Showing papers on "Method of matched asymptotic expansions published in 1992"

Introduction to the general theory of singular perturbations

Abstract: Introduction. General survey Part I.Asymptotic integration of various problems for ordinary differential euqaitons:: The method of regularization of singular perturbations Asymptotic integratin of a boundary value problem Asymptotic integration of linear integro-differential equations Some problems with rapidly oscillating coefficients Problems with an unstable spectrum Singularly perturbed problems for nonlinear equations Part II.Singularly perturbed partial differential equations:: Asymptotic integration of linear parabolic equations Application of the regularization method to some elliptic problems in a cylindrical domain Asymptotic integration of some singularly perturbed evolution equations.

Asymptotic expansions for large closed queueing networks with multiple job classes

Abstract: A closed BCMP queuing network consisting of R job classes (chains), K+1 single-server, fixed-rate nodes, and M/sub j/ class j jobs (j=1, 2, . . ., R) is considered. Asymptotic expansions are constructed for the partition function under assumptions (1) K>>1, (2) M/sub j/>>1 for each j, and (3) K/M/sub j/=O(1). Analytic expressions for performance measures such as the mean queue length are also given. The approach employs the ray method and the method of matched asymptotic expansions. Numerical comparisons illustrate the accuracy of the approximations. >

Tensor invariants of quasihomogeneous systems of differential equations, and the Kovalevskaya-Lyapunov asymptotic method

Abstract: An example is a system with homogeneous quadratic right-hand members: in it, gl = ... = gn = i. Among others, the Euler-Poincare equations describing geodesics on Lie groups with invariant metrics fall into this class. A popular example from dynamics is Kirchoff's problem on the motion of a rigid body in an unbounded volume of an ideal liquid. Quasihomogeneous systems are also exemplified by the equations of the problem of many gravitating bodies and by the Euler-Poisson equations describing the rotation of a heavy rigid body about a fixed point. These remarks show that it is expedient to consider quasihomogeneous systems from the viewpoint of applications.

Cool mass-losing boundary layers of T Tauri accretion disks

Abstract: We investigate the structure of the boundary layer between a thin accretion disk and a T Tauri star. We solve the one-dimensional hydrodynamical equations in axial symmetry by using the method of matched asymptotic expansions and find steady solutions with significant mass loss and with radiative energy transport into the star. Reprocessing of the radial energy flux in the stellar atmospheric layers is expected to give rise to the observed ultraviolet and optical continuum flux. The rates of mass-loss and of kinetic energy carried out by the wind are consistent with observations of protostellar and T tauri winds

Asymptotic expansions of functional inverses

Abstract: We study the automatic computation of asymptotic expansions of functional inverses. Based on previous work on asymptotic expansions, we give an algorithm which computes Hardy-field solutions of equations f(y) = x, with f belonging to a large class of functions.

Scattering of elastic waves by an arbitrary small imperfection in the surface of a half-space

Abstract: This paper, the first of two articles, is concerned with the scattering of elastic waves by arbitrary surface-breaking or near surface defects in an isotropic half-plane. We present an analytical solution, by the method of matched asymptotic expansions, when the parameter e, which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The problem is formulated for a general class of small defects, including cracks, surface bumps and inclusions, and for arbitrary incident waves. As a straightforward example of the asymptotic scheme we specialize the defect to a two-dimensional circular void or protrusion, which breaks the free surface, and assume Rayleigh wave excitation ; this inner problem is exactly solvable by conformal mapping methods. The displacement field is found uniformly to leading order in e, and the Rayleigh waves which are scattered by the crack are explicitly determined. In the second article we use the method given here to tackle the important problem of an inclined edge-crack. In that work we show that the scattered field can be found to any asymptotic order in a straightforward manner, and in particular the Rayleigh wave coefficients are given to O(e2).

On the use of the method of matched asymptotic expansions in propeller aerodynamics and acoustics

Abstract: The applicability of the method of matched asymptotic expansions to both propeller aerodynamics and acoustics is investigated. The method is applied to a propeller with blades of high aspect ratio, in a uniform axial flow. The first two terms of the inner expansion and the first three terms of the outer expansion are considered. The matching yields an expression for the spanwise distribution of the downwash velocity. A numerical application shows that the first two terms of the inner solution do not yield an acceptable approximation for the downwash velocity. However, recasting the analytical expressions into an integral equation, similar to Prandtl's lifting line equation for wings, yields results for both aerodynamic and acoustic quantities, which agree well with experimental results. The method thus constitutes a practical analysis method for conventional propellers.

Scattering of elastic-waves by a small inclined surface-breaking crack

Abstract: I n this paper we examine the scattering of Rayleigh waves by an inclined two-dimensional plane surface-breaking crack in an isotropic elastic half-plane. We follow the method already introduced by the authors (A brahams and W ickham , 1992a, J. Mech. Phys. Solids 40, 1683) to obtain an analytical solution when the parameter e, which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The procedure employed is the method of matched asymptotic expansions, which involves determining the scattered wave field both away from and near the crack. The outer solution is specialized from the general expansion in the first part of this work (A brahams and W ickham , 1992a, J. Mech. Phys. Solids 40, 1683), and the inner problem is exactly solved by the Wiener-Hopf technique. The displacement field and scattered Rayleigh waves are found uniformly to third order in e, and concluding remarks are made about the general method as well as the results presented here.

