Terms Of Asymptotic Expansion Research Articles

Analysis of boundary layer control by heat transfer strips using an asymptotic approach to the PSE

The effect of heating strips on the stability of boundary layer flow over a flat plate is investigated. Heating strips alter the flow stability through the temperature dependence of the fluid properties. A stability study is carried out using the parabolized stability equations (PSE) that calculates the effects of temperature dependent fluid properties in terms of asymptotic expansions based on the total heat input. The leading order influence is obtained as a general result and, for the particular Prandtl number taken, is independent of any special set of property laws. In a fluid for which the intrinsic viscosity increases with temperature and the density decreases with temperature (such as air) the results show that the optimal location for a heating strip to stabilise the flow is upstream of the neutral point. The optimal location moves further upstream as the total heat input level is increased. For a given heat input widening the heating strip further stabilises the flow. Finally, the potential of the asymptotic method as a tool for further analysis of the flow is discussed.

On diffusion theory in turbulence

The Fokker-Planck equation for the probability density of fluid particle position in inhomogeneous unsteady turbulent flow is derived. The equation is obtained starting from the general kinematic relationship between velocity and displacement of a fluid particle and applying exact asymptotic analysis. For (almost) incompressible flow the equation reduces to the convection diffusion equation and the equation pertaining to the scalar gradient hypothesis. In this way the connection is established with eddy diffusivity models, widely used in numerical codes of computational fluid dynamics. It is further shown that within the accuracy of the approximation scheme of the diffusion limit, diffusion constants can equally be based on coarse-grained Lagrangian statistics as defined by Kolmogorov or on Eulerian statistics in a frame that moves with the mean Eulerian velocity as proposed by Burgers. The results presented for diffusion theory are the leading terms of asymptotic expansions. Truncated terms are higher-order spatial derivatives of the probability density or of the scalar mean value with coefficients based on cumulants higher than second order of fluid velocities and their derivatives. The magnitude of these terms has been assessed by employing scaling rules of turbulent flows in pipes and channels, turbulent boundary layers, turbulent jets, wakes and mixing layers, grid turbulence, convective layers and canopy turbulence. It reveals that a true diffusion limit does not exist. Although truncated terms can be of limited magnitude, a limit process by which these terms become vanishingly small and by which the diffusion approximation would become exact does not occur for any of the cases of turbulent flow considered. Applying the concepts of diffusion theory resorts to employing approximate methods of analysis.

Higher-order terms in asymptotic expansion for information loss in quantized random processes

Uniform quantization of random vectors onto e-grids eℤ n is considered. Higherorder terms in asymptotic expansions for the entropy of the e-quantized random vector and for the loss of the mutual information between two random vectors under such quantization as e→0+are obtained. The coefficients in these asymptotics are explicitly calculated for Gaussian distributed vectors. Taken for initial segments of stationary Gaussian sequences, these factors have limit average values per unit of time. For such sequences governed by state-space equations, computation of these average values is reduced to solutions of algebraic matrix Riccati and Lyapunov equations.

On Accuracy of Improved χ 2-Approximations

Abstract For a statistic S whose distribution can be approximated by χ 2-distributions, there is a considerable interest in constructing improved χ 2-approximations. A typical approach is to consider a transformation T = T(S) based on the Bartlett correction or a Bartlett type correction. In this paper we consider two cases in which S is expressed as a scale mixture of a χ 2-variate or the distribution of S allows an asymptotic expansion in terms of χ 2-distributions. For these statistics, we give sufficient conditions for T to have an improved χ 2-approximation. Furthermore, we present a method for obtaining its error bound.

Algebraic methods to compute Mathieu functions

The standard form of the Mathieu differential equation is y''(z) + (a-2 qcos 2z) y(z) = 0, where a is the characteristic number and q is a real parameter. The most useful solution forms are given in terms of expansions for either small or large values of q. In this paper we obtain closed formulae for the generic term of expansions of Mathieu functions in the following cases: (1) standard series expansion for small q; (2) Fourier series expansion for small q; (3) asymptotic expansion in terms of trigonometric functions for large q; and (4) asymptotic expansion in terms of parabolic cylinder functions for large q. We also obtain closed formulae for the generic term of expansions of characteristic numbers and normalization formulae for small and large q. Using these formulae one can efficiently generate high-order expansions that can be used for implementation of the algebraic aspects of Mathieu functions in computer algebra systems. These formulae also provide alternative methods for numerical evaluation of Mathieu functions.

Low-frequency scattering of acoustic waves by a bounded rough surface in a half-plane.

