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Showing papers on "Asymptotology published in 1999"


Journal ArticleDOI
Bixiang Wang1
TL;DR: In this article, the asymptotic behavior of solutions for parabolic non-linear evolution equations in R n is studied and the existence of the global attractor in L 2 (R n ) is established.

272 citations


Journal ArticleDOI
TL;DR: In this article, the authors use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a hyperasymptotic approximation.
Abstract: Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter e which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.

261 citations


Book
28 Oct 1999
TL;DR: In this paper, a boundary value problem for the Laplacian in a multi-structure is introduced. But the boundary value is not a boundary-value problem for 3D-1D multi-structures.
Abstract: 1. Introduction to compound asymptotic expansions 2. A boundary value problem for the Laplacian in a multi-structure 3. Auxiliary facts from mathematical elasticity 4. Elastic multi-structure 5. Non-degenerate elastic multi-structure 6. Spectral analysis for 3D-1D multi-structures Bibliographical remarks Bibliography Index

161 citations


Journal ArticleDOI
TL;DR: Asymptotic formulae for solutions to the Stokes problem in domains which, outside a ball, coincide with the three-dimensional layer are derived in this paper, where the procedure of dimension reduction is employed together with estimates for miscellaneous weighted norms of the solutions.
Abstract: Asymptotic formulae are derived for solutions to the Stokes problem in domains which, outside a ball, coincide with the three-dimensional layer \( {\Bbb R}^2 \times (0,1) \). The properties of detached asymptotic terms differ in the transversal and longitudinal directions. In order to justify the asymptotic expansions the procedure of dimension reduction is employed together with estimates for miscellaneous weighted norms of the solutions.

29 citations



Journal ArticleDOI
01 Jun 1999
TL;DR: In this article, the asymptotics behavior of solutions of the Becker-Doring cluster equations is determined for cases in which coagulation dominates fragmentation, and it is shown that all non-zero solutions tend weak* to zero.
Abstract: The asymptotics behaviour of solutions of the Becker-Doring cluster equations is determined for cases in which coagulation dominates fragmentation. We show that all non-zero solutions tend weak* to zero.

14 citations


Journal ArticleDOI
TL;DR: In this article, the asymptotic behavior of the Whittaker functions Mκ, μ(z) and Wκ, ε for large modulus of the parameter κ is considered.
Abstract: . The asymptotic behavior of the Whittaker functions Mκ, μ(z) and Wκ, μ(z) for large modulus of the parameter κ is considered. Asymptotic expansions in descending powers of √ κ are derived. The κ-independent coefficients of these expansions can be calculated in a simply way making these approximations quite useful in practise. An explicit error bound for the expansion of Mκ, μ(z) is also obtained.

9 citations


Journal ArticleDOI
TL;DR: In this paper, an efficient, flexible and accurate numerical scheme for treating scattering problems involving clamped finite elastic plates is developed, which is applied to a single plate in a rigid baffle and also to a periodic array of elastic plates.

9 citations


Journal ArticleDOI
TL;DR: In this paper, an estimator for regression parameters is studied and consistency and asymptotic normality of the estimator are established both for temporal and for spatial observations in the case of increasing domain.
Abstract: Nonlinear functional errors-in-variables models with error terms satisfying mixing conditions are studied. Both variables X and y are allowed to be vector valued. An estimator for regression parameters is studied. Consistency and asymptotic normality of the estimator are established both for temporal and for spatial observations in the case of increasing domain. Properties of the estimator are studied under infill asymptotics. Simulation results are also presented.

9 citations


Journal ArticleDOI
TL;DR: In this article, the authors generalize the results of Drozdowicz, Popenda, and Migda and show that for every c G R there exists a solution of (Ε) convergent to c. The asymptotic behavior of solutions is investigated.
Abstract: (-E) Δχη = an q. We start our investigations with a useful lemma, given here without proof, which is elementary. LEMMA 1. Assume the series n l °n | is convergent and let rn = ί· Then the series n is absolutely convergent and ^Z^Li \" = η· THEOREM 1. Assume that the functions φ, ψ are continuous and the series Ση=ι n> Σ Γ = 1 are absolutely convergent. Then for every c G R there exists a solution of (Ε) convergent to c. P r o o f . Fix c e i ? . Choose a number a > 0. Let X = [c — a, c + a] χ [c — a, c + a].

9 citations


Journal ArticleDOI
TL;DR: In this article, an asymptotic approximation for solutions of a matrix differential equation with symmetric matrix coefficient, analytic on the real line, was obtained for various singularities near infinity.

