Complete Asymptotic Expansion Research Articles

Asymptotic expansion of the Lebesgue constant for even-order Hermite-Fejér interpolation on Chebyshev nodes

An integral representation is obtained for the Lebesgue constant associated with even-order Hermite-Fejér interpolation on Chebyshev nodes, and this result is extended to give a complete asymptotic expansion with a remainder term.

On a finite sum of cosecants appearing in various problems

Finite trigonometric sums is a challenging and often quite difficult object of study. In this paper we investigate the finite sum of cosecants ∑csc⁡(φ+aπl/n), where the summation index l runs through 1 to n−1 and φ and a are arbitrary parameters, as well as several closely related sums, such as similar sums of a series of secants, of tangents and of cotangents. These trigonometric sums appear in various problems in mathematics, physics, and a variety of related disciplines. Their particular cases were fragmentarily considered in previous works, and it was noted that even a simple particular case ∑csc⁡(πl/n) does not have a closed–form, i.e. a compact summation formula (similar sums of sines, of cosines, of tangents, of cotangents and even of secants, if they exist, possess such expressions). In this paper, we derive several alternative representations for the above–mentioned sums, study their properties, relate them to many other finite and infinite sums, obtain their complete asymptotic expansions for large n and provide accurate upper and lower bounds (e.g. the typical relative error for the upper bound is lesser than 2×10−9 for n⩾10 and lesser than 7×10−14 for n⩾50, which is much better than the bounds we could find in previous works). Our researches reveal that these sums are deeply related to several special numbers and functions, especially to the digamma function (furthermore, as a by-product, we obtain several interesting summation formulæ for the digamma function). Asymptotical studies show that these sums may have qualitatively different behaviour depending on the choice of φ and a. In particular, as n increases some of them may become sporadically large and we identify the terms of the asymptotics responsible for such a behaviour. Finally, we also provide several historical remarks related to various sums considered in the paper. We show that some results in the field either were rediscovered several times or can easily be deduced from various known formulæ, including some formulæ dating back to the XVIIIth century.

Asymptotic analysis of the nonsteady micropolar fluid flow through a system of thin pipes

In this paper, we analyze the time‐dependent flow of an incompressible micropolar fluid in a multiple‐pipe system. Motivated by the applications, we assume that the pipes have circular cross‐section and that the ratio between pipes' thickness and its length is small, denoted by the parameter . Far from the junction, the fluid exhibits different behavior depending on the magnitude of the viscosity coefficients with respect to the small parameter . Focusing on the critical case described by the strong coupling between velocity and microrotation, the complete asymptotic expansion of the solution (up to an arbitrary order) is built. To improve the accuracy of the asymptotic approximation, we introduce the boundary layer correctors near the pipes' ends and take into account the interior layer correction in the vicinity of the junction as well. The convergence is also proved via error estimates, providing the rigorous justification of the proposed effective model.

Complete Uniform Asymptotic Expansion of Finite-Range Integral Representation of Dyadic Green’s Function for Ordinary Dielectric Half-Space

Complete Uniform Asymptotic Expansion of Finite-Range Integral Representation of Dyadic Green’s Function for Ordinary Dielectric Half-Space

Asymptotic expansion for convection-dominated transport in a thin graph-like junction

We consider for a small parameter [Formula: see text] a parabolic convection–diffusion problem with Péclet number of order [Formula: see text] in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order [Formula: see text] connected through a domain (node) of diameter [Formula: see text] Inhomogeneous Robin type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces of the thin cylinders and the boundary of the node. The asymptotic behavior of the solution is studied as [Formula: see text] i.e. when the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energy and uniform pointwise estimates are proved. Depending on the directions of the limit convective fluxes, the corresponding limit problems [Formula: see text] are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are analyzed.

Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: Strong boundary interactions

This article completes the study of the influence of the intensity parameter α in the boundary condition ε ∂ ν ε u ε − u ε V ε → · ν ε = ε α φ ε given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order O ( ε ). Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order O ( ε − 1 ) is considered. The novelty of this article is the case of α < 1, which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φ ε (the cases α = 1 and α > 1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution u ε as ε → 0, i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter ε.

Asymptotic expansions of stable, stabilizable and stabilized means with applications

In this paper we present a complete asymptotic expansion of a symmetric homogeneous stable (balanced), stabilizable and stabilized mean. By including known asymptotic expansions of parametric means it is shown how the obtained coefficients are used to solve the problem of identifying stable means within classes of parametric means under consideration, how to disprove some mean is stabilizable or stabilized and how to obtain best possible parameters such that given mean is sub-stabilizable with a pair of parametric means.

