Terms Of Asymptotic Expansion Research Articles

Diffraction by a narrow circular cone as by a strongly elongated body

Leading order terms of asymptotic expansions in problems of acoustic and electromagnetic waves diffraction by a narrow circular cone are constructed in this paper. By analogy with problems of diffraction by strongly elongated bodies, the derivations are carried out in a special system of coordinates related to the surface, which takes into account the fact that the cone angle is small. Graphs of special functions appearing in the problems of diffraction considered are presented. Bibliography: 9 titles.

On the Infusion of a Therapeutic Agent Into a Solid Tumor Modeled as a Poroelastic Medium

The direct infusion of an agent into a solid tumor, modeled as a spherical poroelastic material with anisotropic dependence of the tumor hydraulic conductivity upon the tissue deformation, is treated both by solving the coupled fluid/elastic equations, and by expressing the solution as an asymptotic expansion in terms of a small parameter, ratio between the driving pressure force in the fluid system, and the elastic properties of the solid. Results at order one match almost perfectly the solutions of the full system over a large range of infusion pressures. Comparison with experimental results is acceptable after the hydraulic conductivity of the medium is properly calibrated. Given the uncertain estimates of some model constants, the order zero solution of the expansion, for which fluid and porous matrix are decoupled, yields acceptable values and trends for all the physical fields of interest, rendering the coupled analysis (in the limit of small displacements) of little use. When the deformation of the tissue becomes large nonlinear elasticity theory must be resorted to.

Bounded and unbounded capillary surfaces in a cusp domain

We study asymptotic behavior of the height of a static liquid surface in a cusp domain as modelled by the Laplace‐Young capillary surface equation. We introduce a new form of an asymptotic expansion in terms of the functions defining the boundary curves forming a cusp. We are able to address the asymptotic behavior of the capillary surface in cusp domains not previously considered, such as an exponential cusp. In addition, we have shown that the capillary surface in a cusp domain is bounded if the contact angles of the boundary walls forming a cusp are supplementary angles, which implies the continuity of the capillary surface at the cusp.

A Justification of Two‐Dimensional Nonlinear Viscoelastic Shells Model

By applying formal asymptotic analysis and Laplace transformation, we obtain two‐dimensional nonlinear viscoelastic shells model satisfied by the leading term of asymptotic expansion of the solution to the three‐dimensional equations.

Effect of higher order asymptotic terms on the competition between crack penetration and debond at a bimaterial interface between aligned orthotropic materials

The effects of the split singularities, i.e. a stronger and a weaker singularity of a crack impinging at bimaterial interface on the competition between crack penetration and debond was recently investigated in [1] for the case of infinite isotropic bimaterial plane. Anisotropy usually lifts a degeneracy of the singularity exponents and strongly influences the direction of crack penetration across the interface. The concept of the Finite fracture mechanics introduced in [2,3–7], conservative line integral derived from Betti’s reciprocal principle, and the matched asymptotic expansion method [3–7] are used to evaluate the energy release rate criteria describing the competition between the crack debond and penetration at a bimaterial interface between aligned orthotropic materials in edge cracked tension specimen. The effect of higher order terms on the crack penetration/debond at a bimaterial interface strongly depends on the elastic mismatch of dissimilar materials and on the crack-specimen configuration. For this reasons, a crack perpendicularly impinging the bimaterial interface and a crack arbitrary inclined to the interface are considered for a wide range of elastically mismatched materials. In the former case, the effect of the T-stress is investigated, in the latter case the combined effect of a weaker singularity and stronger singularity is examined for a range of crack/interface configurations. For the crack arbitrary inclined to the interface, FE solution of near-crack stress–displacement field obtained using a very fine mesh is compared with analytical singular field considering two leading terms of asymptotic expansion. As a side result it is found that the domain of singular field dominance for an arbitrary inclined crack increases with decreasing angle between the crack and the interface.

