Complete Asymptotic Expansion Research Articles

Spectral invariants of Dirichlet-to-Neumann operators on surfaces

We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schrödinger operators on Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For non-zero potentials, we obtain new geometric invariants determined by the spectrum. In particular, for constant potentials, which give rise to the parameter-dependent Steklov problem, the total geodesic curvature on each connected component of the boundary is a spectral invariant. Under the constant curvature assumption, this allows us to obtain some interior information from the spectrum of these boundary operators.

Complete asymptotic expansions for the relativistic Fermi-Dirac integral

Fermi-Dirac integrals appear in problems in nuclear astrophysics, solid state physics or in the fundamental theory of semiconductor modeling, among others areas of application. In this paper, we give new and complete asymptotic expansions for the relativistic Fermi-Dirac integral. These expansions could be useful to obtain a correct qualitative understanding of Fermi systems. The performance of the expansions is illustrated with numerical examples.

Approximation properties related to the Bell polynomials

The authors provide a complete asymptotic expansion for a class of functions in terms of the complete
 Bell polynomials. In particular, they obtain known asymptotic expansions of some Keller type sequences.

Asymptotics and renewal approximation in the online selection of increasing subsequence

We revisit the problem of maximising the expected length of increasing subsequence that can be selected from a marked Poisson process by an online strategy. Resorting to a natural size variable, we represent the problem in terms of a controlled piecewise deterministic Markov process with decreasing paths. We apply a comparison method to the optimality equation to obtain fairly complete asymptotic expansions for the moments of the maximal length, and, with the aid of a renewal approximation, give a novel proof to the central limit theorem for the length of selected subsequence under either the optimal strategy or a strategy sufficiently close to optimality.

Finite Range Integral Representation of Sommerfeld Integral for Homogeneous Dielectric Half-Space and Complete Uniform Asymptotic Expansion

Since the existing integral representations of the Sommerfeld integral (SI) are of the (semi) infinite type, a numerical error is inevitable due to the truncation of the integral range. In this article, a new finite range integral representation of the SI is formulated based on the exact image theory. To derive the new representation, a finite range integral expression of the exact image current is first derived. The radiated field by the exact image current is expressed in terms of a double integral in the complex domain where the exact image current flows. One of the two integrals is a semiinfinite line integral along the current path. This integral is deformed into a more convenient form as the exact image representation for the impedance plane. To convert the line source in the complex domain into a line source in the real domain, the path-deformation technique is again applied. The final line source consists of three line segments along which the exact image current flows vertically from the image source point to ±∞. In addition to the line source, a lateral wave component is also obtained. The impedance exact image representation can be evaluated as a closed-form expression in terms of an incomplete cylindrical function. Thus, the SI can be represented as the finite range integral of the incomplete cylindrical function. To efficiently evaluate the proposed integral, a complete uniform asymptotic expansion is formulated. All the proposed formulations are numerically verified, including, in some cases, near-Earth propagation, and their behaviors are investigated. Also, the electric field is computed for a vertical and horizontal infinitesimal dipole and compared by the exact Sommerfeld and known approximate formulations.

Asymptotic expansions for certain exponential-type operators connected with 2x^{3/2}

In the present paper, we consider the complete asymptotic expansion of certain exponential-type operators connected with 2x^{3/2}. Also, a modification of such exponential-type operators is provided, which preserve the function mathrm{e}^{Ax}.

Reconstruction of a small acoustic inclusion via time-dependent polarization tensors

Abstract This paper aims at introducing the concept of time-dependent polarization tensors (TDPTs) for the wave equation associated to a diametrically small acoustic inclusion, with constitutive parameters different from those of the background and size smaller than the operating wavelength. Firstly, the solution to the Helmholtz equation is considered, and a rigourous systematic derivation of a complete asymptotic expansion of the scattered field due to the presence of the inclusion is presented. Then, by applying the Fourier transform, the corresponding time-domain expansion is readily obtained after truncating the high frequencies. The new concept of TDPTs is shown to be promising for performing imaging. In particular, the optimization approach proposed by Ammari et al. (Ammari, H., Kang, H., Kim, & E. Lee, J.-Y. (2012)The generalized polarization tensors for resolved imaging. Part II: shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements. Math. Comp., 81, 839–860.) is extended to TDPTs. Numerical simulations are presented, showing that the TDPTs reconstructed from noisy measurements allow to image fine shape details of the inclusion.

