A Method for Incorporating Transcendentally Small Terms into the Method of Matched Asymptotic Expansions

The essential ideas behind a method for incorporating exponentially small terms into the method of matched asymptotic expansions are demonstrated using an Ackerberg–O'Malley resonance problem and a spurious solutions problem of Carrier and Pearson. One begins with the application of the standard method of matched asymptotic expansions to obtain at least the leading terms in outer and inner (Poincaré-type) expansions; some, although not all, matching can be carried out at this stage. This is followed by the introduction of supplementary expansions whose gauge functions are transcendentally small compared to those in the standard expansions. Analysis of terms in these expansions allows the matching to be completed. Furthermore, the method allows for the inclusion of globally valid transcendentally small contributions to the asymptotic solution; it is well known that such terms may be numerically significant.     

一类奇摄动非线性激波问题

Using the method of matched asymptotic expansions,the shock solutions for a class of singularly perturbed nonlinear problems are discussed.The relation of the shock solutions and their boundary conditions is obtained.And the known results are generalized.     

On Spurious Solutions of Singular Perturbation Problems

A study is made of several nonlinear boundary-value problems of singular perturbation type for which a straightforward application of boundary-layer theory leads to spurious solutions. It is shown that these problems can be treated successfully by a slight modification of the method of matched asymptotic expansions. The analysis leads to several novel features which are not present in routine singular perturbation problems.     

Matched Asymptotic Expansions in the Two-body Problem with Quick Loss of Mass

In studying models for the two-body problem with quick lossof mass a boundary layer problem arises for a third-order systemof non-linear ordinary differential equations. The models areidentified by a real parameter n with n ? 1. It turns out thatfor n = 1 asymptotic approximations of the solutions can beobtained by applying the method of matched asymptotic expansionsaccouonding to Vasil'eva or a multiple time scales method developedby O'Malley. For n> 1 these methods break down and it isshown that this is due to the occurrence of "unexpected" orderfunctions in the asymptotic expansions. The expansions for n> 1 are obtained by constructing an inner and outer expansionof the solution and matching these by the process of takingintermediate limits. The asymptotic validity of the matched expansions is provedby using an iteration technique; the proof is constructive sothat it provides us at the same time with an alternative wayof constructing approximations without using a matching technique.     

A Variational Approach to Nonlinear Singularly Perturbed Boundary-Value Problems

Carrier and Pearson introduced a nonlinear singularly perturbed boundary value problem that has served as a paradigm for problems where the method of matched asymptotic expansions (MAE) apparently fails. The “failure” of MAE is its inability to select the location of possible internal layers, though their structure is determined. Thus, a straightforward application of MAE leaves the positions of any internal layers arbitrary, though the asymptotic expansion of the exact solution to the problem exhibits internal layers only at specific locations. For this reason the solutions produced by MAE have been referred to as spurious solutions. We resolve the question of finding the positions of the interior layers by employing the variational approach of Grasman and Matkowsky. In addition, we show that this method tells how solutions bifurcate as the boundary values are varied, and give an alternative motivation for the variational approach via Newton”s method.     

Asymptotic Solutions for Australian Options with Low Volatility

Abstract

In this paper we derive asymptotic expansions for Australian options in the case of low volatility using the method of matched asymptotics. The expansion is performed on a volatility scaled parameter. We obtain a solution that is of up to the third order. In case that there is no drift in the underlying, the solution provided is in closed form, for a non-zero drift, all except one of the components of the solutions are in closed form. Additionally, we show that in some non-zero drift cases, the solution can be further simplified and in fact written in closed form as well. Numerical experiments show that the asymptotic solutions derived here are quite accurate for low volatility.     

The non-relativistic limit of Euler–Maxwell equations for two-fluid plasma

This paper is concerned with two-fluid time-dependent non-isentropic Euler–Maxwell equations in a torus for plasmas or semiconductors. By using the method of formal asymptotic expansions, we analyze the non-relativistic limit for periodic problems with the prepared initial data. It is shown that the small parameter problems have unique solutions existing in the finite time interval where the corresponding limit problems (compressible Euler–Poisson equations) have smooth solutions. Moreover, the formal limit is rigorously justified by an iterative scheme and an analysis of asymptotic expansions up to any order.     

Self-adjoint Extensions for the Neumann Laplacian and Applications

A new technique is proposed for the analysis of shape optimization problems. The technique uses the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains. The asymptotics of solutions are derived in the framework of compound and matched asymptotics expansions. The analysis involves the so-called interior topology variations. The asymptotic expansions are derived for a model problem, however the technique applies to general elliptic boundary value problems. The self-adjoint extensions of elliptic operators and the weighted spaces with detached asymptotics are exploited for the modelling of problems with small defects in geometrical domains, The error estimates for proposed approximations of shape functionals are provided.     

The vanishing viscosity method for the sensitivity analysis of an optimal control problem of conservation laws in the presence of shocks

In this article we study, by the vanishing viscosity method, the sensitivity analysis of an optimal control problem for 1-D scalar conservation laws in the presence of shocks. It is reduced to investigate the vanishing viscosity limit for the nonlinear conservation law, the corresponding linearized equation and its adjoint equation, respectively. We employ the method of matched asymptotic expansions to construct approximate solutions to those equations. It is then proved that the approximate solutions, respectively, satisfy those viscous equations in the asymptotic sense, and converge to the solutions of the corresponding inviscid problems with certain convergent rates. A new equation for the variation of shock positions is derived. It is also discussed how to identify descent directions to find the minimizer of the viscous optimal control problem in the quasi-shock case.     