The problem of the scattering of harmonic plane waves by a rough half-plane is studied here. The surface roughness is finite. The slope of the irregularity is taken as arbitrary. Two boundary conditions are considered, those of Dirichlet and Neumann. An asymptotic solution is obtained, when the wavelength lambda of the incident wave is much larger than the characteristic length of the roughness iota, by means of the method of matched asymptotic expansions in terms of the small parameter epsilon= 2piiota/lambda. For the Dirichlet problem, the solution of the near and far fields is obtained up to O(epsilon2). The far field solution is given in terms of a coefficient that have a simple explicit expression, which also appears in the corresponding solution to the Neumann problem, already solved. Also the scattering cross section is given by simple formulas to O(epsilon3). It is noted that, for the Dirichlet problem, the leading term is of order epsilon3 which, by contrast, is different from that of the circular cylinder in full space, that is, of order epsilon(-1) (log epsilon)(-2). Some examples display the simplicity of the general results based on conformal mapping, which involve arcs of circle, polygonal lines, surface cracks and the like.

On the Distribution Density of the First Exit Point of a Diffusion Process form a Small Neighborhood of Its Initial Position

Considering a diffusion process in a d-dimensional space, we study the distribution of the first exit point of a process from a small neighborhood of its initial state. A weak asymptotic expansion in terms of the small scale parameter is obtained for the density of the distribution. In the case of spherical neighborhoods, simple formulas are deduced for the first three coefficients of the expansion, reflecting the probabilistic sense of the coefficients of an elliptic partial differential equation.

Singularities of Ionization-Field Instability and Structurization of Plasma of High-Pressure Non-Self-Maintained Free-Localized Discharge in the Field of a Quasi-Monochromatic Microwave

A self-consistent set of the equations for the electromagnetic field and balance of plasma concentration in view of ambipolar diffusion and the equation for the fast oscillating component of electron motion in a variable field are used to investigate the linear stage of ionization-field instability of the plasma of high-pressure non-self-maintained discharge in the approximation of unbounded plasma in a plane quasi-monochromatic microwave packet. Green‘s function is constructed. In the simplest one-dimensional formulation of the problem, assuming that all parameters of the plasma and field vary only in the direction of propagation of the wave packet, the saddle point approximation is used to derive the dominant terms of asymptotic expansion of Green‘s function in two limiting cases; namely, for t→ ∞, |z– z0| → 0, and for t≥ 0, |z– z0 | → ∞. It is demonstrated that, contrary to the existing notion of insignificant importance of ionization-field instability in a high-pressure plasma, this instability may produce a high rate of development of perturbations in such a plasma and have a decisive effect on the structure of plasma inhomogeneity being formed. A method is suggested of stabilizing ionization-field instability of a microwave plasma by selecting a proper shape of the wave packet, including an increase in the effective width of the frequency band emitted by the source.

Diffraction by a line of curvature jump (a special case)

We consider the problem of diffraction of a short-wave field by a cylindrical body with the boundary that consists of a half plane and a convex-cylindrical surface jointed together along a straight line. The main feature of the problem under consideration is the jump of curvature of the boundary on the line. The problems with the Dirichlet, Neumann, and impedance boundary conditions are considered. The main terms of asymptotic expansion of wave fields for the problems are constructed.

Filtration combustion of a gas in a semiinfinite porous medium

An asymptotic analysis of the combustion of a gas moving in a semiinfinite porous medium is performed for large values of the Zel’dovich parameter. The case of a highly porous medium with high gas permeability is considered. The basic terms of asymptotic expansions of the main parameters of the process for combustion and separation regimes are obtained by the method of joined asymptotic expansions. The effect of the rate of motion of the gas and heat transfer from the surface of the skeleton into the ambient medium on the combustion parameters is analyzed. The critical conditions of failure of stationary combustion at the external surface of the layer and the conditions of transition of the process to the separation and induction regimes are determined.

Polyhomogeneity and zero-rest-mass fields with applications to Newman-Penrose constants

A discussion of polyhomogeneity (asymptotic expansions in terms of 1/r and lnr) for zero-rest-mass fields and gravity and its relation with the Newman-Penrose (NP) constants is given. It is shown that for spin-s zero-rest-mass fields propagating on Minkowski spacetime, the logarithmic terms in the asymptotic expansion appear naturally if the field does not obey the `peeling theorem'. The terms that give rise to the slower fall-off admit a natural interpretation in terms of an advanced field. The connection between such fields and the NP constants is also discussed. The case when the background spacetime is curved and polyhomogeneous (in general) is considered. The free fields have to be polyhomogeneous, but the logarithmic terms due to the connection appear at higher powers of 1/r. In the case of gravity, it is shown that it is possible to define a new auxiliary field, regular at null infinity, and containing some relevant information on the asymptotic behaviour of the spacetime. This auxiliary zero-rest-mass field `evaluated at future infinity (i+)' yields the logarithmic NP constants.

A heat Kernel associated to Ramanujan's tau function

We prove a conjecture of Zagier, that the inverse Mellin transform of the symmetric square L-function attached to Ramanujan's tau function has an asymptotic expansion in terms of the zeros of the Riemann function.

Allowance for mutual effect of ‘Hot spots’ in solution of problems of site thermal explosion

Local effect of intensification of substance heating depending on the distance between ‘hot spots’ is studied within the framework of a site thermal explosion in the case of a zeroth-order reaction. A combination of the values of the Arrhenius number Ar, Frank-Kamenetskii criterion Fk, distance between ‘hot spots’ a, and their width w at which two nearest points give a nonlinear effect of local intensification of the total heating of a substance is found. The method is based on the use of an approximate analytical solution of the corresponding Cauchy problem which is presented in the form of asymptotic expansion in terms of Poincare.

