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Showing papers on "Asymptotology published in 1994"


Journal ArticleDOI
TL;DR: In this paper, the authors show how to obtain the full asymptotic expansion for solutions of integrable wave equations to all orders, as t→∞, without relying on an a priori ansatz for the form of the solution.
Abstract: The authors show how to obtain the full asymptotic expansion for solutions of integrable wave equations to all orders, as t→∞. The method is rigorous and systematic and does not rely on an a priori ansatz for the form of the solution.

76 citations


Book ChapterDOI
TL;DR: In this paper, the authors present the distributional theory of asymptotic expansions for functions of one variable, where the multidimensional expansions are studied in the central part of the book.
Abstract: The purpose of this chapter is to present the distributional theory of asymptotic expansions for functions of one variable. This chapter and the next, where the multidimensional expansions are studied, are the central part of the book.

48 citations


Journal ArticleDOI
TL;DR: In this paper, the optimal remainder terms in the well-known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one are obtained.
Abstract: Re-expansions are found for the optimal remainder terms in the well-known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one. The re-expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions, as well being considerably more accurate and providing a smooth interpretation of the Stokes phenomenon. They are also of strikingly simple form. In addition, explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained.

29 citations



Journal ArticleDOI
TL;DR: In this article, the second terms of the asymptotic expansions of Toth's formulae are extended by specifying the second term of the first terms of their expansions.
Abstract: L. Fejes Toth gave asymptotic formulae as n → ∞ for the distance between a smooth convex disc and its best approximating inscribed or circumscribed polygons with at most n vertices, where the distance is in the sense of the symmetric difference metric. In this paper these formulae are extended by specifying the second terms of the asymptotic expansions. Tools are from affine differential geometry.

19 citations


Journal ArticleDOI
TL;DR: In this article, the existence of solutions of the Kadomtsev-Petviashvili equation whose temporal asymptotics can be represented as a superposition of curved solitons was proved.
Abstract: We prove the existence of solutions of the Kadomtsev—Petviashvili equation whose temporal asymptotics can be represented as a superposition of curved solitons.

18 citations


Book ChapterDOI
TL;DR: Asymptotic formulas are obtained for the partition function of the generalized multiclass Engset model which has applications to ISDN design and to current I/O computer architectures and a good accuracy is obtained even when these numbers are of order 10.
Abstract: Asymptotic formulas are obtained for the partition function of the generalized multiclass Engset model which has applications to ISDN design and to current I/O computer architectures. The derivation of the asymptotic expansion is based on an integral representation of the partition function in complex space through its generating function and evaluation of the integral by the saddle point method. The generating function is also used to derive an efficient exact recursive algorithm. Numerical results show that the accuracy of the asymptotic approximation increases when the number of sources in all classes and the number of servers grow simultaneously. Moreover, a good accuracy is obtained even when these numbers are of order 10.

17 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that for large n, the coefficients an,i, and an>2 can be expanded in asymptotic series of inverse factorials with explicit coefficients.
Abstract: In the neighborhood of an irregular singularity of rank one at infinity, a differential equation of the form dw „, x dw , N has well-known asymptotic solutions of the form oo oo n=0 n—Q in which Ai, A2, Ml? i ^2 are constants. It is proved that for large n, the coefficients an,i, and an>2 can be expanded in asymptotic series of inverse factorials with explicit coefficients.

16 citations


Journal ArticleDOI
TL;DR: In this paper, asymptotic methods of nonlinear mechanics and the method of Fokker-Planck-Kolmogorov equations were applied to the investigation of random multifrequency oscillations in systems with many degrees of freedom.
Abstract: We study applications of asymptotic methods of nonlinear mechanics and the method of Fokker-Planck-Kolmogorov equations to the investigation of random multifrequency oscillations in systems with many degrees of freedom.

15 citations


Journal ArticleDOI
TL;DR: In this paper, the modified Chapman-Enskog expansion procedure is applied to find the asymptotic solution of the Fokker-Planck equation related to Brownian motion.
Abstract: In this paper we apply the modified Chapman-Enskog expansion procedure to find the asymptotic solution of the Fokker-Planck equation related to Brownian motion. We prove that the asymptotic solution is defined by the diffusion equation and show that the difference between the exact and asymptotic solutions is of order e2 where 1/e is related to the magnitude of the collision operator.

10 citations




Book ChapterDOI
01 Jan 1994
TL;DR: In this article, the role of asymptotics in a variety of fields, most of which are related to mechanics but are not restricted to fluids, is discussed and discussed.
Abstract: This is not a review paper. It is, rather, a collection of thoughts on the role of asymptotics in a variety of fields, most of which are related to mechanics but are not restricted to fluids. The emphasis is more on the achievements of asymptotics, than on the description of the methods used to achieve these. The topics have been chosen so as to illustrate the versatility of asymptotic methods concerning the type of problems solved and the degree of sophistication required by each particular solution. No internal logic other than that illustrating this efficiency and versatility should be expected in this paper.