Equilibria of toroidal plasmas with toroidal and poloidal flow in high-beta reduced magnetohydrodynamic models

A reduced set of magnetohydrodynamic equilibrium equations for high-beta tokamaks is derived from the fluid moment equations for collisionless, magnetized plasmas. Effects of toroidal and poloidal flow comparable to the poloidal-sound velocity, two-fluid, ion finite Larmor radius (FLR), pressure anisotropy and parallel heat fluxes are incorporated into the Grad–Shafranov equation by means of asymptotic expansions in terms of the inverse aspect ratio of a torus. The two-fluid effects induce the diamagnetic flows, which result in asymmetry of the equilibria with respect to the sign of the E × B flow. The gyroviscosity and other FLR effects cause the so-called gyroviscous cancellation of the convection due to the ion diamagnetic flow. The qualitative difference between the equilibria with and without the parallel heat fluxes is shown to stem from characteristics of the sound waves. Higher order terms of quantities like the pressures and the stream functions show the shift of their isosurfaces from the magnetic surfaces due to effects of flow, two-fluid and pressure anisotropy. The reduced form of the diamagnetic current associated with pressure anisotropy is also obtained.

Uniform asymptotics for Meixner–Pollaczek polynomials with varying parameters

In this Note, we study the uniform asymptotics of the Meixner–Pollaczek polynomials P n ( λ n ) ( z ; ϕ ) with varying parameter λ n = ( n + 1 2 ) A as n → ∞ , where A > 0 is a constant. Uniform asymptotic expansions in terms of parabolic cylinder functions and elementary functions are obtained for z in two overlapping regions which together cover the whole complex plane.

On the spectrum of a two-dimensional periodic operator with a small localized perturbation

We consider a two-dimensional periodic self-adjoint second-order differential operator on the plane with a small localized perturbation. The perturbation is given by an arbitrary (not necessarily symmetric) operator. It is localized in the sense that it acts on a pair of weighted Sobolev spaces and sends functions of sufficiently rapid growth to functions of sufficiently rapid decay. By studying the spectrum of the perturbed operator, we establish that the essential spectrum is stable, the residual spectrum is absent, and the set of isolated eigenvalues is discrete. We obtain necessary and sufficient conditions for the existence of new eigenvalues arising from the ends of lacunae in the essential spectrum. In the case when such eigenvalues exist, we construct the first terms of asymptotic expansions of these eigenvalues and the corresponding eigenfunctions.

Spectral theory of large Wiener–Hopf operators with complex-symmetric kernels and rational symbols

AbstractThis paper is devoted to the asymptotic behaviour of individual eigenvalues of truncated Wiener–Hopf integral operators over increasing intervals. The kernel of the operators is complex-symmetric and has a rational Fourier transform. Under additional hypotheses, the main result describes the location of the eigenvalues and provides their asymptotic expansions in terms of the reciprocal of the length of the truncation interval. Also determined is the structure of the eigenfunctions.

ASYMPTOTIC BEHAVIOR OF DISTRIBUTION DENSITIES IN MODELS WITH STOCHASTIC VOLATILITY. I

We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of a geometric Brownian motion and the density of the stock price process in the Hull–White model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the price of an Asian option are obtained.

On the Hörmander Classes of Bilinear Pseudodifferential Operators

Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hormander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue spaces for the classes that yield operators with bilinear Calderon–Zygmund kernels. Some boundedness properties for other classes with estimates in the form of Leibniz’ rule are presented as well.

Asymptotic Error Expansions for the Finite Element Method for Second Order Elliptic Problems in $R^N$, $N\geq2$. I: Local Interior Expansions

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].

Asymptotic Behavior of the Stock Price Distribution Density and Implied Volatility in Stochastic Volatility Models

We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Stein-Stein and the Heston model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the implied volatility in the Stein-Stein and the Heston model are obtained.

Complete asymptotic expansion for multivariate Bernstein–Durrmeyer operators and quasi-interpolants

We establish complete asymptotic expansions in terms of certain differential operators for the Bernstein–Durrmeyer operators with Jacobi weights on the d -dimensional simplices, and for their so-called natural quasi-interpolants. Our method extensively uses spectral properties of the involved operators. In particular, we prove new identities for the eigenvalues of the operators. We also show that the obtained asymptotic expansions can be differentiated term-by-term.

The Riesz energy of the N th roots of unity: an asymptotic expansion for large N

We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$ equally spaced points on the unit circle as $N\to \infty$. For $s\ge -2$, such points form optimal energy $N$-point configurations with respect to the Riesz potential $1/r^{s}$, $s\neq0$, where $r$ is the Euclidean distance between points. By analytic continuation we deduce the expansion for all complex values of $s$. The Riemann zeta function plays an essential role in this asymptotic expansion.

