Showing papers on "Method of matched asymptotic expansions published in 1977"

On the Uniform Asymptotic Stability of Certain Linear Nonautonomous Differential Equations

Abstract: In this paper we give a simple characterization of the uniform asymptotic stability of equations $\dot x = - P(t)x$ where $P(t)$ is a bounded piecewise continuous symmetric positive semi-definite matrix. In the course of developing this characterization, a new and general sufficient condition is given for uniform asymptotic stability in terms of Lyapunov functions. The stability of this type of equation has come up in various control theory contexts (identification, optimization and filtering).

Singular Perturbations of Bifurcations

Abstract: An asymptotic theory is presented to analyze perturbations of bifurcations of the solutions of nonlinear problems. The perturbations may result from imperfections, impurities, or other inhomogeneities in the corresponding physical problem. It is shown that for a wide class of problems the perturbations are singular. The method of matched asymptotic expansions is used to obtain asymptotic expansions of the solutions. Global representations of the solutions of the perturbed problem are obtained when the bifurcation solutions are known globally. This procedure also gives a quantitative method for analyzing singularities of nonlinear mappings and their unfoldings. Applications are given to a simple elasticity problem, and to nonlinear boundary value problems.

A General Theorem in the Theory of Asymptotic Expansions as Approximations to the Finite Sample Distributions of Econometric Estimators

Abstract: THERE HAS RECENTLY BEEN A GROWING INTEREST in the use of asymptotic series expansions of the Edgeworth type to approximate finite sample distributions in econometrics. Working in the framework of a conventional simultaneous equations model, a number of authors [1, 2, 6, 7, and 13] have derived such expansions for various single-equation estimators and Sargan [10] has considered the problem of developing an expansion of the distribution of the full information maximum likelihood estimator (FIML). In addition, Sargan [11] has recently established an important general theorem on the validity of Edgeworth expansions for sample distributions of statistics which can be represented as very general functions of sample data, imposing only weak conditions on the class of functions. This result covers a wide variety of econometric estimators and test statistics. Nevertheless, work in this field to date has been based on two limiting assumptions: normally distributed structural disturbances and nonrandom exogenous variables. The latter is particularly unfortunate since models in practice usually involve lagged variables in the regressor set. On the other hand, there is no reason in principle, at least, why valid expansions cannot be obtained in more general models. The present paper, therefore, is concerned with extending Sargan's approximation theorem in [11] to include such cases. The central result of the paper is stated and proved in Section 2. In Section 3 we provide some discussion of the theorem and its conditions and attempt to relate them to the contemporaneous work of Sargan in [12].

Theory of viscous transonic flow over airfoils at high Reynolds number

Abstract: This paper considers viscous flows with unseparated turbulent boundary layers over two-dimensional airfoils at transonic speeds. Conventional theoretical methods are based on boundary layer formulations which do not account for the effect of the curved wake and static pressure variations across the boundary layer in the trailing edge region. In this investigation an extended viscous theory is developed that accounts for both effects. The theory is based on a rational analysis of the strong turbulent interaction at airfoil trailing edges. The method of matched asymptotic expansions is employed to develop formal series solutions of the full Reynolds equations in the limit of Reynolds numbers tending to infinity. Procedures are developed for combining the local trailing edge solution with numerical methods for solving the full potential flow and boundary layer equations. Theoretical results indicate that conventional boundary layer methods account for only about 50% of the viscous effect on lift, the remaining contribution arising from wake curvature and normal pressure gradient effects.

Mode conversion and tunneling in an inhomogeneous plasma

Abstract: A theory is presented describing mode conversion and tunneling in a warm, collisionless, toroidal plasma near the first ion‐cyclotron harmonic frequency when the waves are propagating perpendicular to the static magnetic field. The coupling region may be characterized by a fourth‐order differential equation with one turning point. Employing the method of Laplace integrals and the method of matched asymptotic expansions, transmission, reflection, and conversion coefficients are calculated assuming a fast wave is incident from either side of the coupling region. For a fast wave incident from the low magnetic field side, transmission, reflection, and conversion occur, while for a fast wave incident from the high magnetic field side, only transmission and conversion occur with no reflection, independent of the thickness of the tunneling layer, which implies complete conversion of the fast wave to the slow wave as the tunneling layer becomes thick. The problem of cavity resonance is discussed.

