Theory Of Asymptotic Expansions Research Articles

Asymptotic profiles of zero points of solutions to the heat equation

In this paper, we consider the asymptotic profiles of zero points for the spatial variable of the solutions to the heat equation. By giving suitable conditions for the initial data, we prove the existence of zero points by extending the high-order asymptotic expansion theory for the heat equation. This reveals a previously unknown asymptotic profile of zero points diverging at O ( t ) O(t) . In a one-dimensional spatial case, we show the zero point’s second and third-order asymptotic profiles in a general situation. We also analyze a zero level set in high-dimensional spaces and obtain results that extend the results for the one-dimensional spatial case.

High order asymptotic expansion for Wiener functionals

Combining the Malliavin calculus with Fourier techniques, we develop a high-order asymptotic expansion theory for general Wiener functionals. Our method gives an expansion of the characteristic functional and of the local density of a Wiener functional up to an arbitrary order. The asymptotic expansion is distributional. Except for the non-degeneracy of the limit covariance matrix, we do not assume any condition of non-degeneracy of the Malliavin covariance like a non-degeneracy condition for temporally local characteristic functions so far assumed in the theory for mixing processes, that corresponds to the Cramér condition in the classical setting. Moreover, our method does not require the Markovian property used in the mixing approach. An application to the stochastic wave equation with space–time white noise is discusses.

A Unified Asymptotic Theory of Supersonic, Transonic, and Hypersonic Far Fields

The problems of steady, inviscid, isentropic, irrotational supersonic plane flow passing a body with a small thickness ratio was solved by the linearized theory, which is a first approximation at and near the surface but fails at far fields from the body. Such a problem with far fields was solved by W.D. Hayes’ “pseudo-transonic” nonlinear theory in 1954. This far field small disturbance theory is reexamined in this study first by using asymptotic expansion theory. A systematic approach is adopted to obtain the nonlinear Burgers’ equation for supersonic far fields. We also use the similarity method to solve this boundary value problem (BVP) of the inviscid Burgers’ equation and obtain the nonlinear flow patterns, including the jump condition for the shock wave. Secondly, the transonic and hypersonic far field equations were obtained from the supersonic Burgers’ equation by stretching the coordinate in the y direction and considering an expansion of the freestream Mach number in terms of the transonic and hypersonic similarity parameters. The mathematical structures of the far fields of the supersonic, transonic, and hypersonic flows are unified to be the same. The similar far field flow patterns including the shock positions of a parabolic airfoil for the supersonic, transonic, and hypersonic flow regimes are exemplified and discussed.

Asymptotic expansion and estimates of Wiener functionals

Asymptotic expansion of a variation with anticipative weights is derived by the theory of asymptotic expansion for Skorohod integrals having a mixed normal limit. The expansion formula is expressed with the quasi-torsion, quasi-tangent and other random symbols. To specify these random symbols, it is necessary to classify the level of the effect of each term appearing in the stochastic expansion of the variable in question. To solve this problem, we consider a class L of certain sequences (In)n∈N of Wiener functionals and give a systematic way of estimation of the order of (In)n∈N. Based on this method, we introduce a notion of exponent of the sequence (In)n∈N, and investigate the stability and contraction effect of the operators Dun and D on L, where un is the integrand of a Skorohod integral. After constructed these machineries, we derive asymptotic expansion of the variation having anticipative weights. An application to robust volatility estimation is mentioned.

Higher-order semi-rational solutions for the coupled complex modified Korteweg-de Vries equation

We explore the Darboux-dressing transformation of the coupled complex modified Korteweg-de Vries equation. Next, with the aid of an asymptotic expansion theory, we derive the concrete forms of three types of semi-rational solutions. In particular, the seed solution is related to the normalized distance and retarded time. Interestingly, we construct a kind of novel rogue wave called as curve rogue wave. More importantly, the kinetics of semi-rational solutions are discussed in detail. We hope that these results would shed more light on comprehending of the solutions occurring in multi-component coupled systems.