Self-similar crossover in statistical physics

An analytical method is advanced for constructing interpolation formulae for complicated problems of statistical mechanics, in which just a few terms of asymptotic expansions are available. The method is based on the self-similar approximation theory, being its variant where control functions are defined from asymptotic crossover conditions. Several examples from statistical physics demonstrate that the suggested method results in rather simple and surprisingly accurate formulae.

A note on an asymptotic solution for the contact of two biphasic cartilage layers in a loaded synovial joint at rest

Retaining the first terms of asymptotic expansions and assuming zero gradient of the contact pressure at the contact edge in the perpendicular direction to the edge, Ateshian et al. (1994, J. Biomechanics 27, 1347–1360; 1992, Adv. Bioeng. ASME 22, 191–194) have presented an asymptotic axially symmetric and plane strain solutions to the contact problems of the two thin biphasic cartilage layers in the synovial joint (with the synovial fluid film in between already depleted), subjected to a sudden normal load. Both the immediate and an early time response to a step loading have been analysed. The present note shows that the contact width thus obtained immediately after the load application differs from the numerical values for a dry frictionless contact of incompressible single-phase elastic layers available in the literature. The difference increases with the decreasing contact width-to-layer thickness ratio. It is proposed to improve the above solutions by releasing zero pressure gradient condition at the contact edge and calculating the instantaneous contact widths from the equations proposed for the corresponding case by Matthewson (1981, J. Mech. Phys. Solids 29, 89–113) and Meijers (1967, Appl. Sci. Res. 18, 353–383) that approximate better the numerical values. The instantaneous rate of change in the contact width is then obtained for a step load varying in time.

Anisotropic particle spectra and momentum sharing in multicomponent collision cascades

Sigmund’s analytical sputtering theory has been used to explain various Monte Carlo simulations given by Urbassek and Conrad. It has been demonstrated that only Sigmund’s theory can predict all the simulation results. The present author has shown that all the transport equations describing the statistics of linear random collision cascades are suitable for a complete use of the exact scattering cross-section of power potential (V∝r−1/m) interaction collisions. Asymptotic expansions of an anisotropic particle flux (L=0,1) are derived here for an arbitrary multicomponent medium. Several new explicit expressions are further reported for various statistical distribution functions. Asymptotic formulae for energy and momentum sharing are presented. The correction terms in asymptotic expansions of isotropic solutions (L=0) are evaluated for a binary target by the use of both scattering cross-sections [dσ]p=C0dT/T and [dσ]z=σdT/Tm. It has been shown that [dσ]z greatly enlarges the range of validity of the asymptotic solutions as compared to [dσ]p. In some cases, [dσ]z can even yield exact solutions. As usual, the electronic stopping is ignored in the analysis.

Second-order expansion for the expected regret risk in classification of one-parametric distributions

The plug-in rule is used for the classification of random observations into one of two regular one-parametric distributions. The maximum likelihood estimates of unknown parameters obtained from the stratified training sample are used. The second-order asymptotic expansion in terms of the inverses of the training sample sizes is derived for the expected regret risk. The closed-form expressions of the expansion coefficients are applicable for the performance evaluation of the proposed classification rule.

Asymptotics of the Expected Posterior

Let (X1,...,X,...,Xn) be independently and identically distributed observations from an exponential family pθ equipped with a smooth prior density w on a real d-dimensional parameter θ. We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, θ0, admits an asymptotic expansion in terms of the Fisher information I(θ0), the prior w, and their first two derivatives. The leading term of the expansion is of the form nd/2c1(d, θ0) and the second order term is of the form n4/2-1c2(d, θ0>, w), with an error term that is o(nd/2-1). We identify the functions c1 and c2 explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at θ0. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.

Unified approach to crossover phenomena

A general analytical method is developed for describing crossover phenomena of arbitrary nature. The method is based on the algebraic self-similar renormalization of asymptotic series, with control functions defined by crossover conditions. The method can be employed for such difficult problems for which only a few terms of asymptotic expansions are available, and no other techniques are applicable. As an illustration, analytical solutions for several important physical problems are presented.

Precise estimates of the difference between the homogenized solution with its first corrector and the original One

The goal of this paper is to estimate the error done while approximating the solutions of an elliptic boundary problem with periodic structures by the two first asymptotic expansion terms in the case of non smooth data. After some preliminary technical results in Section 2, we obtain the convergence theorems in Section 3. The method used herein is inspired from [1] and from [2] where numerical scheme and differential equation solutions were studied with nonsmooth data Classically, weak convergence follows from the G — convergence as for instance in [3] or [4] and estimates for smooth data are obtained like for instance in [5] or [6]. We start in this paper by studying the case with some regularity assumptions on the data, in a second forthcoming paper we are studying the case with less regularity assumptions.