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Showing papers on "Asymptotology published in 1998"


Book
01 Jan 1998
TL;DR: The asymptotic approaches in nonlinear dynamics is one of the literary work in this world in suitable to be reading material and it will show the amazing benefits of reading a book.
Abstract: Now, we come to offer you the right catalogues of book to open. asymptotic approaches in nonlinear dynamics is one of the literary work in this world in suitable to be reading material. That's not only this book gives reference, but also it will show you the amazing benefits of reading a book. Developing your countless minds is needed; moreover you are kind of people with great curiosity. So, the book is very appropriate for you.

73 citations


Journal ArticleDOI
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


Journal Article
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


Journal ArticleDOI
TL;DR: In this paper, the complete asymptotic expansion for the Kantorovich polynomials Kn as n→∞ is presented in a form convenient for applications, where all coefficients of n−k (k=1,2,...) are calculated explicitly in terms of Stirling numbers of the first and second kind.
Abstract: We present the complete asymptotic expansion for the Kantorovich polynomials Kn as n→∞. The result is in a form convenient for applications. All coefficients of n−k (k=1,2,...) are calculated explicitly in terms of Stirling numbers of the first and second kind.

28 citations


Journal ArticleDOI
TL;DR: Systematic methods are given for simplification of the normal form of a perturbed dynamical system near a rest point which can be rigorously justified as a universal unfolding with respect to asymptotic equivalence.

20 citations


Journal Article
TL;DR: In this article, the authors studied the asymptotic behavior of solutions to the Korteweg-deVries-Burgers equation in the case when the initial data has different scaling factors at different scales.
Abstract: In this work we study the asymptotic behaviour of solutions to the Korteweg--deVries--Burgers equation in the case when the initial data has different asymptotic limits at $\pm\infty $ The method used is the one developed by Kawashima and Matsumura to discuss the asymptotic behaviour of travelling-wave solutions to Burgers equation

18 citations


Proceedings ArticleDOI
16 Dec 1998
TL;DR: In this article, a generalized Liapunov theorem was proposed to guarantee practical asymptotic stability for periodic solutions of time-invariant systems in terms of the Poincare map.
Abstract: We prove a generalized Liapunov theorem which guarantees practical asymptotic stability. Based on this theorem, we show that if the averaged system x/spl dot/=f/sub av/(x) corresponding to x/spl dot/=f(x,t) is globally asymptotically stable then, starting from an arbitrarily large set of initial conditions, the trajectories of x/spl dot/=f(x, t//spl epsiv/) converge uniformly to an arbitrarily small residual set around the origin when /spl epsiv/>0 is taken sufficiently small. In other words, the origin is semi-globally practically asymptotically stable. As another application of the generalized Liapunov theorem, one may recover the classical asymptotic stability result for periodic solutions of time-invariant systems x/spl dot/=f(x) in terms of the Poincare map.

17 citations


Posted Content
TL;DR: In this paper, the first three coefficients of the asymptotic expansion of Zelditch were computed and it was shown that in general, the $k$-th coefficient is a polynomial of the curvature and its derivative of weight.
Abstract: In this paper, we computed the first three coefficients of the asymptotic expansion of Zelditch We also proved that in general, the $k$-th coefficient is a polynomial of the curvature and its derivative of weight $k$

14 citations



Journal ArticleDOI
TL;DR: The problem of optimal control of solutions of an elliptic equation in a domain with a small cavity is discussed in this article, where uniform asymptotic formulae for the solutions are obtained up to an arbitrary degree of the small parameter by the method of matching asymPTotic expansions.
Abstract: The problem of optimal control of solutions of an elliptic equation in a domain with a small cavity is discussed. Uniform asymptotic formulae for the solutions are obtained up to an arbitrary degree of the small parameter by the method of matching asymptotic expansions. Other aspects of the same problem have been considered by Kapustyan.

11 citations


Journal ArticleDOI
TL;DR: In this article, regular and singular asymptotic methods are applied to one-and two-dimensional integral equations of the first kind with irregular kernels that arise in the treatment of various 2D axisymmetric and 3D problems in contact mechanics.


