Large degree asymptotics of generalized Bessel polynomials

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].

The topological derivative provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal perturbations. Therefore, as a natural extension of this concept, we can consider higher-order terms in the expansion. In particular, the next one we recognize as the second-order topological derivative, which allows us to deal with perturbations of finite sizes. This term depends explicitly on higher-order gradients of the solution associated to the non-perturbed problem and also implicitly through the solution of an auxiliary variational problem. In this article, we calculate the explicit as well as implicit terms of the second-order topological asymptotic expansion for the total potential energy associated to the Laplace equation in the two-dimensional domain. The domain perturbation is done by the insertion of a small inclusion with a thermal conductivity coefficent value different from the bulk material. Finally, we present some numerical experiments showing the influence of the second-order term in the topological asymptotic expansion for several values of the thermal conductivity coefficent of the inclusion.

Periodic homogenization for inertial particles

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.

The problem of the ground state electron density pattern of an electron system in the field of nuclei is recast in inverse form, and tied to the expression of the "internal" energy as a density functional. Various formal expansions in terms of linear response functions are carried out. These motivate the construction of a model internal energy which tests quite well on "mock" He and Ne.

We consider a two-species competition-diffusion system involving a small parameter $\varepsilon>0$ and discuss the validity of formal asymptotic expansions of solutions near the sharp interface limit $\varepsilon\approx0$. We assume that the corresponding ODE system has two stable equilibria. As in the scalar Allen--Cahn equation, it is known that the motion of the sharp interfaces of such systems is governed by the mean curvature flow with a driving force. The formal expansion also suggests that the profile of the transition layers converges to that of a traveling wave solution as $\varepsilon\rightarrow0$. In this paper, we rigorously verify this latter ansatz for a large class of initial data. The proof relies on a rescaling argument, the super--subsolution method, and a Liouville type theorem for eternal solutions of parabolic systems. Roughly speaking, the Liouville type theorem states that any eternal solution that lies between two traveling waves is itself a traveling wave. The same Liouville type theorem was established for the scalar Allen--Cahn equation by Berestycki and Hamel. In view of their importance, we prove the Liouville type theorems in a rather general framework, not only for two-species competition-diffusion systems but also for $m$-species cooperation-diffusion systems possibly with time periodic or spatially periodic coefficients.

A uniform asymptotics of the surface Love modes for a special case of anisotropy (tranverse isotropy) of an elastic media is obtained. In constructing the asymptotics of surface waves, the space-time (ST) ray method is employed. The wave field of each Love mode is represented as the sum of the ST caustic expansion involving the Airy functions with a real eikonal and two correction terms that are ST ray solutions, which in fact are inhomogeneous waves with complex eikonals. The eikonals and coefficients of the caustic and ray series are sought in the form of expansions in powers of two variables. The first variable is the distance from the surface, whereas the other characterizes the proximity of the caustic of a ray field to the boundary surface. Thanks to the specific structure of the elasticity tensor for a transversely isotropic medium, the boundary surface is necessarily a plane. A recursion process of computation of higher terms of the asymptotic expansion allows one to trace the conversion of the formulas obtained to the known ray solutions for isotropic elastic media. Relations between the elasticity parameters of a medium are obtained that ensure the existence of SH Love waves in a transversely isotropic medium and that are consistent with the conditions of the positiveness of the elastic energy of deformation. Bibliography: 6 titles.

This paper presents an analytic formulation of the near- and far-field distributions due to periodic spherical arrays of dipole antennas on two spherical biisotropic substrates, i.e., on the chiral sphere and on a perfectly conducting sphere coated with one-layer chiral medium, respectively. For an arbitrary orientation of the dipole antenna array, the dyadic Green's functions in the form of an eigenfunction expansion in terms of the normalized spherical vector wave functions are directly employed. Thereby, exact representations of the radiated fields in both near and far zones are obtained in closed form. They are all in a superposition of right- and left-handed circularly polarized waves, and are also applicable for the cases of spherical arrays of dipole antennas on spherical isotropic substrates. Furthermore, the constitutive parameter effects of the chiral medium on the three-dimensional patterns of the far-zone field and its polarized feature are demonstrated numerically.