Book ChapterDOI
A. Jeffrey1
01 Jan 1994
TL;DR: In this paper, the concepts of linear dispersive and dissipative wave propagation are reviewed, and then extended to travelling waves characterized by nonlinear evolution equations, and the general effect of nonlinearity on the development of a wave is examined and the propagation speed of a discontinuous solution (shock) is derived.
Abstract: In Section 1 the concepts of linear dispersive and dissipative wave propagation are reviewed, and then extended to travelling waves characterized by nonlinear evolution equations. The general effect of nonlinearity on the development of a wave is examined and the propagation speed of a discontinuous solution (shock) is derived. Various travelling wave solutions are discussed and soliton solutions of the KdV equation are mentioned. Section 2 reviews different ways of finding travelling wave solutions for the KdVB equation and comments on their equivalence. The ideas of weak and strong dispersion are then defined. The notion of a far field is introduced and hyperbolicity is discussed. Hyperbolic systems and waves form the topic of Section 3, which reviews Riemann invariants and simple waves, and their generalization. Shocks, the Riemann problem and entropy conditions are introduced. Sections 4 and 5 are concerned with the asymptotic derivation of far field equations both for systems and for scalar equations. The reductive perturbation method is described in Section 4 for weakly dispersive systems, while in Section 5 the multiple scale method is introduced and used to derive both the nonlinear Schrodinger and the KdV equation from a model nonlinear dispersive equation. Two physical examples with different evolution equations are given.

Book ChapterDOI
01 Jan 1994
TL;DR: In this paper, the authors propose a simplification procedure for the electrical behavior of semiconductor devices in a nonlinear system of partial differential equations posed on domains with complicated geometries.
Abstract: In circuit simulation, device models should be as simple as possible. On the other hand, physically sound models for the electrical behaviour of semiconductor devices involve nonlinear systems of partial differential equations posed on domains with complicated geometries. Therefore simplifications have to be introduced corresponding to certain idealizing assumptions. By the use of asymptotic methods the simplification procedure can be carried out in a mathematically justifiable way.

Journal ArticleDOI
TL;DR: In this article, uniform asymptotic approximations for solutions of Mathieu's equation were derived by an application of a theory of a coalescing turning point and simple pole in the complex plane.
Abstract: Uniform asymptotic approximations are derived for solutions of Mathieu's equation —Y = {2qcos(2z) ajw, for a and q real, and z complex. These are uniformly valid for q large and a lying in the interval —2q < a < (2 — d)q, (d > 0), for all real or complex values of z. The approximations involve both elementary functions (LiouvilleGreen) and Whittaker functions. These results are derived by an application of a recent asymptotic theory of a coalescing turning point and simple pole in the complex plane. The new asymptotic approximations are then analytically continued around infinity, to derive a uniform asymptotic approximation between the characteristic exponent v and the parameters a and q. Error bounds are either included or available for all approximations.


Journal ArticleDOI
TL;DR: In this article, an extension of V-statistics called infinite order V-Statistics (IOVSs) is studied and their L 1 consistency and asymptotic normality are proved.

Book ChapterDOI
01 Jan 1994
TL;DR: The relationship between the existence of self-similar asymptotic solutions of Einstein's equations and equations of state is investigated in this paper, where it is shown that if the spacetime is selfsimilar, then the resulting equations must be of this same dimensionless form.
Abstract: The relationship between the existence of self-similar asymptotic solutions of Einstein’s equations and equations of state is investigated. For instance, imperfect fluid Bianchi models with ‘dimensionless’ equations of state are shown to have self-similar asymptotic solutions. Conversely, it is also shown that if the spacetime is self-similar, then the resulting equations of state must be of this same ‘dimensionless’ form. The conditions under which solutions are asymptotically self-similar are discussed, and it is noted that this is not a generic property of Einstein’s equations.

Journal ArticleDOI
TL;DR: In this article, the asymptotic behavior of solutions of model problems for a coupled system is studied and the stability of the steady-state solutions of the corresponding time-dependent system is discussed.



Journal ArticleDOI
TL;DR: In this article, an algorithm for constructing asymptotic expansions is presented, and properties of the coefficients of the expansion coefficients are investigated for functions of series, and moment bounds for solutions of linear stochastic differential equations.
Abstract: CONTENTS §1 Introduction §2 Algorithm for constructing asymptotic expansions Formulation of the main results §3 Properties of the coefficients of asymptotic series §4 Proof of Theorem 3 §5 Proof of Theorem 4 §6 Auxiliary lemmas 61 Asymptotic series for functions of series 62 Asymptotic properties of solutions of matrix linear differential equations 63 Strong solutions of stochastic differential equations 64 Moment bounds for solutions of linear stochastic differential equations References

Journal ArticleDOI
TL;DR: In this article, conditions under which solutions of some systems of differential equations are bounded are established and their asymptotic properties are studied under the assumption that the solutions of each system are known.
Abstract: We establish conditions under which solutions of some systems of differential equations are bounded and study their asymptotic properties.

Journal ArticleDOI
TL;DR: The asymptotic behavior of the eigenfunctions of the Schrodinger equation in the limith→0 case when the corresponding classical motion is ergodic was studied in this article.
Abstract: The asymptotic behavior of the eigenfunctions of the Schrodinger equation in the limith→0 is found in the case when the corresponding classical motion is ergodic. New equations for a self-consistent field and the plasma waves corresponding to it are derived.


Journal ArticleDOI
TL;DR: In this paper, the authors considered the nonlinear Sturm-Liouville problem with 1 ≤ p and studied the asymptotic behavior of the optimal remainder estimator.
Abstract: We consider the following nonlinear Sturm-Liouville problem where 1 ≤ p. In thir paper we shall study the asymptotic behavior of . We shall establirh an asymptotic formula with the optimal remainder estimste.

Journal ArticleDOI
TL;DR: In this paper, asymptotic properties of nonoscillating solutions of opreator-differential equations of arbitrary order are investigated, where the authors consider the non-oscillation of ODEs with arbitrary order.
Abstract: Some asymptotic properties of the nonoscillating solutions of opreator-differential equations of arbitrary order are investigated.