Journal ArticleDOI
TL;DR: In this paper, the Dirichlet problem in a two-dimensional domain with a narrow slit is studied, and the complete asymptotic expansion for the eigenvalue of the perturbed problem converging to a simple eigen value of the limiting problem is constructed by means of the method of matched asmptotic expansions.
Abstract: The Dirichlet problem in a two-dimensional domain with a narrow slit is studied. The width of the slit is a small parameter. The complete asymptotic expansion for the eigenvalue of the perturbed problem converging to a simple eigenvalue of the limiting problem is constructed by means of the method of matched asymptotic expansions. It is shown that the regular perturbation theory can formally be applied in a natural way up to terms of order {epsilon}{sup 2}. However, the result obtained in that way is false. The correct result can be obtained only by means of an inner asymptotic expansion.

Journal ArticleDOI
TL;DR: For the perturbed nonlinear Klein-Gordon equation, this paper constructed an asymptotic solution by using Ateb-functions for both autonomous and non-autonomous cases.
Abstract: For the perturbed nonlinear Klein-Gordon equation, we construct an asymptotic solution by using Ateb-functions. We consider autonomous and nonautonomous cases.

Journal ArticleDOI
TL;DR: In this paper, the existence and uniqueness theorem of the Riccati difference equation on the infinite index set was proved under natural assumptions, and the existence of the asymptotic expansion of the solution and computability of its coefficients were shown.
Abstract: Matrix Riccati difference equations are investigated on the infinite index set. Under natural assumptions an existence and uniqueness theorem is proven. The existence of the asymptotic expansion of the solution and computability of its coefficients are shown, provided the coefficients of the equation have such an expansion.


Journal ArticleDOI
TL;DR: In this article, the authors studied the asymptotic behavior of solutions for a class of nonlinear difference equations and gave the results about the solutions of the equations convergent to a constant.
Abstract: In this paper, we study the asymptotic behavior of solutions for a class of nonlinear difference equations and give the results about the solutions of the equations convergent to a constant. Our results generalize all conclusions obtained in [1].

Book
01 Dec 1998
TL;DR: The notion of asymptotic convergence was introduced in this paper for solving multidimensional inverse gravimetry problems on the basis of an iterative method for solution of the problem.
Abstract: Part 1 The notion of asymptotic convergence. Part 2 Asymptotic models for equations of mathematical physics: asymptotic the Helmholtz equation asymptotic models for the equation of the contact gravimetry problem asymptotic models for equations of geometric optics asymptotic models for Maxwell equation system the dual asymptotic models for the elliptic system the dual asymptotic models for the Maxwell equation system. Part 3 The iterative methods for solution of multidimensional inverse problems on the basis of the asymptotic models: on quasi-solution uniqueness of operator equations asymptotic approximation to the inverse problem solution for the Helmholtz equation conditions for applications of the iterative asymptotic method for solution of multidimensional problems the asymptotic method for solution of the inverse gravimetry problem (Part contents).

Journal ArticleDOI
TL;DR: Asymptotic representations of the solutions of boundary value problems for a second-order equation with rapidly oscillating coefficients in a domain with a small cavity (of diameter comparable with the period of oscillation) are found and substantiated.
Abstract: Asymptotic representations of the solutions of boundary-value problems for a second-order equation with rapidly oscillating coefficients in a domain with a small cavity (of diameter comparable with the period of oscillation) are found and substantiated. Dirichlet or Neumann conditions are set at the boundary of the domain. In addition to an asymptotic series of structure standard for homogenization theory there occur terms describing the boundary layer phenomenon near the opening, while the solutions of the homogenized problem and their rapidly oscillating correctors acquire singularities at the contraction point of the openings. The dimension of the domain and some other factors influence even the leading term of the asymptotic formula. Some generalizations, including ones to the system of elasticity theory, are discussed.

Journal Article
TL;DR: In this article, the authors study a generalized asymptotic equivalence between the solutions of the difference equations, and show that the equivalence can be obtained by a generalized linear equivalence.
Abstract: The purpose of this paper is the study of a generalized asymptotic equivalence between the solutions of the difference equations

Journal ArticleDOI
TL;DR: In this paper, the application of asymptotic methods of nonlinear mechanics (the Krylov-Bogolyubov-Mitropol’skii method) and the method of separation of motions in nonlinear systems for the construction of an approximate solution of a nonlinear equation that describes a nonstationary wave process was considered.
Abstract: We consider the application of asymptotic methods of nonlinear mechanics (the Krylov-Bogolyubov-Mitropol’skii method) and the method of separation of motions in nonlinear systems for the construction of an approximate solution of a nonlinear equation that describes a nonstationary wave process.

