is a Fourier multiplier operator with symbol |ξ|. It is our purpose here to investigate the solitary-wave solutions of Benjamin’s model. For a class of equations that include Benjamin’s equation, which feature conflicting contributions to dispersion from dynamic effects on the interface and surface tension, we establish existence of travelling-wave solutions. This is complished by using P.L. Lions concentrated-compactness principle. Using the recently developed theory of Li and Bona, we are also able to determine rigorously the spatial asymptotics of these solutions. Department of Mathematics, The University of Texas at Austin, Austin, TX 78712. Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, Austin, TX 78712.
50 citations
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TL;DR: Degond and Raviart as mentioned in this paper provided a mathematical framework to this physical theory, by successively investigating the reduced problem (when the perturbation parameter e is set equal to zero) and the boundary layer problem.
Abstract: 187 Degond, P. and P.A. Raviart, An asymptotic analysis of the one-dimensional Vlasov-Poisson system: the Child-Langmuir law, Asymptotic Analysis 4 (1991) 187-214. 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 e is set equal to zero) and the boundary layer problem, which gives a sharp qualitative information.
50 citations