The Borel Transform and Its Use in the Summation of Asymptotic Expansions

The solution of connection problems on the real line (the x axis) often give asymptotic expansions which are either even or odd. This gives rise to 'identically zero' expansions, that is, an asymptotic expansion in which all terms are identically zero at the origin. We show that the Borel transform of these problems have solutions that provide integral representations of the solution. The evaluation of these integrals, as x →0, allows us to compute the exponentially small term that these 'identically zero' expansions represent.     

Innovative solution of a 2D elastic transmission problem¶

This article is concerned with a boundary-field equation approach to a class of boundary value problems exterior to a thin domain. A prototype of this kind of problems is the interaction problem with a thin elastic structure. We are interested in the asymptotic behavior of the solution when the thickness of the elastic structure approaches to zero. In particular, formal asymptotic expansions will be developed, and their rigorous justification will be considered. As will be seen, the construction of these formal expansions hinges on the solutions of a sequence of exterior Dirichlet problems, which can be treated by employing boundary element methods. On the other hand, the justification of the corresponding formal procedure requires an independence on the thickness of the thin domain for the constant in the Korn inequality. It is shown that in spite of the reduction of the dimensionality of the domain under consideration, this class of problems are, in general, not singular perturbation problems, because of appropriate interface conditions.     

一类非线性奇摄动Dirichlet边值问题的匹配解法

利用匹配渐近展开法,讨论一类形如εy″+(xn-k)(y′+ym)=0的非线性奇摄动方程的Dirichlet边值问题,并且通过对参数k的五种不同取值的分类探讨,得到了该问题必有左边界层、右边界层或内部层之一的结论(其中左、右边界层又各分为两种类型).进而给出该问题解的零次渐近展开式,推广并改进了已有的结果.     

On qualitative properties of differential-algebraic equations

Linear systems of ordinary differential equations with identically degenerate coefficient matrix before the derivative of the unknown vector function are considered. The structure of general solutions and the notion of singular point of such systems are discussed. From the comparison of the properties of the “perturbed” and original problems, a sufficient criterion for the Lyapunov asymptotic stability of the zero solution is obtained.     

Asymptotic analysis of a singular Sturm-Liouville boundary value problem

Asymptotic expansions are given for the eigenvalues λ_n and eigenfunctions u_n of the following singular Sturm-Liouville problem with indefinite weight: $$\begin{gathered} - ((1 - x^2 )u'(x))' = \lambda xu(x) on ( - 1,1), \hfill \\ lim_{| x | \to 1} u(x) finite \hfill \\ \end{gathered} $$ This eigenvalue problem arises if one separates variables in a partial differential equation which describes electron scattering in a one-dimensional slab configuration. Asymptotic expansions of the normalization constants of the eigenfunctions are also given. The constants in these asymptotic expansions involve complete elliptic integrals. The asymptotic results are compared with the results of numerical calculations.     

Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter

In previous papers [6-8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not considered in Olver"s theory [Olver, 1974]. In this paper we go one step forward and consider linear differential equations of the third order: $y"+a\Lambda^2 y"+b\Lambda^3y=f(x)y"+g(x)y$, with $a,b\in\mathbb{C}$ fixed, $f"$ and $g$ continuous, and $\Lambda$ a large positive parameter. We propose two different techniques to handle the problem: (i) a generalization of Olver"s method and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter. As an application of the theory, we obtain new convergent and asymptotic expansions of the Pearcey integral $P(x,y)$ for large $|x|$.     

Numerical methods and asymptotic error expansions for the Emden-Fowler equations

In the present paper we analyse a numerical method for computing the solution of some boundary-value problems for the Emden-Fowler equations. The differential equations are discretized by a finite-difference method and we derive asymptotic expansions for the discretization error. Based on these asymptotic expansions, we use an extrapolation algorithm to accelerate the convergence of the numerical method.     

Homogenization of a Thick Plate Model with Inclusions or Openings Periodically Distributed in the Thickness

AMS (MOS): 65 M-N

Using Hencky-Mindlin thermoelastic linear model for plates with a large number of small identical inclusions or openings in their thickness, we obtain equilibrium equations in which coefficients depend on x and periodically on x/ε where ε is the diameter of the cell of the periodic structure.

Formal asymptotic expansions give “homogenized” equations with coefficients independant of ε corresponding to an equivalent homogeneous plate.

Solution of these homogenized equations is proved to be (in a weak sense) the limit, when ε tends to zero, of original equations solution.

Moreover this result leads to a method giving explicit computation of thermal and elastic coefficients of the equivalent homogeneous plate in the case of right cylindrical openings. Numerical results for different shapes of openings are given.     

Stability criteria of the Bendixon type

Summary In this paper we consider a system of two first order differential equations {x′=P(x,y). y′=Q(x,y)}. We usually assume that ∂P/∂x+∂Q/∂y vanish identically in a certain region. A number of conditions are then given to insure boundedness of solutions or asymptotic stability of the zero solution. Entrata in Redazione il 5 gennaio 1972.     

Extension of Edgeworth type expansion of the distribution of the sums of I.I.D. random variables in non-regular cases

The asymptotic expansions of the distributions of the sums of independent identically distributed random variables are given by Edgeworth type expansions when moments do not necessarily exist, but when the density can be approximated by rational functions. Supported in part by the Sakkokai Foundation.     

The asymptotic approximations to linear weakly singular Volterra integral equations via Laplace transform

Numerical Algorithms - In this paper, the asymptotic expansions for the solution about zero and infinity are formulated via Laplace transform for linear Volterra integral equation with weakly...     