Journal ArticleDOI
TL;DR: The equivalent, second equivalent and (simply) modified equations for the implicit midpoint rule are shown to be asymptotically equivalent in the sense that anAsymptotic analysis of these equations with the time step size as small parameter yields exactly the same results; for linear problems with constant coefficients, they are also equivalent to the original finite difference scheme.

Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of asymptotic decay as t \rightarrow + \infty $ of solutions of an abstract evolution equation of second order with a nonlinear and nonmonotone feedback.
Abstract: We consider the problem of asymptotic decay as t \rightarrow + \infty $ of solutions of an abstract evolution equation of second order with a nonlinear and nonmonotone feedback. Weak asymptotic stability of the global solutions is proved. This abstract result can be applied to different types of equations (wave, beam, and plate equations) and to different types of controls (interior, boundary, or pointwise controls). In particular, we significantly improve several earlier results on the asymptotic stability of the wave equation in a bounded domain with an interior or boundary control.

Journal ArticleDOI
TL;DR: In this article, the authors considered the asymptotic behavior with respect to time of the solution to the initial problem for an ordinary differential equation with a small parameter ∈ and constructed an approximation that is valid for time valuest≫∈ up to any order in ∈.
Abstract: We consider the asymptotic behavior with respect to time of the solution to the initial problem for an ordinary differential equation with a small parameter ∈. We construct an asymptotic approximation that is valid for time valuest≫∈ up to any order in ∈.

Journal ArticleDOI
TL;DR: In this paper, the periodic Ateb-functions were applied to construct single-frequency asymptotic approximations of solutions of problems for the nonlinear nonautonomous wave equation.
Abstract: Applying the periodic Ateb-functions we construct single-frequency asymptotic approximations of solutions of problems for the nonlinear nonautonomous wave equation.

Journal ArticleDOI
TL;DR: In this article, the properties of asymptotic solutions of evolution equations are studied under fairly general assumptions on the map associating a pair, with an ǫ-asymptotic formula.
Abstract: For various evolution equations for an element of a Hilbert space one uses different asymptotic methods to construct approximate solutions of these equations, which are expressed in terms of points (that are time-dependent and satisfy certain equations) in a smooth manifold and elements of a Hilbert space . In the present paper the properties of asymptotic solutions are studied under fairly general assumptions on the map associating a pair , with an asymptotic formula. An analogue of the concept of complex Maslov germ is introduced in the abstract case and its properties are studied. An analogue of the theory of Lagrangian manifolds with complex germ is discussed. The connection between the existence of an invariant complex germ and the stability of the solution of the equation for a point in the smooth manifold is investigated. The results so obtained can be used for the construction and geometric interpretation of new asymptotic solutions of evolution equations in the case when some class of asymptotic solutions is already known.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the asymptotic evaluation of an acoustic field propagating on the surface of a piezoelectric material covered with conductors and constructed uniform expansions based on the Bleistein-Handelsman theory.

Journal ArticleDOI
TL;DR: In this article, the asymptotic behavior for large time solutions of the Cauchy problem for the complex Landau-Ginzburg equation is described, where the initial data are assumed to be small in the multidimensional case and can be arbitrary in the one-dimensional case.
Abstract: The asymptotic behaviour for large time of solutions of the Cauchy problem for the complex Landau-Ginzburg equation is described. The initial data are assumed to be small in the multidimensional case (relative to the space variables), and they can be arbitrary in the one-dimensional case. In both cases the leading term is explicitly presented and an estimate for the remainder in the uniform metric is given.

Journal ArticleDOI
TL;DR: In this paper, the authors construct meromorphic functions with asymptotic power series expansion in at on an Arakelyan set having prescribed zeros and poles outside the set of approximation.
Abstract: We construct meromorphic functions with asymptotic power series expansion in at on an Arakelyan set having prescribed zeros and poles outside . We use our results to prove approximation theorems where the approximating function fulfills interpolation restrictions outside the set of approximation.

Journal ArticleDOI
TL;DR: In this paper, an asymptotic theory for a general fourth-order differential equation with large coefficients is developed. But the theory is applied with large numbers of coefficients, and the forms of the asymPTotic solutions are given under general conditions on the coefficients.
Abstract: An asymptotic theory is developed for a general fourth-order differential equation. The theory is applied with large coefficients. The forms of the asymptotic solutions are given under general conditions on the coefficients.