A variational approach to singularly perturbed boundary value problems for ordinary and partial differential equations with turning points

Abstract: In studying singularity perturbed boundary value problems for second order linear differential equations with a simple turning point, R. C. Ackerberg and R. E. O’Malley [2] pointed out a number of interesting anomalies. In particular they observed that standard application of the method of matched asymptotic expansions did not suffice to uniquely determine the asymptotic expansion of the solution. They further noted that the standard construction in that method led to boundary layers at both ends of the interval, even for problems where in fact there is only one boundary layer located at one or other of the endpoints. In this paper we employ a variational formulation of the problem to resolve the question of the number and location of the boundary layers as well as to uniquely determine the asymptotic expansion of the solution. The results are then extended to analogous problems for partial differential equations, and new results are obtained for a class of singularly perturbed elliptic boundary value pro...

Diffraction of plane elastic waves by ellipsoidal inclusions

Abstract: The method of matched asymptotic expansions has been used to obtain a low‐frequency solution for the diffraction of a plane compressional or shear wave by an elastic ellipsoidal inclusion. Attention has been focused on the scattered farfield, angular dependences of which are illustrated graphically for ellipsoidal cavities of different axes ratios.

Asymptotic solutions of nonlinear wave equations using the methods of averaging and two-timing

Abstract: The method of averaging and the two-time procedure are outlined for a class of hyperbolic second-order partial differential equations with small nonlinearities. It is shown that they both lead to the same integro-differential equation for the lowest-order approximation to the solution. For the special case of the nonlinear wave equation, this lowest-order solution consists of the superposition of two modulated travelling waves, and separate integro-differential equations are derived for the amplitudes of these two waves. As an example, the wave equation with Van der Pol type of nonlinearity is considered.

Singular Perturbation Techniques: A Comparison of the Method of Matched Asymptotic Expansions with that of Multiple Scales

Abstract: A comparison is made between the uniformly valid asymptotic representations which can be developed for the solution of a singular perturbation boundary value problem involving a linear second order differential equation by using both the technique of matched asymptotic expansions and the method of multiple scales. Next, there is a discussion of some of the subtle features as well as the relative advantages, limitations, logical extensions, and typical applications of these two methods of obtaining uniformly valid asymptotic representations when applied to slightly more general singular perturbation problems which arise from investigations of various phenomena in the natural sciences. Finally, a parameter identification example relevant to biological population dynamics is presented to illustrate the fact that the mere knowledge of these singular perturbation techniques can be a powerful analytical tool.

Turbulent flow over a wavy boundary

Abstract: The linearized, two-dimensional flow of an incompressible fully turbulent fluid over a sinusoidal boundary is solved using the method of matched asymptotic expansions in the limit of vanishing skin-friction. A phenomenological turbulence model due to Saffman (1970, 1974) is utilized to incorporate the effects of the wavy boundary on the turbulence structure. Arbitrary lowest-order wave speed is allowed in order to consider both the stationary wavy wall, and the water wave moving with arbitrary positive or negative velocity. Good agreement is found with measured tangential velocity profiles and surface normal stress coefficients. The phase shift of the surface normal stress exhibits correct qualitative behavior with both positive and negative wave speeds, although predicted values are low.

Pulsatile flow in a tube with wall injection and suction

Abstract: This paper studies similarity solutions for pulsatile flow in a tube with wall injection and suction. The Navier-Stokes equations are reduced to a system of three ordinary differential equations. Two of the equations represent the effects of suction and injection on the steady flow while the third represents the effects of suction and injection on pulsatile flow. Since the equations for steady flow have been studied previously, the analysis centers on the third equation. This equation is solved numerically and by the method of matched asymptotic expansions. The exact numerical solutions compare well with the asymptotic solutions. The effects of suction and injection on pulsatile flow are the following: a) Small values of suction can cause a resonance-like effect for low frequency pulsatile flow. b) The annular effect still occurs but for large injection or suction the frequency at which this effect becomes dominant depends on the cross-flow Reynolds number. c) The maximum shear stress at the wall is decreased by injection, but may be increased or decreased by suction.

Forces and moments on a ship moving in a shallow channel

Abstract: The method of matched asymptotic expansions is applied to the problem of a ship moving with constant velocity in a channel of rectangular cross section. The depth of the channel is assumed small so that a linearized shallow-water solution is developed. The ship is assumed to be slender and is replaced in the outer region by a distribution of sources and vortices. As in earlier theories for slender bodies translating in shallow water, the source strength is proportional to the change in cross-sectional area of the ship. An integral equation must be solved to find the vortex strength. Using the source and vortex strengths, expressions are developed for the side force, yaw moment, sinkage, and trim. Numerical results are presented which show good agreement with experimental results.

