Time evolution kernels: uniform asymptotic expansions

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9].

Our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h, valid for h sufficiently small. Such expansions are obtained from the so-called asymptotic expansion inequalities valid in $\mathbf{R}^N$ for $N\geq2$, studies by Schatz in [Math. Comp., 67 (1998), pp. 877–899] and [SIAM J. Numer. Anal., 38 (2000), pp. 1269–1293].

Modelling physical problems in mathematical form yields the governing equations that may be linear or nonlinear for known and unknown boundaries. The exact solution for those equations may or may not be obtained easily. Hence we seek an analytical approximation solution in terms of asymptotic expansion. In this study, we focus on a singular perturbation in second order ordinary differential equations. Solutions to several perturbed ordinary differential equations are obtained in terms of asymptotic expansion. The aim of this work is to find an approximate analytical solution using the classical method of matched asymptotic expansion (MMAE). The Mathematica computer algebra system is used to perform the algebraic computations. The details procedures will be discussed and the underlying concepts and principles of the MMAE will be clarified. Perturbation problem for linear equation that occurs at one boundary and two boundary layers are discussed. Approximate analytical solution obtained for both cases are illustrated by graph using selected parameter by showing the outer, inner and composite solution separately. Then, the composite solution will be compare to the exact solution to show their accuracy by graph. By comparison, MMAE is found to be one of the best methods to solve singular perturbation problems in second order ordinary differential equation since the results obtained are very close to the exact solution.

An asymptotic model of a nonlinear adaptive elastic plate

Wallis ratio can be expressed as an asymptotic expansion using Stirling series and Bernoulli numbers. We prove the general inequalities for Wallis ratio for arbitrary number of terms in the asymptotic expansion. We show that the coefficients in the asymptotic expansion are the best possible.

We present a comprehensive study of interpolation inequalities for periodic functions with zero mean, including the existence of and the asymptotic expansions for the extremals, best constants, various remainder terms, etc. Most attention is paid to the critical (logarithmic) Sobolev inequality in the two-dimensional case, although a number of results concerning the best constants in the algebraic case and different space dimensions are also obtained.

The existence of asymptotic expansions of the global discretization error for a general class of nonlinear stiff differential equations \[ y'(t) = A(t)y(t) + \varphi (t,y(t)), \] where $A(t)$ has a “stiff spectrum” characterized by a small parameter $\varepsilon $ and where $\varphi (t,y)$ is smooth, is discussed. The following methods are considered: implicit Euler, implicit midpoint, and trapezoidal rules. In strongly stiff situations ($\varepsilon $ significantly smaller than the stepsize h) the implicit Euler scheme admits a full asymptotic expansion; the same is true for the midpoint rule and for the trapezoidal rule under certain coupling conditions. In those strongly stiff cases where a full expansion does not exist for the midpoint or trapezoidal rule, the remainder term is of a reduced order but shows a regular, oscillating behavior that is described in detail. In mildly stiff situations, order reductions of the remainder term inevitably occur in any case after the start or after the change of stepsize but—as can be shown by discrete singular perturbation techniques—these order reductions are rapidly damped out as the integration proceeds. Results are illustrated by various numerical examples; in particular, numerical experience with extrapolation and defect correction is reported.

On the Accuracy of Gaussian Approximation to the Distribution Functions of Sums of Independent Variables

Large degree asymptotics of generalized Bessel polynomials

We consider the second order linear differential equation where Λ is a large complex parameter and g is a continuous function. In previous works we have considered the case α ∈ (−∞, 2] and designed a convergent and asymptotic method for the solution of the corresponding initial value problem with data at t = 0. In this paper we complete the research initiated in those works and analyze the remaining case α ∈ (2, ∞). We use here the same fixed point technique; the main difference is that for α ∈ (2, ∞) the convergence of the method requires that the initial datum is given at a point different from the origin; for convenience we choose the point at the infinity. We obtain a sequence of functions that converges to the unique solution of the problem. This sequence has also the property of being an asymptotic expansion for large Λ (not of Poincare-type) of the solution of the problem. The generalization to non-linear problems is ´ straightforward. An application to a quantum mechanical problem is given as an illustration.

Asymptotic expansions of two-loop Feynman diagrams in the Sudakov limit

We continue our work [F. Bornemann, Asymptotic expansions of the limit laws of Gaussian and Laguerre (Wishart) ensembles at the soft edge, preprint (2024), arXiv:2403.07628] on asymptotic expansions at the soft edge for the classical [Formula: see text]-dimensional Gaussian and Laguerre random matrix ensembles. By revisiting the construction of the associated skew-orthogonal polynomials in terms of wave functions, we obtain concise expressions for the level densities that are well suited for proving asymptotic expansions in powers of a certain parameter [Formula: see text]. In the unitary case, the expansion for the level density can be used to reconstruct the first correction term in an established asymptotic expansion of the associated generating function. In the orthogonal and symplectic cases, we can even reconstruct the conjectured first and second correction terms.

We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω e that is e-periodically perforated by small holes of order . The holes are divided into three e-periodical sets depending on boundary conditions. The homogeneous Dirichlet boundary conditions are imposed for holes of one set, whereas, for holes in the remaining sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For a solution to the quasilinear problem we find the leading terms of the asymptotic expansion and prove asymptotic estimates that show the influence of perturbation parameters. In the linear case, we construct and justify a complete asymptotic expansion of the solution by using the two-scale asymptotic expansion method. Bibliography: 25 titles. Illustrations: 1 figure.

An interactive computer system’ service is characterized by the random waiting (or response) time perceived by users. This paper presents a novel solution to the problem of efficiently computing the second moment of the waiting time of a class of large interactive systems. The physical system consists of a bank of terminals, each of which asynchronously alternates between “thinking” and waiting for service from a CPU which operates under the processor-sharing discipline. The problem of obtaining higher order waiting time moments is quite different from that of obtaining CPU queue statistics. Only the first moment of the waiting time is obtainable from the CPU queue statistics. The technique for arriving at the second moment of the waiting time consists of developing an asymptotic expansion in inverse powers of the number of terminals. Hence, as the system grows, quite fortuitously, fewer terms of the series require computation to achieve the desired accuracy. Beside its numerical advantages, the results give new insight since the leading terms of the series, which contain most of the information, are obtained explicitly. The novelty also rests on the fact that instead of solving matrix equations, the problem is turned into one of solving a second order differential equation. A simple two-dimensional recursion yields all the terms of the asymptotic expansion.