Journal ArticleDOI
TL;DR: In this article, the asymptotic behavior of solutions of second-order difference equations is discussed, where the authors show that the second order difference equation can be solved by a second order differential equation.

Journal ArticleDOI
TL;DR: In this paper, an approach to derive analytical approximations for the free modes of motion of an aircraft is presented, based on an asymptotic analysis, which eliminates the need for a priori assumptions regarding the characteristic motion of each mode, and is thus particularly suitable for an introductory exposition.
Abstract: Abstract A new approach to deriving analytical approximations for the free modes of motion of an aircraft is presented. Based on an asymptotic analysis, it eliminates the need for a priori assumptions regarding the characteristic motion of each mode, and is thus particularly suitable for an introductory exposition. In its basic form, it yields expressions for natural frequencies, damping factors and time constants which agree, to leading order, with well established results. At a more advanced level, it allows us to assess the validity of these results. In particular, we find that the two degree-of-freedom approximations to the short period oscillation and Dutch roll are asymptotically incorrect. This is unlikely to have a significant effect on the accuracy of the former, but is the reason for the poor damping predictions of the latter.

Journal ArticleDOI
TL;DR: In this paper, the asymptotic behaviour of widths of classes of entire functions in the uniform metric on a compact set is studied and a logarithmic formula is obtained which contains not only the natural parameters defining the growth at infinity of functions, but also the capacity of the compact set.
Abstract: The asymptotic behaviour of widths of classes of entire functions in the uniform metric on a compact set is studied. A logarithmic asymptotic formula is obtained which contains not only the natural parameters defining the growth at infinity of functions in the class under consideration, but also the capacity of the compact set. Under certain additional conditions a weak asymptotic formula is obtained. An example of the calculation of the strong asymptotics is presented.



Journal ArticleDOI
TL;DR: In this article, the relation of the coefficients of multidimensional exponential series to the asymptotic behavior of its sum was studied using the R-order of the growth.
Abstract: We study the relation of the asymptotic behavior of the coefficients of multidimensional exponential series to the asymptotic behavior of its sum by using theR-order of the growthp QR (a 1,...,a n ) in an octantQ(a 1,...,a n ).

Journal ArticleDOI
TL;DR: In this paper, asymptotic solutions of singularly perturbed homogeneous and heterogeneous systems of integro-differential Fredholm-type equations with degenerate matrix at the derivative were constructed.
Abstract: We construct asymptotic solutions of singularly perturbed homogeneous and heterogeneous systems of integro-differential Fredholm-type equations with degenerate matrix at the derivative.

Journal ArticleDOI
TL;DR: In this article, an asymptotic expansion is constructed and substantiated for the boundary value problem for the two-dimensional elliptic system of Dirac equations with rapidly oscillating coefficients, which holds uniformly with respect to the complex variable and the two real variables.
Abstract: An asymptotic expansion is constructed and substantiated for the solution of the boundary-value problem for the two-dimensional elliptic system of Dirac equations with rapidly oscillating coefficients, which holds uniformly with respect to the complex variable and the two real variables.

Journal Article
TL;DR: In this article, a transition function Pt(x,Γ), t ≥ 0, x ∈ E, Γ ∈ B(E), is a mapping from [0,+∞)×E × B(e) into [0 1] such that Pt(ex, ·)
Abstract: 1. Invariant measures for dynamical systems Let (E, ρ) be a separable, complete metric space, B(E) the σ-field of its Borel subsets. The space of all probability measures on B(E), equipped with the metric topology of weak convergence will be denoted by P1(E) or shortly P1. The space P1 is separable and complete, see [B]. A transition function Pt(x,Γ), t ≥ 0, x ∈ E, Γ ∈ B(E), is a mapping from [0,+∞)×E × B(E) into [0, 1] such that: i) Pt(x, ·) ∈ P1, for all t ≥ 0 and x ∈ E, ii) Pt(·,Γ) is B(E)-measurable for all t ≥ 0 and Γ ∈ B(E), iii) For all s, t ≥ 0, x ∈ E and Γ ∈ B(E),

Journal ArticleDOI
TL;DR: Theorems on asymptotic behaviour of a general intrgral transform of functions and distributions are proved in this article. And the concept of pseudo-asymptotic expansion (p.a.e) is introduced and a characterization is given.
Abstract: Theorems on asymptotic behaviour of a general intrgral transform of functions and distributions are proved. The concept of pseudo-asymptotic expansion (p.a.e) is introduced and a characterization is given. The p.a.e of the general inetgral transform is investigated.