Radiation from the open end of a cylindrical or conical pipe and scattering from the end of a rod or slab

Abstract: The radiation of sound through the open end of a cylindrical or conical pipe of any cross section, or through a hole in a plane wall, is analyzed theoretically. The scattering of a sound wave by the end of a rod or slab is also treated. Only the case in which the wavelength is large compared with a typical radial dimension of the opening or of the end is considered. The method of matched asymptotic expansions is employed. Results on end corrections and reflection coefficients previously obtained by Helmholtz (1860), Rayleigh (1945), and Bazer and Karp (1954), using intuitive arguments, are recovered and verified. Agreement is found with the exact results of Levine and Schwinger (1948) and Vainstein (1948), as well as with the small radial-dimension/wavelength results of Lesser and Lewis (1972), in the cases they treated. In addition various new results are obtained.

Elastico-viscous flow around a circular cylinder executing small amplitude, high frequency oscillations

Abstract: An analysis is presented of incompressible elastico-viscous flow around a circular cylinder executing small amplitude, high frequency oscillations. A description of the flow field, and its dependence on elasticity, is obtained by means of the method of matched asymptotic expansions. It is shown that the presence of elasticity can result in a reversal of the induced steady streaming, a result in qualitative agreement with recently reported experimental work.

Magnetohydrodynamic flow past a circular cylinder

Abstract: This paper deals with the slow-flow problem of an incompressible, viscous, electrically conducting fluid past a circular cylinder in an aligned magnetic field. The solutions for the velocity and magnetic fields are sought by the method of matched asymptotic expansions under the assumptions:R, Rm≪M≪1, such thatR=O(Rm). The influence ofR andRm on this solution is studied toO(R/(logM)2) andO(R/logM), respectively.

The scattering of electromagnetic surface waves by circular or elliptic cylinders

Abstract: A transverse magnetic surface wave is normally incident upon an infinitely long elliptic cylinder which may be a perfect conductor or a dielectric. The possibility of zero reflexion is investigated by the method of matched asymptotic expansions when the width of the cylinder is small compared with the other length scales of the problem. The radiated wave and the reflected and transmitted surface waves are calculated for various combinations of geometrical and material properties.

On the integrability of solutions of perturbed non-linear differential equations

Abstract: Sufficient conditions are given to insure that all solutions of a perturbed non-linear second-order differential equation have certain integrability properties. In addition, some continuability and boundedness results are given for solutions of this equation.

Application of Matched Expansion Methods to Problems in Acoustics.

Abstract: : This report describes how the method of Matched Asymptotic Expansions (MAE) can be used safely and systematically (1) to indicate the appropriate from taken by the inner (near field) and outer (wave field) series and (2) to determine all unknown functions and constants appearing in both series by matching the series according to a clearcut rule. These points are illustrated by detailed study of several very simple problems in low-frequency acoustic scattering problems which serve to demonstrate that physical arguments are unreliable in these problems and that they are no substitute for the unambiguous matching rule. Two-dimensional scattering problems are used to introduce logarithmic gauge functions; it is shown that the matching rule can easily accommodate these functions and moreover, that insistence upon satisfaction of the matching rule can in some cases be used to greatly improve the rapidly of convergence of series involving logarithmic functions. The report emphasizes the very widespread applicability of the MAE method to problem in classical and modern, linear and nonlinear acoustics and related fields.

Asymptotic solution for coupled-channel Coulomb excitation

Abstract: A new method for solving the coupled-channel equations for inelastic scattering of charged particles is described. It is based on a new type of asymptotic expansion of the inhomogeneous problem. It is shown that in cases where the asymptotic series is inadequate, it is possible to use an analytic continuation in the form of continued fraction.

Asymptotic expansions for estimates of a signal parameter in gaussian white noise

Abstract: We consider asymptotic expansions of estimates of an unknown parameter based on observation of a diffusion process : Bibliography: 11 titles.

An estimate for the small parameter in the asymptotic analysis of non-linear systems by the method of averaging

Abstract: The Method of Averaging is an asymptotic method that can be used to obtain approximate solutions for many parameter dependent non-linear systems. The resulting approximate solutions are as accurate as desired provided the system parameter is sufficiently small. In this paper, a general technique is developed to obtain a relationship between the magnitude of the small parameter and the permissible error in the approximate solution. The technique is then demonstrated by its application to two examples.

Heave and pitch motions in shallow water including the effect of forward speed

Abstract: The problem of small heave and pitch motions of a slender ship in shallow water including the effect of forward speed is analysed using the method of matched asymptotic expansions. Formulae valid to first order in slenderness are given for the added-mass and damping coefficients in terms of the frequency and subcritical Froude number.