Analytic high-beta tokamak equilibria with poloidal-sonic flow

A reduced set of equations for high-beta tokamak equilibria with flow comparable to the poloidal sound velocity is solved analytically. The solution includes higher-order terms of asymptotic expansions in terms of the inverse aspect ratio and indicates the modification of the magnetic structure and the departure of the pressure surfaces from the magnetic surfaces by sub- or super-poloidal-sonic flows. The analytical representation for the shift of the magnetic axis from the geometrical axis (the Shafranov shift) and that of the pressure maximum and the equilibrium beta limit are also obtained. The Shafranov shift is enhanced by a slightly super-poloidal-sonic flow and it produces a forbidden region of equilibrium by the poloidal-sonic flow. The physical mechanism of the shift of the pressure maximum from the magnetic axis due to the poloidal-sonic flow is discussed in analogy to those of the geodesic acoustic mode and the slow magnetosonic wave.

Asymptotic expansions for Yosida approximations of semigroups

In this paper we provide asymptotic expansions for Yosida approximations of contraction semigroups. We also obtain optimal bounds for convergencerate and remainder terms of asymptotic expansions. We use a method introduced in [2] for analysis of errors in Central Limit Theorem and in approximations by accompanying laws. This method was applied in [3] to obtain optimal convergence rates in some approximation formulas for operators and in [10] to obtain asymtotic expansions and optimal error bounds for Euler’s approximations of semigroups.

Numerical simulation of a thermoacoustic wave amplification

We performed a numerical study of the thermal and physical phenomena occurring in thermoacoustic wave generators. The goal of the simulation is to predict the amplification due to thermoacoustics of a wave initially of small amplitude. Therefore, we focus on the stack and the two heat exchangers, which we call the active cell, which is acoustically compact. The resonator area is split into two parts: the active cell, in which heat transfer takes place, and a resonator, in which the flow is acoustic. The flow in the two‐dimensional active cell can be approximated as a low Mach number viscous, conductive flow, subjected to spatially uniform pressure fluctuations. This model is formally derived using asymptotic expansions in terms of Mach number. The focus here is heat transfer between two successive stack plates. The two‐dimensional time‐dependent problem resulting from this model is solved numerically. Outside the active cell, flow in the resonator is described by a reversible acoustic one‐dimensional model. The coupling between the two zones is obtained by matching the velocity at the interface. The acoustics in the resonance tube can be solved using the d'Alembert solution, relating velocities at the interface to velocity values at an earlier time.

High-Beta Axisymmetric Equilibria with Flow in Reduced Single-Fluid and Two-Fluid Models

Reduced single-fluid and two-fluid equations for axisymmetric toroidal equilibria of high-beta plasmas with flow are derived by using asymptotic expansions in terms of the inverse aspect ratio. Two different orderings for the flow velocity, comparable to the poloidal Alfven velocity and comparable to the poloidal sound velocity, are considered. For a poloidal-Alfvenic flow, the two-fluid equilibrium equations with hot ion effects are shown to have a singularity that is shifted by the gyroviscous cancellation from the Alfven singularity found in single- fluid magnetohydrodynamics (MHD) when the poloidal flow velocity equals the poloidal Alfven velocity. For a poloidal-sonic flow, a reduced single-fluid model is used to derive a set of equilibrium equations that includes higher-order terms. The singularity at a poloidal flow velocity equal to the poloidal sound velocity is recovered in the higher order equations. c

Graph representation for asymptotic expansion in homogenisation of nonlinear first-order equations

Homogenisation of a linear transport equation leads to an integro-differential equation with the differential part of the same type as the starting equation. The (non-periodic) homogenisation of semilinear transport equations is open. In order to pinpoint technical difficulties, as a first step in that direction, following the approach of Tartar we consider an ordinary differential equation with an oscillating coefficient a $$\left\{\begin{array}{ll} u' +au^2 &=f \\ u(0) &=v \end{array}\right. $$ instead, and expand the solution in terms of a small parameter (the amplitude of oscillations in a). The crucial observation we made is a correspondence between multiple integrals representing the terms in asymptotic expansion of the solution and certain graphs, which allows easy manipulation of otherwise highly complicated expressions, and leads to efficient computation of the terms in expansion.