Vector breather waves and higher-order rouge waves to the coupled higher-order nonlinear Schrödinger equations

ABSTRACT We investigate the vector breather waves and higher-order rouge waves of the coupled higher-order nonlinear Schrödinger equations via the Darboux-dressing transformation. Firstly, based on the Darboux-dressing transformation, we introduce the asymptotic expansion theory, which allows the solution of the equation to be iterated through the same spectral parameters. Then, the matrix exponential function and the Taylor multi-series expansion method are used to derive the vector breather wave solutions and the higher-order rouge wave solutions. Moreover, we analyze the characteristic lines of the breather waves, which determine the dynamic behaviours of the breather waves. Finally, we give some conclusions of this work.

The radiated acoustic pressure and time scales of a spherical bubble

Numerical simulations of violent bubble dynamics are often associated with numerical instabilities at the end of collapse, when a shock wave is emitted. Based on the Keller–Miksis equation, we show that this is caused by two time scales associated with the phenomenon. Nonsingular equations are thus formed based on asymptotic expansion theory and the time derivatives of the bubble radius are shown to have algebraic singularities in the Mach number. The period of oscillation is shown to divide into two asymptotic layers: a long and short time scale. The short time scale, on which significant acoustic radiation is emitted from the bubble, has been determined to be , where c is the speed of sound in the liquid, the maximum bubble radius, ρ the liquid density, the hydrostatic pressure of the liquid, the vapour pressure of the liquid and κ the polytropic index of the bubble gas. Using the scalings for this short time scale, the radiated acoustic pressure scale has been deduced to be , where is the radial distance from the bubble centre to the point of measurement. The results are validated by comparison with experimental results.

Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equations

This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in [11]. We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential equations with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are coherent combinations of exponential, power and iterated logarithmic functions. We prove that any decaying solution admits an asymptotic expansion, as time tends to infinity, corresponding to the asymptotic structure of the forcing function. Moreover, these expansions can be generated by more than two base functions and go beyond the polynomial formulation imposed in previous work.

A hybrid reduced-order modeling technique for nonlinear structural dynamic simulation

Thin-walled structures are always subjected to a large range of extreme loading cases leading to obvious geometric nonlinearities in structural dynamic response, such as vibration with large amplitudes in aeronautics and astronautics field. Various dynamic reduced-order models have been investigated from detailed finite element models, to largely reduce the computational burden of the structural dynamic responses. However, to construct these low-order models, applying a series of nonlinear static simulations to the full-order model is necessary. This paper aims to develop a hybrid reduced-order modeling method by combining the dynamic and static reduced-order models together, to estimate the dynamic transient response caused by geometrically nonlinear finite element models. A few free-vibration modes of interest are selected to reduce basis vectors of dynamic reduced-order model. Based on Koiter asymptotic expansion theory, the constructed static reduced-order model is applied to the nonlinear static analyses. Therefore, not only does the proposed method make it possible to calculate the nonlinear dynamic response far more efficiently than full-order modeling methods, but computational burdens in construction of dynamic reduced-order model are also largely reduced compared to existing approaches. Various engineering numerical examples with hardening and/or softening nonlinearities are carefully tested to validate the good quality and efficiency of the proposed method.

Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos

We develop the asymptotic expansion theory for vector-valued sequences (FN)N≥1 of random variables in terms of the convergence of the Stein–Malliavin matrix associated with the sequence FN. Our approach combines the classical Fourier approach and the recent Stein–Malliavin theory. We find the second order term of the asymptotic expansion of the density of FN and we illustrate our results by several examples.

Asymptotic Behaviors of Wronskians and Finite Asymptotic Expansionsin the Real Domain - Part II: Mixed Scales and Exceptional Cases

In this second Part of our work we study the asymptotic behaviors of Wronskians involving both regularly- and rapidly-varying functions, Wronskians of slowly-varying functions and other special cases. The results are then applied to the theory of asymptotic expansions in the real domain.