Modelling and simulation of acoustic pulse interaction with a fluid-filled hollow elastic sphere through numerical Laplace inversion

This paper investigates the achievable rates of an additive white Gaussian noise energy-harvesting (EH) channel with an infinite battery. The EH process is characterized by a sequence of blocks of harvested energy, which is known causally at the source. The harvested energy remains constant within a block while the harvested energy across different blocks is characterized by a sequence of independent and identically distributed random variables. The blocks have length $L$ , which can be interpreted as the coherence time of the energy-arrival process. If $L$ is a constant or grows sublinearly in the blocklength $n$ , we fully characterize the first-order term in the asymptotic expansion of the maximum transmission rate subject to a fixed tolerable error probability $\varepsilon $ . The first-order term is known as the $\varepsilon $ -capacity. In addition, we obtain lower and upper bounds on the second-order term in the asymptotic expansion, which reveal that the second order term is proportional to $-({L/{n}})^{1/2}$ for any $\varepsilon $ less than 1/2. The lower bound is obtained through analyzing the save-and-transmit strategy. If $L$ grows linearly in $n$ , we obtain lower and upper bounds on the $\varepsilon $ -capacity, which coincide whenever the cumulative distribution function of the EH random variable is continuous and strictly increasing. In order to achieve the lower bound, we have proposed a novel adaptive save-and-transmit strategy, which chooses different save-and-transmit codes across different blocks according to the energy variation across the blocks.

An asymptotic model of a nonlinear adaptive elastic plate

For a wide class of self-adjoint Schrödinger Hamiltonians, a detailed description of the time evolution kernel is obtained. In a setting of a d-dimensional Euclidean space without boundaries, the Schrödinger Hamiltonian H is the sum of the negative Laplacian plus a real-valued local potential v(x). The class of potentials studied is the family of bounded and continuous functions that are formed from the Fourier transforms of complex bounded measures. These potentials are suitable for the N-body problem, since they do not necessarily decrease as ‖x‖→∞. An asymptotic expansion in the complex parameter z, around z=0, is derived for the family of kernels Uz(x,y) corresponding to the analytic semigroup {e−zH:Re z>0}, which is uniform in the coordinate variables x and y. The asymptotic expansion has a simple semiclassical interpretation. Furthermore, an explicit bound for the remainder term in the asymptotic expansion is found. The expansion and the remainder term bound continue to the time axis boundary z=it/ℏ (t≠0) of the analytic semigroup domain.

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.

A generalization of the Braginskii ion fluid description [S. I. Braginskii, Sov. Phys. - JETP 6, 358 (1958)] to the case of an unmagnetized collisional plasma with multiple ion species is presented. An asymptotic expansion in the ion Knudsen number is used to derive the individual ion species continuity, as well as the total ion mass density, momentum, and energy evolution equations accurate through the second order. Expressions for the individual ion species drift velocities with respect to the center of mass reference frame, as well as for the total ion heat flux and viscosity, which are required to close the fluid equations, are evaluated in terms of the first-order corrections to the lowest order Maxwellian ion velocity distribution functions. A variational formulation for evaluating such corrections and its relation to the plasma entropy are presented. Employing trial functions for the corrections, written in terms of expansions in generalized Laguerre polynomials, and maximizing the resulting functionals produce two systems of linear equations (for “vector” and “tensor” portions of the corrections) for the expansion coefficients. A general matrix formulation of the linear systems as well as expressions for the resulting transport fluxes are presented in forms convenient for numerical implementation. The general formulation is employed in Paper II [A. N. Simakov and K. Molvig, Phys. Plasmas 23, 032116 (2016)] to evaluate the individual ion drift velocities and the total ion heat flux and viscosity for specific cases of two and three ion species plasmas.