Exact closed-form and asymptotic expressions for the electrostatic force between two conducting spheres

We present exact closed-form expressions and complete asymptotic expansions for the electrostatic force between two charged conducting spheres of arbitrary sizes. Using asymptotic expansions of the force, we confirm that even like-charged spheres attract each other at sufficiently small separation unless their voltages/charges are the same as they would be at contact. We show that for sufficiently large size asymmetries, the repulsion between two spheres increases when they separate from contact if their voltages or their charges are held constant. Additionally, we show that in the constant voltage case, this like-voltage repulsion can be further increased and maximized through an optimal lowering of the voltage on the larger sphere at an optimal sphere separation.

The swallowtail integral in the highly oscillatory region III

We consider the swallowtail integral for large values of and bounded values of and . The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the modified saddle point method introduced in López et al., A systematization of the saddle point method application to the Airy and Hankel functions. J Math Anal Appl. 2009;354:347–359. The analysis is more straightforward with this method and it is possible to derive complete asymptotic expansions of for large and fixed x and y. The asymptotic analysis requires the study of three different regions for separated by three Stokes lines in the sector . The asymptotic approximation is a certain combination of two asymptotic series whose terms are elementary functions of x, y and z. They are given in terms of an asymptotic sequence of the order when , and it is multiplied by an exponential factor that behaves differently in the three mentioned sectors. The accuracy and the asymptotic character of the approximations are illustrated with some numerical experiments.

Метод обработки данных гидродинамических исследований скважин

Horner’s traditional method of processing well test data can be improved by a special transformation of the pressure curves, which reduces the time the converted curves reach the asymptotic regimes necessary for processing these data. In this case, to take into account the action of the «skin factor» and the effect of the wellbore, it is necessary to use a more complete asymptotic expansion of the exact solution of the conductivity equation at large values of time. At the same time, this method does not allow to completely eliminate the influence of the wellbore, since the used asymptotic expansion of the solution for small values of time is limited by the existence of a singular point, in the vicinity of which the asymptotic expansion ceases to be valid. To solve this problem, a new method of processing well test data is proposed, which allows completely eliminating the influence of the wellbore. The method is based on the introduction of a modified inflow function to the well, which includes a component of the boundary condition corresponding to the influence of the wellbore.

Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions

We derive a complete asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions on arbitrary symplectic manifolds, characterizing the coefficients in the expansion as integrals over the symplectic strata of the corresponding Marsden-Weinstein reduced space and distributions on the Lie algebra. The obtained coefficients involve singular contributions of the lower-dimensional strata related to numerical invariants of the fixed-point set.

Asymptotic behavior of acoustic waves scattered by very small obstacles

The direct numerical simulation of the acoustic wave scattering created by very small obstacles is very expensive, especially in three dimensions and even more so in time domain. The use of asymptotic models is very efficient and the purpose of this work is to provide a rigorous justification of a new asymptotic model for low-cost numerical simulations. This model is based on asymptotic near-field and far-field developments that are then matched by a key procedure that we describe and demonstrate. We show that it is enough to focus on the regular part of the wave field to rigorously establish the complete asymptotic expansion. For that purpose, we provide an error estimate which is set in the whole space, including the transition region separating the near-field from the far-field area. The proof of convergence is established through Kondratiev’s seminal work on the Laplace equation and involves the Mellin transform. Numerical experiments including multiple scattering illustrate the efficiency of the resulting numerical method by delivering some comparisons with solutions computed with a finite element software.

A complete asymptotic expansion for operators of exponential type with pleft( xright) =xleft( 1+xright) ^{2}

In the year 1978, Ismail and May studied operators of exponential type and proposed some new operators which are connected with a certain characteristic function pleft( xright) . Several of these operators were not separately studied by researchers due to its unusual behavior. The topic of the present paper is the local rate of approximation of a sequence of exponential type operators R_{n} belonging to pleft( xright) =xleft( 1+xright) ^{2}. As the main result we derive a complete asymptotic expansion for the sequence left( R_{n}fright) left( xright) as n tends to infinity.