Asymptotic Behaviors of Wronskians and Finite Asymptotic Expansions in the Real Domain - Part I: Scales of Regularly- or Rapidly-Varying Functions

In a previous series of papers we established a general theory of finite asymptotic expansions in the real domain for functions f of one real variable sufficiently-regular on a deleted neighborhood of a point x0 ∈ R, a theory based on the use of a uniquely-determined linear differential operator L associated to the given asymptotic scale and wherein various sets of asymptotic expansions are characterized by the convergence of improper integrals involving both the operator L applied to f and certain weight functions constructed by means of Wronskians of the given scale. Very special cases apart, Wronskians have quite complicated expressions and unrecognizable asymptotic behaviors; however in the present work, split in two parts, we highlight some approaches for determining the exact asymptotic behavior of a Wronskian when the involved functions are regularly- or rapidly-varying functions of higher order. This first part contains: (i) some preliminary results on the asymptotic behavior of a determinant whose entries are asymptotically equivalent to the entries of a Vandermonde determinant; (ii) the fundamental results about the asymptotic behaviors of Wronskians involving scales of functions all of which are either regularly (or, more generally, smoothly) varying or rapidly varying of a suitable higher order. A lot of examples and applications to the theory of asymptotic expansions in the real domain are given.

The Role of Asymptotic Mean in the Geometric Theory of Asymptotic Expansions in the Real Domain

We call “asymptotic mean” (at +∞) of a real-valued function the number, supposed to exist, , and highlight its role in the geometric theory of asymptotic expansions in the real domain of type (*) where the comparison functions , forming an asymptotic scale at +∞, belong to one of the three classes having a definite “type of variation” at +∞, slow, regular or rapid. For regularly varying comparison functions we can characterize the existence of an asymptotic expansion (*) by the nice property that a certain quantity F（t) has an asymptotic mean at +∞. This quantity is defined via a linear differential operator in f and admits of a remarkable geometric interpretation as it measures the ordinate of the point wherein that special curve , which has a contact of order n - 1 with the graph of f at the generic point t, intersects a fixed vertical line, say x = T. Sufficient or necessary conditions hold true for the other two classes. In this article we give results for two types of expansions already studied in our current development of a general theory of asymptotic expansions in the real domain, namely polynomial and two-term expansions.

Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-C: Constructive Algorithms for Canonical Factorizations and a Special Class of Asymptotic Scales

This part II-C of our work completes the factorizational theory of asymptotic expansions in the real domain. Here we present two algorithms for constructing canonical factorizations of a disconjugate operator starting from a basis of its kernel which forms a Chebyshev asymptotic scale at an endpoint. These algorithms arise quite naturally in our asymptotic context and prove very simple in special cases and/or for scales with a small numbers of terms. All the results in the three Parts of this work are well illustrated by a class of asymptotic scales featuring interesting properties. Examples and counterexamples complete the exposition.

Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-A: The Factorizational Theory for Chebyshev Asymptotic Scales

This paper, divided into three parts (Part II-A, Part II-B and Part II-C), contains the detailed factorizational theory of asymptotic expansions of type (?) , , , where the asymptotic scale , , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of . It follows two pre-viously published papers: the first, labelled as Part I, contains the complete (elementary but non-trivial) theory for ; the second is a survey highlighting only the main results without proofs. All the material appearing in §2 of the survey is here reproduced in an expanded form, as it contains all the preliminary formulas necessary to understand and prove the results. The remaining part of the survey—especially the heuristical considerations and consequent conjectures in §3—may serve as a good introduction to the complete theory.

Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-B: Solutions of Differential Inequalities and Asymptotic Admissibility of Standard Derivatives

Part II-B of our work continues the factorizational theory of asymptotic expansions of type (*) , , where the asymptotic scale , , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x0. The main result states that to each scale of this type it remains as-sociated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion (*), if valid, is automatically formally differentiable n ? 1 times in the two special senses characterized in Part II-A. A second result shows that formal applications of ordinary derivatives to an asymptotic expansion are rarely admissible and that they may also yield skew results even for scales of powers.

A note on approximate Bayesian credible sets based on modified loglikelihood ratios

Higher-order asymptotic arguments for a scalar parameter of interest have been widely investigated for Bayesian inference. In this paper the theory of asymptotic expansions is discussed for a vector parameter of interest. A modified loglikelihood ratio is suggested, which can be used to derive approximate Bayesian credible sets with accurate frequentist coverage. Three examples are illustrated.

Anisotropic dynamics of optical vortex-beam propagating in biaxial crystals: a numerical method based on asymptotic expansion

One difficulty of angular spectrum representation method in studying optical propagation inside anisotropic crystals is to calculate the double integrals containing highly oscillating functions. In this paper, we introduce an accuracy and numerically cheap method based on asymptotic expansion theory to overcome this difficulty, which therefore allows to compute optical fields with arbitrary incident beam and is not restricted to the paraxial limit. This numerical method is benchmarked against the analytical solutions in uniaxial crystals and excellent agreements between them are obtained. As an application, we adopt it to investigate the propagation of a Gaussian vortex-beam in a biaxial crystal. The general features of anisotropic dynamics and power conversion between field components are revealed. The numerical results is interpreted by making appropriate analytical approximation to the wave equations.

Influence of the non-singular stress on the crack extension and fatigue life

Abstract The complete elasticity stress field at a crack tip region can be presented by the sum of the singular stress and several non-singular stress terms according to the Williams asymptotic expansion theory. The non-singular stress has a non-negligible influence on the prediction of the crack extension direction and crack growth rate under the fatigue loading. A novel method combining the boundary element method and the singularity characteristic analysis is proposed here to evaluate the complete stress field at a crack tip region. In this new method, any non-singular stress term in the Williams series expansion can be evaluated according to the computational accuracy requirement. Then, a modified Paris law is introduced to predict the crack propagation under the mixed-mode loading for exploring the influence of the non-singular stress on the fatigue life duration. By comparing with the existed experimental results, the predicted crack fatigue life when the non-singular stress is taken into consideration is more accurate than the predicted ones only considering the singular stress.

Analytic theory of finite asymptotic expansions in the real domain. Part I: two-term expansions of differentiable functions

We establish a general analytic theory of asymptotic expansions of type 1 $$f(x) = a_1 \varphi _1 (x) + \cdots + a_n \varphi _n (x) + o(\varphi _n (x)) x \to x_0 ,$$ , where the given ordered n-tuple of real-valued functions (ϕ 1, ..., ϕ n ) forms an asymptotic scale at x 0 ∈ . By analytic theory, as opposed to the set of algebraic rules for manipulating finite asymptotic expansions, we mean sufficient and/or necessary conditions of general practical usefulness in order that (*) hold true. Our theory is concerned with functions which are differentiable (n − 1) or n times and the presented conditions involve integro-differential operators acting on f, ϕ 1, ..., ϕ n . We essentially use two approaches; one of them is based on canonical factorizations of nth-order disconjugate differential operators and gives conditions expressed as convergence of certain improper integrals, very useful for applications. The other approach starts from simple geometric considerations and gives conditions expressed as the existence of finite limits, as x → x 0, of certain Wronskian determinants constructed with f, ϕ 1, ..., ϕ n . There is a link between the two approaches and it turns out that some of the integral conditions found via the factorizational approach have geometric meanings. Our theory extends to more general expansions the theory of real-power asymptotic expansions thoroughly investigated in previous papers. In the first part of our work we study the case of two comparison functions ϕ 1, ϕ 2 because the pertinent theory requires a very limited theoretical background and completely parallels the theory of polynomial expansions.