Complete Minimal Logarithmic Energy Asymptotics for Points in a Compact Interval: A Consequence of the Discriminant of Jacobi Polynomials

AbstractThe electrostatic interpretation of zeros of Jacobi polynomials, due to Stieltjes and Schur, enables us to obtain the complete asymptotic expansion as $$n \rightarrow \infty $$ n → ∞ of the minimal logarithmic potential energy of n point charges restricted to move in the interval $$[-1,1]$$ [ - 1 , 1 ] in the presence of an external field generated by endpoint charges. By the same methods, we determine the complete asymptotic expansion as $$N \rightarrow \infty $$ N → ∞ of the logarithmic energy $$\sum _{j\ne k} \log (1/| x_j - x_k |)$$ ∑ j ≠ k log ( 1 / | x j - x k | ) of Fekete points, which, by definition, maximize the product of all mutual distances $$\prod _{j\ne k} | x_j - x_k |$$ ∏ j ≠ k | x j - x k | of N points in $$[-1,1]$$ [ - 1 , 1 ] . The results for other compact intervals differ only in the quadratic and linear term of the asymptotics. Explicit formulas and their asymptotics follow from the discriminant, leading coefficient, and special values at $$\pm 1$$ ± 1 of Jacobi polynomials. For all these quantities we derive complete Poincaré-type asymptotics.

Complete asymptotic expansions and the high-dimensional Bingham distributions

Complete asymptotic expansions and the high-dimensional Bingham distributions

Classical wave methods and modern gauge transforms: spectral asymptotics in the one dimensional case

In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let H: L^{2}(mathbb{R})to L^{2}(mathbb{R}) have the form \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ H:=-\\frac{d^{2}}{dx^{2}}+Q, $$\\end{document} where Q is a formally self-adjoint first order differential operator with smooth coefficients, bounded with all derivatives. We show that the kernel of the spectral projector, {1}_{(-infty ,rho ^{2}]}(H), has a complete asymptotic expansion in powers of ρ. This settles the 1-dimensional case of a conjecture made by the last two authors.

Lambert series of logarithm, the derivative of Deninger’s function and a mean value theorem for

Abstract An explicit transformation for the series $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ , or equivalently, $\sum \limits _{n=1}^{\infty }d(n)\log (n)e^{-ny}$ for Re $(y)>0$ , which takes y to $1/y$ , is obtained for the first time. This series transforms into a series containing the derivative of $R(z)$ , a function studied by Christopher Deninger while obtaining an analog of the famous Chowla–Selberg formula for real quadratic fields. In the course of obtaining the transformation, new important properties of $\psi _1(z)$ (the derivative of $R(z)$ ) are needed as is a new representation for the second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$ evaluated at $b=1$ , all of which may seem quite unexpected at first glance. Our transformation readily gives the complete asymptotic expansion of $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ as $y\to 0$ which was also not known before. An application of the latter is that it gives the asymptotic expansion of $ \displaystyle \int _{0}^{\infty }\zeta \left (\frac {1}{2}-it\right )\zeta '\left (\frac {1}{2}+it\right )e^{-\delta t}\, dt$ as $\delta \to 0$ .

On the Barnes double gamma function

We aim to achieve the following three goals. First of all, we collect all known definitions, transformation properties and functional identities of Barnes double gamma function . Second, we derive an algorithm for numerically computing the double gamma function and present its complete asymptotic expansion as . Third, we derive some new properties, including a new product identity and new representations of the gamma modular forms and .

Asymptotic Solution for a Visco-Elastic Thin Plate: Quasistatic and Dynamic Cases

The Kelvin–Voigt model for a thin stratified two-dimensional visco-elastic strip is analyzed both in the quasistatic and in the dynamic cases. The Neumann boundary conditions on the upper and the lower parts of the boundary and periodicity conditions with respect to the longitudinal variable are stated. A complete asymptotic expansion of the solution is constructed in both cases, by using the dimension reduction combined with a homogenization technique. The error between the exact solution and the asymptotic one is evaluated in each case and the obtained results fully justify the asymptotic construction. The results were partially (quasistatic case) announced in the short note in C.R. Acad. Sci. Paris; the present article contains the complete proofs and generalizations in the dynamic case.

The rate of convergence of a generalization of Post–Widder operators and Rathore operators

In this paper, we study local approximation properties of certain gamma-type operators. They generalize the Post–Widder operators and the Rathore operators, and approximate locally integrable functions satisfying a certain growth condition on the infinite interval [0,infty ). We derive the complete asymptotic expansion for these operators and prove a localization result. Also, we estimate the rate of convergence for functions of bounded variation.

