A comparison of the plate theories in the sense of Kirchhoff–Love and Reissner–Mindlin

In this article we compare the two plate theories in the sense of Kirchhoff–Love and Reissner–Mindlin for several different settings of the physical system. We establish existence, uniqueness and regularity of solutions to the respective boundary and initial boundary value problems. Moreover, we give asymptotic expansions of the solutions in the limit of a vanishing plate thickness, ϵ→0, whenever this is possible. Finally, we compare the solutions in the sense of Kirchhoff–Love and Reissner–Mindlin in that very limit. Copyright © 1999 John Wiley & Sons, Ltd.     

Asymptotic analysis and boundary homogenization in linear elasticity

We describe the asymptotic behaviour of the solution of a linear elastic problem posed in a domain of ℝ³, with homogeneous Dirichlet boundary conditions imposed on small zones of size less than ϵ distributed on the boundary of this domain, when the parameter ϵ goes to 0. We use epi‐convergence arguments in order to establish this asymptotic behaviour. We then specialize this general situation to the case of identical strips of size r_ϵ ϵ‐periodically distributed on the lateral surface of an axisymmetric body. We exhibit a critical size of the strips through the limit of the non‐negative quantity −1/(ϵ ln r_ϵ) and we identify the different limit problems according to the values of this limit. Copyright © 2000 John Wiley & Sons, Ltd.     

Poly-p-Bernoulli polynomials and generalized Arakawa–Kaneko zeta function

In this paper, we first obtain several properties of poly-p-Bernoulli polynomials. In particular, we achieve some new results for poly-Bernoulli polynomials. We next define a generalization of the Arakawa–Kaneko zeta function associated with poly-p-Bernoulli polynomials, investigate some its particular values, and give asymptotic and series expansions.

     

Error Bounds for the Large‐Argument Asymptotic Expansions of the Lommel and Allied Functions

In this paper, we reconsider the large‐z asymptotic expansion of the Lommel function and its derivative. New representations for the remainder terms of the asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re‐expansions for these remainder terms and provide their error estimates. Applications to the asymptotic expansions of the Anger–Weber‐type functions, the Scorer functions, the Struve functions, and their derivatives are provided. The sharpness of our error bounds is discussed in detail, and numerical examples are given.     

New asymptotic representations of the noncentral t-distribution

New asymptotic approximations of the noncentral t distribution are given a generalization of the Student's t distribution. Using new integral representations, we give new asymptotic expansions not only for large values of the noncentrality parameter but also for large values of the degrees of freedom parameter. In some cases, we accept more than one large parameter. These results are not only in terms of elementary functions, but also in terms of the complementary error function and the incomplete gamma function. A number of numerical tests demonstrate the performance of the asymptotic approximations.     

Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones

In this paper we deal with several issues concerning variable-stepsize linear multistep methods. First, we prove their stability when their fixed-stepsize counterparts are stable and under mild conditions on the stepsizes and the variable coefficients. Then we prove asymptotic expansions on the considered tolerance for the global error committed. Using them, we study the growth of error with time when integrating periodic orbits. We consider strongly and weakly stable linear multistep methods for the integration of first-order differential systems as well as those designed to integrate special second-order ones. We place special emphasis on the latter which are also symmetric because of their suitability when integrating moderately eccentric orbits of reversible systems. For these types of methods, we give a characterization for symmetry of the coefficients, which allows their construction, and provide some numerical results for them.

     

Asymptotic analysis of a spectral problem in a periodic thick junction of type 3:2:1

Convergence theorems and asymptotic estimates (as ϵ→0) are proved for eigenvalues and eigenfunctions of a mixed boundary value problem for the Laplace operator in a junction Ω_ϵ of a domain Ω₀ and a large number N² of ϵ‐periodically situated thin cylinders with thickness of order ϵ=O(N⁻¹). We construct an extension operator that is only asymptotically bounded in ϵ on the eigenfunctions in the Sobolev space H¹. An approach based on the asymptotic theory of elliptic problem in singularly perturbed domains is used. Copyright © 2000 John Wiley & Sons, Ltd.     

Scattering of a scalar time-harmonic wave by <Emphasis Type="Italic">N</Emphasis> small spheres by the method of matched asymptotic expansions

In this paper, we construct an asymptotic expansion of a time-harmonic wave scattered by N small spheres. This construction is based on the method of matched asymptotic expansions. Error estimates give a theoretical background to the approach.     

The Second Appell Function for one Large Variable

We consider the Mellin convolution integral representation of the second Appell function given in [8]. Then, we apply the asymptotic method designed in [12] for this kind of integrals to derive new asymptotic expansions of the Appell function F ₂ for one large variable in terms of hypergeometric functions. For certain values of the parameters, some of these expansions involve logarithmic terms in the asymptotic variables. The accuracy of the approximations is illustrated with numerical experiments.     

The coupling of hyperbolic and elliptic boundary value problems with variable coefficients

We consider a one‐dimensional coupled problem for elliptic second‐order ODEs with natural transmission conditions. In one subinterval, the coefficient ϵ>0 of the second derivative tends to zero. Then the equation becomes there hyperbolic and the natural transmission conditions are not fulfilled anymore. The solution of the degenerate coupled problem with a flux transmission condition is corrected by an internal boundary layer term taking into account the viscosity ϵ. By using singular perturbation techniques, we show that the remainders in our first‐order asymptotic expansion converge to zero uniformly. Our analysis provides an a posteriori correction procedure for the numerical treatment of exterior viscous compressible flow problems with coupled Navier–Stokes/Euler models. Copyright © 2000 John Wiley & Sons, Ltd.     