Convective wall plume in power-law fluid: Second-order correction for the adiabatic wall

Abstract: This paper considers the steady-state free convection flow arising from an infinitely long horizontal line source of heat embedded in the base of a vertical adiabatic surface when the ambient fluid is a non-Newtonian fluid for moderately large values of the generalized Grashof numbers by the method of matched asymptotic expansions In particular, the second-order corrections to account for the non-boundary layer effects have been predicted A family of numerical solutions for the power-law fluid behavior indexn ranging from 04 to 20 and for the Prandtl numberPr=10 and 100 are reported

Transient Natural Convection in a Horizontal Concentric Annulus Filled With a Porous Medium

Abstract: The method of matched asymptotic expansions is applied to study the problem of transient natural convection in a horizontal concentric porous annulus with inner and outer cylinders maintained at uniform temperatures. Asymptotic solutions for the inner layer, the outer layer, and the core are obtained for small times. A uniformly valid solution for stream function, tangential velocity, and temperature, which is valid for the whole domain, is constructed. 13 refs., 2 figs.

The interface crack in a tension field : an eigenvalue problem for the gap

Abstract: The problem of the interface crack in a tension field is reexamined with the objective of obtaining simple and explicit expressions for all quantities of physical interest. The known exact solution to this problem is complicated and difficult to analyze. Here the gap is shown to satisfy an eigenvalue problem. The condition that the gap is nonnegative implies that the smallest eigenvalue is the correct one. A simple uniform approximation for the gap is obtained by using the method of matched asymptotic expansions to take advantage of the smallness of the contact zones. As the tangential shift satisfies the same differential equation as the gap, a uniform approximation for this quantity is obtained by the same method. The tractions are also shown to satisfy a second order ordinary differential equation, and uniform approximations are given for the quantities.

Asymptotic analysis of the equichordal problem.

Abstract: The equichordal problem asks whether there exists an equichordal curve, i.e. a simple closed curve in the plane with two equichordal points. An equichordalpoint of a simple closed curve is a point such that every line passing through the point meets the curve in exactly two points and all chords determined in this way have the same length. According to Klee [5] the problem was first raised by Fujiwara [4] in 1916 and independently by Blaschke, Rothe and Weitzenb ck [1] in 1917. The problem has been investigated by S ss [10], Dirac [2], Wirsing [11], Ehrhart [3] and others. Several of these authors pointed out that the existence of an equichordal curve is very unlikely but no proof demonstrating the nonexistence has been given so far.

Rapidly oscillating asymptotic solution of magnetohydrodynamic equations in the Tokamak approximation

Abstract: An asymptotic solution of the magnetohydrodynamic equations for large Reynolds numbers is constructed in the approximation of a strong longitudinal magnetic field. Model equations that generalize the Kadomtsev—Pogutze equations to the case of toroidal symmetry are derived. The asymptotic stability of the constructed solution is demonstrated.

The Asymptotic Theory of Rapidly Rotating Sound Fields

Abstract: A rapidly rotating sound field, such as that produced by a supersonic propeller, may contain several types of diffraction pattern, each in a different region of space. This paper determines these patterns and their locations for the field around a propeller of simple type; the method used is stationary phase analysis of a certain double integral, and leads to asymptotic formulae valid when the number of blades is not too small or the high harmonics are being investigated. Physically, the results describe propagation along rays: each stationary phase point is a `loud spot', producing a ray which points directly at the observer; most of the noise comes from these loud spots, because extensive cancellation takes place everywhere else. At most two interior and two boundary stationary points may be present: the number and type depend on the position of the observer in relation to a cusped torus and two hyperboloids of one sheet. As these surfaces are crossed, the acoustic field changes in character. For example, when two stationary points coalesce and annihilate each other, as they do at a caustic, an Airy function describes the transition from a loud zone of rapid oscillation to a quiet zone of exponential decay; and when an interior stationary point crosses the boundary of the disc the transition region is described either by a Fresnel integral or by a generalized Airy function. Separate analyses are given for regions close to and well away from the transition surfaces, and inner and outer limits are calculated for use in the method of matched asymptotic expansions. In all cases, an overlap region is found in which the leading terms agree. The results of the paper determine completely the geometry of the acoustic field, because the different regions have boundaries at known positions and cover the whole of space.

Evaluation of an optimal guidance algorithm for aero-assisted orbit transfer

Abstract: A detailed evaluation is performed of a guidance algorithm for aeroassisted orbit transfer that was developed earlier based on the method of matched asymptotic expansions. It is shown that, by exploiting the structure of the matched asymptotic expansion solution procedure, the original problem which requires the solution of a set of 20 implicit algebraic equations can be reduced to a problem of six implicit equations in six unknowns. The main contribution here is that it was possible to obtain a solution that is near optimal, requires a minimum of computation, and thus can be implemented in real time and on-board the vehicle. Guidance law implementation entails treating the current state as a new initial state and repetitively solving the matched asymptotic expansion problem to obtain the feedback controls.

Influence of the position of the opening and its shape on the properties of a Helmholtz resonator

Abstract: The method of matched asymptotic expansions is used to influence the choice of the position of the opening and its shape on the properties of a Helmholtz acoustic resonator. It is shown that these parameters influence the imaginary part of the pole of the analytic continuation of the Green's function of the Helmholtz resonator, which, in its turn, has a strong influence on the behavior of the solutions of the corresponding scattering and radiation problems.

The recursive reduced-order solution of an open-loop control problem of linear singularly perturbed systems

Abstract: A reduced-order method with an arbitrary degree of accuracy is obtained for solving the linear-quadratic optimal open-loop control problem. The original two-point boundary value problem is transformed into the pure-slow and pure-fast reduced-order completely decoupled initial value problems. By doing this, the stiffness of the singularly perturbed two-point boundary value problem is converted into the problem of an ill-defined linear system of algebraic equations. >