Asymptotic Expansions of Large Deviations for Sums of Nonidentically Distributed Random Variables

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in Cramer zones and Linnik power zones for the distribution function of sums of independent nonidentically distributed random variables (r.v.). In this scheme of summation of r.v., the results are obtained first by mainly using the general lemma on large deviations considering asymptotic expansions for an arbitrary r.v. with regular behaviour of its cumulants [11]. Asymptotic expansions in the Cramer zone for the distribution function of sums of identically distributed r.v. were investigated in the works [1,2]. Note that asymptotic expansions for large deviations were first obtained in the probability theory by J. Kubilius [3].     

Eliminating Indeterminacy in Singularly Perturbed Boundary Value Problems with Translation Invariant Potentials

Solutions exhibiting an internal layer structure are constructed for a class of nonlinear singularly perturbed boundary value problems with translation invariant potentials. For these problems, a routine application of the method of matched asymptotic expansions fails to determine the locations of the internal layer positions. To overcome this difficulty, we present an analytical method that is motivated by the work of Kath, Knessl and Matkowsky [4]. To construct a solution having n internal layers, we first linearize the boundary value problem about the composite expansion provided by the method of matched asymptotic expansions. The eigenvalue problem associated with the homogeneous form of this linearization is shown to have n exponentially small eigenvalues. The condition that the solution to the linearized problem has no component in the subspace spanned by the eigenfunctions corresponding to these exponentially small eigenvalues determines the internal layer positions. These “near” solvability conditions yield algebraic equations for the internal layer positions, which are analyzed for various classes of nonlinearities.     

Matched Asymptotic Expansions in the Two-body Problem with Quick Loss of Mass

In studying models for the two-body problem with quick lossof mass a boundary layer problem arises for a third-order systemof non-linear ordinary differential equations. The models areidentified by a real parameter n with n ? 1. It turns out thatfor n = 1 asymptotic approximations of the solutions can beobtained by applying the method of matched asymptotic expansionsaccouonding to Vasil'eva or a multiple time scales method developedby O'Malley. For n> 1 these methods break down and it isshown that this is due to the occurrence of "unexpected" orderfunctions in the asymptotic expansions. The expansions for n> 1 are obtained by constructing an inner and outer expansionof the solution and matching these by the process of takingintermediate limits. The asymptotic validity of the matched expansions is provedby using an iteration technique; the proof is constructive sothat it provides us at the same time with an alternative wayof constructing approximations without using a matching technique.     

具非线性边界条件的泛函微分方程边值问题奇摄动(英文)

研究一类具非线性边界条件的泛函微分方程边值问题εx″( t) =f ( t,x( t) ,x( t-τ) ,x′( t) ,ε) ,　 t∈ ( 0 ,1 ) ,x( t) =φ( t,ε) ,　 t∈ [-τ,0 ],　 h( x( 1 ) ,x′( 1 ) ,ε) =A(ε) .我们利用微分不等式理论证明了边值问题解的存在性 ,并给出了解的一致有效渐近展开式     

Strong Disorder for a Certain Class of Directed Polymers in a Random Environment

We study a model of directed polymers in a random environment with a positive recurrent Markov chain, taking values in a countable space Σ. The random environment is a family () of independent and identically distributed real-valued variables. The asymptotic behaviour of the normalized partition function is characterized: when the common law of the g(·,·) is infinitely divisible and the Markov chain is exponentially recurrent we prove that the normalized partition function converges exponentially fast towards zero at all temperatures.     

关于一类非线性微分方程组的边值问题的渐近解（Ⅰ）

应用新的方法 ,研究一类非线性微分方程组　　u″ =v,εv″ f(x ,u ,u′)v′ -g(x ,u ,u′)v =0　　 (0 <ε 1) ,的边值问题的解的渐近性质·　构造出解的渐近展开式 ,和估计了余项 ,改进并拓广了前人的工作·     

Iterative methods for ill-posed problems and semiconvergent sequences

Iterative schemes, such as LSQR and RRGMRES, are among the most efficient methods for the solution of large-scale ill-posed problems. The iterates generated by these methods form semiconvergent sequences. A meaningful approximation of the desired solution of an ill-posed problem often can be obtained by choosing a suitable member of this sequence. However, it is not always a simple matter to decide which member to choose. Semiconvergent sequences also arise when approximating integrals by asymptotic expansions, and considerable experience and analysis of how to choose a suitable member of a semiconvergent sequence in this context are available. The present note explores how the guidelines developed within the context of asymptotic expansions can be applied to iterative methods for ill-posed problems.     

一类非线性奇摄动问题的匹配解法

利用匹配渐近展开法,讨论了一类非线性奇摄动问题的解,得出了奇摄动边值问题的零次渐近展开式.     

Asymptotic stability in singular perturbation problems. II: Problems having matched asymptotic expansion solutions

The stability of systems of ordinary differential equations of the form

$\text{dx}\text{dt}\text{= f(t, x, y, ?), ?}\text{dy}\text{dt}\text{= g(t, x, y, ?)}$

, where ? is a real parameter near zero, is studied. It is shown that if the reduced problem

$\text{dx}\text{dt}\text{= f(t, x, y, 0), 0 = g(t, x, y, 0)}$

, is stable, and certain other conditions which ensure that the method of matched asymptotic expansions can be used to construct solutions are satisfied, then the full problem is asymptotically stable as t → ∞, and a domain of stability is determined which is independent of ?. Moreover, under certain additional conditions, it is shown that the solutions of the perturbed problem have limits as t → ∞. In this case, it is shown how these limits can be calculated directly from the equations

$\text{f(∞, x, y, ?) = 0 g(∞, x, y, ?) = 0}$

as expansions in powers of ?.