Asymptotic Behavior of Remainders of Special Number Series

We consider a one-parameter family of number series involving the generalized harmonic series and study asymptotic properties of the remainders. Using $$ R\left(N,p\right)\equiv \sum \limits_{n=N}^{\infty }1/{n}^p $$ as an example, we describe the typical obtained results: we obtain the integral representation, find the complete asymptotic expansion with respect to the parameter 2N − 1 as N →∞, and prove that R(N, p) is enveloped by its asymptotic series. The possibilities of the proposed approach are demonstrated by the problem of exact two-sided estimates for the central binomial coefficient.

A Complete Asymptotic Expansion for Bernstein\u2013Chlodovsky Polynomials for Functions on mathbb {R}

We consider a variant of the Bernstein–Chlodovsky polynomials approximating continuous functions on the entire real line and study its rate of convergence. The main result is a complete asymptotic expansion. As a special case we obtain a Voronovskaja-type formula previously derived by Karsli [11].

Asymptotic expansions for the radii of starlikeness of normalised Bessel functions

The asymptotic behaviour, with respect to the large order, of the radii of starlikeness of two types of normalised Bessel functions is considered. We derive complete asymptotic expansions for the radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums and asymptotic inversion. The techniques employed in the paper could be useful to treat similar problems where inversion of asymptotic expansions is involved.

Dispersion of waves in two and three-dimensional periodic media

We consider the propagation of acoustic time-harmonic waves in homogeneous media containing periodic lattices of spherical or cylindrical inclusions. It is assumed that the wavelength has the order of the periods of the lattice while the radius a of inclusions is small. A new approach is suggested to derive the complete asymptotic expansions of the dispersion relations in two- and three-dimensional cases as and first several terms of the expansions are evaluated explicitly. Our method is based on the reduction of the original singularly perturbed (by inclusions) problem to the regular one. The Dirichlet, Neumann, and transmission boundary conditions are considered. In the former case, we estimate the cutoff wavelength supported by the periodic medium in two and three dimensions. The effective wave speed is obtained as a function of the wave frequency, the filling fraction of the inclusions, and the physical properties of the constituents of the mixture. Dependence of the asymptotic formulas obtained in the paper on geometric and material parameters is illustrated by graphs.

Asymptotics of the Solution to a Singularly Perturbed Time-Optimal Control Problem with Two Small Parameters

The paper continues the authors’ previous studies. We consider a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball \(\left\{\begin{array}[]{llll}\phantom{\varepsilon^{3}}\dot{x}=y,&x,\,y\in \mathbb{R}^{2},\quad u\in\mathbb{R}^{2},\\ \varepsilon^{3}\dot{y}=Jy+u,&\,\|u\|\leq 1,\quad 0<\varepsilon,\mu\ll 1,\\ x(0)=x_{0}(\varepsilon,\mu)=(x_{0,1},\varepsilon^{3}\mu\xi)^{*},\quad y(0)=y_{ 0},\\ x(T_{\varepsilon,\mu})=0,\quad y(T_{\varepsilon,\mu})=0,\quad T_{\varepsilon, \mu}\to\min,&\end{array}\right.\) where \(J=\left(\begin{array}[]{rr}0&1\\ 0&0\end{array}\right).\) The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix \(J\) at the fast variables is the second-order Jordan block with zero eigenvalue and, thus, does not satisfy the standard asymptotic stability condition. Continuing the research, we consider initial conditions depending on the second small parameter \(\mu\). We derive and justify a complete asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to the asymptotic sequence \(\varepsilon^{\gamma}(\varepsilon^{k}+\mu^{k})\), \(0<\gamma<1\).

Asymptotic Expansions at Nonsymmetric Cuspidal Points

We study the asymptotics of solutions to the Dirichlet problem in a domain $$\mathcal{X} \subset \mathbb{R}^3$$ whose boundary contains a singular point $$O$$ . In a small neighborhood of this point, the domain has the form $$\{ z > \sqrt{x^2 + y^4} \}$$ , i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat’ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point.

Asymptotics of approximation of functions by conjugate Poisson integrals

Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha $, i.e. the class of continuous $ 2\pi $-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0&lt;\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha $. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.