A Systematic “Saddle Point Near a Pole” Asymptotic Method with Application to the Gauss Hypergeometric Function

In recent works [ 1 ] and [ 2 ], we have proposed more systematic versions of the Laplace’s and saddle point methods for asymptotic expansions of integrals. Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions. In this work we apply the ideas introduced in [ 1 ] and [ 2 ] to the uniform method “saddle point near a pole.” We obtain a computationally more systematic version of that uniform asymptotic method for integrals having a saddle point near a pole that, in many interesting examples, gives a closed algebraic formula for the coefficients. The asymptotic sequence is given, in general, in terms of exponential integrals of fractional order (or incomplete gamma functions). In particular, when the order of the saddle point is two, the basic approximant is given in terms of the error function (as in the standard method). As an application, we obtain new asymptotic expansions of the Gauss Hypergeometric function ₂F₁(a, b, c; z) for large b and c with c > b + 1 .     

Some asymptotic expansions on hyperfactorial functions and generalized Glaisher–Kinkelin constants

In this paper, by the Bernoulli numbers and the exponential complete Bell polynomials, we establish two general asymptotic expansions on the hyperfactorial functions \(\prod _{k=1}^nk^{k^q}\) and the generalized Glaisher–Kinkelin constants \(A_q\), where the coefficient sequences in the expansions can be determined by recurrences. Moreover, the explicit expressions of the coefficient sequences are presented and some special asymptotic expansions are discussed. It can be found that some well-known or recently published asymptotic expansions on the factorial function n!, the classical hyperfactorial function \(\prod _{k=1}^nk^k\), and the classical Glaisher–Kinkelin constant \(A_1\) are special cases of our results, so that we give a unified approach to dealing with such asymptotic expansions.     

A multiresolution method for fitting scattered data on the sphere

In this work, we propose an efficient multiresolution method for fitting scattered data functions on a sphere S, using a tensor product method of periodic algebraic trigonometric splines of order 3 and quadratic polynomial splines defined on a rectangular map of S. We describe the decomposition and reconstruction algorithms corresponding to the polynomial and periodic algebraic trigonometric wavelets. As application of this method, we give an algorithm which allows to compress scattered data on spherelike surfaces. In order to illustrate our results, some numerical examples are presented.     

The distribution |f|λ, oscillating integrals and principal value integrals

We study the coefficients of asymptotic expansions of oscillating integrals. We also consider the connection with the coefficients of Laurent expansions at candidate poles of the distribution |f|^λ and show that some of these coefficients vanish. Next, we express some of the most important of these coefficients as the so-called principal value integrals, first introduced by Langlands. Together with our results on principal value integrals, this leads to new results on the vanishing of these coefficients.     

Asymptotic stability of the numerical solution for an integrodifferential reaction-diffusion system

This paper presents the study of the numerical solution of a reaction-diffusion system involving a reaction term of integral type arising from biological models. By means of a monotone approach we introduce upper and lower solutions and then we show the existence and the asymptotic behavior of nonnegative numerical solutions. To this end, we require the positivity of the numerical scheme and so we can use some properties of positive and M-matrices. Finally we give some sufficient conditions to verify the asymptotic stability of the numerical solution.     

Analysis of multiple scattering iterations for high-frequency scattering problems. I: the two-dimensional case

We present an analysis of a recently proposed integral-equation method for the solution of high-frequency electromagnetic and acoustic scattering problems that delivers error-controllable solutions in frequency-independent computational times. Within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities and on suitable extensions of the method of stationary phase. The extension to multiple-scattering configurations, in turn, is attained through consideration of an iterative (Neumann) series that successively accounts for further geometrical wave reflections. As we show, for a collection of two-dimensional (cylindrical) convex obstacles, this series can be rearranged into a sum of periodic orbits (of increasing period), each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers when these are transversed sequentially in a periodic manner. Here, we analyze the properties of these periodic orbits in the high-frequency regime, by deriving precise asymptotic expansions for the “currents” (i.e. the normal derivative of the fields) that they induce on the surface of the obstacles. As we demonstrate these expansions can be used to provide accurate estimates of the rate at which their magnitude decreases with increasing number of reflections, which defines the overall rate of convergence of the multiple-scattering series. Moreover, we show that the detailed asymptotic knowledge of these currents can be used to accelerate this convergence and, thus, to reduce the number of iterations necessary to attain a prescribed accuracy.     

Asymptotic growth bounds for the 3‐D Vlasov‐Poisson system with point charges

In this article, we consider the asymptotic behavior of the classical solution to the 3‐dimensional Vlasov‐Poisson plasma interacting repulsively with N point charges. The large time behavior in terms of diameters of its velocity‐spatial supports is improved to O(t^2/3+ϵ) for any ϵ>0.     

Periodic Homogenization of Green and Neumann Functions

For a family of second‐order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result we obtain asymptotic expansions of Poisson kernels and the Dirichlet‐to‐Neumann maps as well as optimal convergence rates in L^p and W^1,p for solutions with Dirichlet or Neumann boundary conditions. © 2014 Wiley Periodicals, Inc.     

On the linearity of ω-primality in numerical monoids

In an atomic, cancellative, commutative monoid, the ω-value measures how far an element is from being prime. In numerical monoids, we show that this invariant exhibits eventual quasilinearity (i.e., periodic linearity). We apply this result to describe the asymptotic behavior of the ω-function for a general numerical monoid and give an explicit formula when the monoid has embedding dimension 2.     

Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions

Expansions in terms of Bessel functions are considered of the Kummer function ₁ F ₁(a; c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the expansions is explained for large values of the parameter a of the Kummer function.