Asymptotic expansions in the local limit theorem for a branching Wiener process

Consider a supercritical branching Wiener process in Rd. Let Zn(A) be the number of the nth generation particles located in a given set A⊂Rd. Under the moment condition of EX(lnX)1+λ type, the complete asymptotic expansions of Zn(A) as n tends to infinity are obtained. This result gives an alternative version of the work by Révész, Rosen and Shi (2005), hence generalizing theirs by weakening the second moment condition therein.

Complete Nonuniform Asymptotic Expansion of Sommerfeld Integral for Dielectric Half-Space

Generally, the Sommerfeld integral (SI) for a dielectric half-space is approximated based on the steepest descent method. A higher order approximation of the integral requires higher order derivatives of the integrand in the steepest descent path in the complex domain, whose explicit formulation may be cumbersome. Recently, a new finite-range integral representation of the SI has been proposed, whose integrand is written in terms of the SI for an impedance half-plane. Based on the known complete nonuniform asymptotic expansion of the SI for an impedance half-plane, the complete nonuniform asymptotic expansion can be analytically formulated for the dielectric half-space. Moreover, the completeness is mathematically proven such that the expansion satisfies the recurrence relation for the Wilcox expansion. The accuracy of the proposed expansion is numerically examined for several propagation scenarios. Furthermore, a numerical procedure is proposed to calculate the higher order terms of the expansion by partially using the impedance approximation for a large relative permittivity of the half-space.

Asymptotics of the Solution of a Singularly Perturbed System of Equations with a Single-Scale Internal Layer

We consider a boundary value problem for a singularly perturbed system of two second-order ordinary differential equations with different powers of the small parameter multiplying the second derivatives. A specific feature of the problem is that one of the two equations of the degenerate system has a double root and the other has three nonintersecting simple roots. It is proved that for sufficiently small values of the small parameter the problem has a solution that has a fast transition in a neighborhood of some interior point of the interval. A complete asymptotic expansion of this solution is constructed and justified. It qualitatively differs from the well-known expansion in the case where all the roots of the degenerate equations are simple but also does not coincide with the expansions in the previously studied problems with double roots; in particular, the inner transition layer turns out to be single-scale.

A case of impulsive singularity

The paper considers an impulsive system with singularities. Different types of problems with singular perturbations have been discussed in many books. In Bainov and Kovachev’s book [4]several articles cited therein consider impulse systems with small parameter involving only differential equations. The parameter is not in the impulsive equation of the systems. In our present the small parameter is inserted into the impulse equation. This is the principal novelty of our study. Furthermore, for the impulsive function, we found a condition that prevents the impulsive function to blow up as the parameter tends to zero. So we have significantly extended the singularity concept for discontinuous dynamics. The singularity of the impulsive part of the system can be treated in the manner of perturbation theory methods. This article is a continuation of [1] work. In our present research, we apply the method of the paper [1]. Our goal is to construct an approximation with higher accuracy and to obtain the complete asymptotic expansion. We construct a uniform asymptotic approximation of the solution that is valid in the entire close interval by using the method of boundary functions [22]. An illustrative example using numerical simulations is given to support the theoretical results.

Asymptotics of Solutions of Linear Singularly Perturbed Optimal Control Problems with a Convex Integral Performance Index and a Cheap Control

We consider an optimal control problem for a linear system with constant coefficients with an integral convex performance index containing a small parameter multiplying the integral term in the class of piecewise continuous controls with smooth geometric constraints. Such problems are called cheap control problems. It is shown that the limit problem will be a problem with a terminal performance index. It is established that if the terminal term of the performance index is a convex (strictly convex) and continuously differentiable function, then the performance functional in the limit problem has similar properties. It is proved that, in the general case, convergence with respect to the performance functional is valid, and under the condition of strict convexity of the terminal term of the performance index in the original problem, convergence to the minimum point of the terminal summand of the performance index in the limit problem is valid. The limit of the defining vector in the original problem is found as the small parameter tends to zero. In particular, it is shown that the first component of the defining vector in the original problem converges to the defining vector in the limit problem. The problems of controlling a point of low mass in a medium with and without resistance with a terminal part depending on both slow and fast variables are considered in detail, and complete asymptotic expansions of the defining vectors in these problems are constructed.

Complete asymptotic expansion of the compound means with applications

Complete asymptotic expansion of the compound means with applications