Journal ArticleDOI
TL;DR: In this article, the asymptotic properties of attainable sets of singularly-perturbed linear autonomous control systems are investigated and it is shown that if an explicitly given linear scaling operator is applied to the attainable set, the resulting sets converge (as the small parameter tends to zero).

Journal ArticleDOI
TL;DR: In this paper, the authors examined a general technique for deriving the small time asymptotic expansion of a correlation from the large frequency asymPTotic form of the associated spectrum.
Abstract: We examine a general technique for deriving the small time asymptotic expansion of a correlation from the large frequency asymptotic form of the associated spectrum ( conjugate asymptotic properties). Our analysis explicitly takes into account the form of approach of the spectrum to its asymptotic limit (i.e. asymptotic convergence), and the resulting impact on the correlation asymptotic expansion. We fully evaluate the two lowest-order terms in the small time asymptotic expansion of the correlation for the important special case of the large frequency asymptotic behaviour of the spectrum being a negative power of frequency. Included in our analysis is a determination of sufficient conditions on the rapidity of approach of the spectrum towards its asymptotic form (i.e. convergence rate), for the derived correlation asymptotic approximation to be accurate to second order. We comment on how small time must be for our correlation asymptotic approximations to be valid. To motivate this analysis we propose circumstances under which these results could be of utility in physics.


Journal ArticleDOI
01 Jan 1998-Analysis
TL;DR: In this paper, a simplified asymptotic expansion for large positive values of s is constructed for the integral transform, where the modified Bessel function of the third kind of purely imaginary order is defined.
Abstract: A simplified asymptotic expansion valid for large positive values of s is constructed for the integral transform ^dx m = Kis{x)f{x) ^ where Ki,{x) denotes the modified Bessel function of the third kind of purely imaginary Order. The expansion applies to functions f{x) that are analytic in some sector containing the half plane Re{x) > 0 and are exponentially damped as a: —̂ oo in this half plane.

Journal ArticleDOI
TL;DR: In this article, conditions of global asymptotic stability of solutions of stochastic functional-differential equations with Poisson switchings were obtained for solutions of functional differential equations.
Abstract: We obtain conditions of global asymptotic stability of solutions of stochastic functional-differential equations with Poisson switchings.

Journal ArticleDOI
TL;DR: In this article, the asymptotic expansions for the distribution functions of Pickands-type estimators in extreme statistics are obtained, and several useful results on regular variation and intermediate order statistics are presented.
Abstract: The asymptotic expansions for the distribution functions of Pickands-type estimators in extreme statistics are obtained. In addition, several useful results on regular variation and intermediate order statistics are presented.

01 Jan 1998
TL;DR: In this article, it was shown that transport in high contrast conductive media has a discrete behavior and that the transport problem has an asymptotic, resistor-independant-capacitor network approximation.
Abstract: We show that transport in high contrast conductive media has a discrete behavior. In the asymptotic limit of infinitely high contrast, the effective impedance and the magnetic field in such media are given by discrete min-max variational principles. Furthermore, we show that the transport problem has an asymptotic, resistor-inductor-capacitor network approximation. We use new variational formulations of the effective impedance of the media, and we assess the accuracy of the asymptotic approximation by numerical computations.

Journal ArticleDOI
TL;DR: In this paper, a method for the construction of complete asymptotic expansions for eigenvalues and eigenfunctions of spectral boundary-value problems for differential equations with rapidly varying coefficients in the case of multiple spectra of the averaged problem is presented.
Abstract: We suggest a method for the construction of complete asymptotic expansions for eigenvalues and eigen-functions of spectral boundary-value problems for differential equations with rapidly varying coefficients in the case of multiple spectra of the averaged problem. The effect of splitting of multiple eigen-values is illustrated by an example of a special fourth-order problem.

Journal ArticleDOI
TL;DR: In this article, asymptotic theory and appropriate symbolic computer code are developed to compute the asymPTotic expansion of the solution of an n-th order ODE with a double eigenvalue.
Abstract: This paper is part of a series of papers in which the asymptotic theory and appropriate symbolic computer code are developed to compute the asymptotic expansion of the solution of an n-th order ordinary differential equation. The paper examines the situation when the matrix that appears in the Levinson expansion has a double eigenvalue. Application is made to a fourth-order ODE with known special function solution.