Quasi-classical asymptotics of solutions to the matrix factorization problem with quadratically oscillating off-diagonal elements

The paper investigates the asymptotic behavior of solutions to the 2 × 2 matrix factorization (Riemann-Hilbert) problem with rapidly oscillating off-diagonal elements and quadratic phase function. A new approach to study such problems based on the ideas of the stationary phase method and M. G. Krein’s theory is proposed. The problem is model for investigating the asymptotic behavior of solutions to factorization problems with several turning points. Power-order complete asymptotic expansions for solutions to the problem under consideration are found. These asymptotics are used to construct asymptotics for solutions to the Cauchy problem for the nonlinear Schrödinger equation at large times.     

一类具非线性边界条件的高阶方程的奇摄动问题

通过引入伸展变量和非常规的渐近序列{∈}）,运用合成展开法,对一类具非线性边界条件的非线性高阶微分方程的奇摄动问题构造了形式渐近解,再运用微分不等式理论证明了原问题解的存在性及所得渐近近似式的一致有效性．     

A method for constructing asymptotic expansions of bisingularly perturbed problems

In this paper we propose an analog of the method of boundary functions for constructing uniform asymptotic expansions of solutions to bisingularly perturbed problems. With the help of this method we construct uniform asymptotic expansions of solutions to the Dirichlet problem for bisingularly perturbed ordinary differential equations and elliptic equations of the second order. By the use of the maximum principle we obtain estimates for the remainder terms.     

Eliminating Indeterminacy in Singularly Perturbed Boundary Value Problems with Translation Invariant Potentials

Solutions exhibiting an internal layer structure are constructed for a class of nonlinear singularly perturbed boundary value problems with translation invariant potentials. For these problems, a routine application of the method of matched asymptotic expansions fails to determine the locations of the internal layer positions. To overcome this difficulty, we present an analytical method that is motivated by the work of Kath, Knessl and Matkowsky [4]. To construct a solution having n internal layers, we first linearize the boundary value problem about the composite expansion provided by the method of matched asymptotic expansions. The eigenvalue problem associated with the homogeneous form of this linearization is shown to have n exponentially small eigenvalues. The condition that the solution to the linearized problem has no component in the subspace spanned by the eigenfunctions corresponding to these exponentially small eigenvalues determines the internal layer positions. These “near” solvability conditions yield algebraic equations for the internal layer positions, which are analyzed for various classes of nonlinearities.     

Singular perturbations for the exterior three-dimensional slow viscous flow problem

In this paper the rigorous justification of the formal asymptotic expansions constructed by the method of matched inner and outer expansions is established for the three-dimensional steady flow of a viscous, incompressible fluid past an arbitrary obstacle. The justification is based on the series representation of the solution to the Navier-Stokes equations due to Finn, and it involves the reductions of various exterior boundary value problems for the Stokes and Oseen equations to boundary integral equations of the first kind from which existence as well as asymptotic error estimates for the solutions are deduced. In particular, it is shown that the force exerted on the obstacle by the fluid admits the asymptotic representation F = A₀ + A₁Re + O(Re² ln Re⁻¹) as the Reynolds number Re → 0⁺, where the vectors A₀ and A₁ can be obtained from the method of matched inner and outer expansions.     

奇摄动半线性Robin问题的多重解

研究了一类奇摄动半线性Robin问题.在适当的条件下,分析了该问题出现多重解现象.利用合成展开法构造出问题的形式渐近解,并应用微分不等式理论证明了解的存在性以及当ε→0时解的渐近性质.     

一类非线性奇摄动问题的匹配解法

利用匹配渐近展开法,讨论了一类非线性奇摄动问题的解,得出了奇摄动边值问题的零次渐近展开式.     

Matching and Singularity Distributions in Inviscid Flow

This paper describes a technique for obtaining explicit approximatesolutions for some inviscid-flow problems. Using simple ideasfrom the theories of matched asymptotic expansions and generalizedfunctions, solutions can be obtained for problems arising insuch areas as submarine hydrodynamics and aerodynamics. We illustratethe ideas by considering the flows induced by the arbitrarymotion of (i) a sphere or translating slender spheroid deeplysubmerged beneath a free surface and (ii) a wing of high aspectratio whose chord is in a fixed direction. In each case, explicitlift and drag formulae are written down.     

Singular Perturbations of Limit Points with Application to Tubular Chemical Reactors

Singular perturbation techniques are used to study solutions of certain nonlinear boundary-value problems defined on domains with a circular hole of radius ε, in the limit ε → 0. Asymptotic expansions are constructed to describe the behavior of solutions at and near simple and double limit points (cusps). In particular, the behavior of axisymmetric solutions in an annular domain at limit points is investigated. The results are applied to two model problems arising in chemical-reactor theory. The asymptotic analysis predicts a surprisingly large sensitivity of limit points to the ε-domain perturbation considered here.     

Closure method and asymptotic expansions for linear stochastic systems

A closure procedure for the hierarchy of moment equations related to linear systems of ordinary differential equations with a random parametric excitation is introduced. A generalization of Pringsheim's theorem for continued fractions is used in a proof of the procedure convergence. The boundary function method for singular perturbation problems is applied to obtain asymptotic expansions for the moments of the solutions of such systems.