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Showing papers on "Method of matched asymptotic expansions published in 1976"


Journal ArticleDOI
TL;DR: In this article, a viscous incompressible fluid past a slender body of circular crosssection is treated by the method of matched asymptotic expansions, and the main result is an integral equation for the force per unit length exerted on the body by the fluid.
Abstract: Slow flow of a viscous incompressible fluid past a slender body of circular crosssection is treated by the method of matched asymptotic expansions. The main result is an integral equation for the force per unit length exerted on the body by the fluid. The novelty is that the body is permitted to twist and dilate in addition to undergoing the translating, bending and stretching, which have been considered by others. The method of derivation is relatively simple, and the resulting integral equation does not involve the limiting processes which occur in the previous work.

323 citations



Book ChapterDOI
TL;DR: In this article, asymptotic expansions for the power of distributionfree tests in the two-sample problem were established, and these expansions were then used to obtain deficiencies in the sense of Hodges and Lehmann for distribution-free tests with respect to their parametric competitors and for the estimators of shift associated with these tests.
Abstract: Asymptotic expansions are established for the power of distributionfree tests in the two-sample problem. These expansions are then used to obtain deficiencies in the sense of Hodges and Lehmann for distributionfree tests with respect to their parametric competitors and for the estimators of shift associated with these tests.

154 citations


Journal ArticleDOI
TL;DR: In this article, the singular perturbation technique was used to obtain an approximate solution to the aircraft minimum time-to-climb problem, which is the same as the one we consider in this paper.
Abstract: Application of singular perturbation techniques to trajectory optimization problems of flight mechanics is discussed. The method of matched asymptotic expansions is used to obtain an approximate solution to the aircraft minimum time-to-climb problem. Outer, boundary-layer, and composite solutions are obtained to zeroth and first orders. A stability criterion is derived for the zeroth-order boundary-layer solutions (the theory requires a form of boundary-layer stability). A numerical example is considered for which it is shown that the stability criterion is satisfied and a useful numerical solution is obtained. The zeroth-order solution proves to be a poor approximation, but the first-order solution gives a good approximation for both the trajectory and the minimum time-to-climb. The computational cost of the singular perturbation solution is considerably less than that of a steepest descent solution. Thus singular perturbation methods appear to be promising for the solution of optimal control problems.

110 citations


Journal ArticleDOI
TL;DR: The dominant asymptotic behavior of the solution of the nonlinear Schrodinger equation when there is one soliton and decaying oscillations has been shown in this paper, where the method of solution uses the conservation laws, rather than the integral equations.
Abstract: We find the dominant asymptotic behavior of the solution of the nonlinear Schrodinger equation when there is one soliton and decaying oscillations. The solution behaves like the soliton near the soliton, and like the solution found in the preceding paper (I) elsewhere. The method of solution uses the conservation laws, rather than the integral equations.

71 citations


Journal ArticleDOI
TL;DR: In this article, the problem of uniform transverse flow past a prolate spheroid of arbitrary aspect ratio at low Reynolds numbers has been analysed by the method of matched asymptotic expansions.
Abstract: The problem of a uniform transverse flow past a prolate spheroid of arbitrary aspect ratio at low Reynolds numbers has been analysed by the method of matched asymptotic expansions. The solution is found to depend on two Reynolds numbers, one based on the semi-minor axis b, R[sub]b = Ub/v, and the other on the semi-major axis a, R[sub]a = Ua/v (U being the free-stream velocity at infinity, which is perpendicular to the major axis of the spheroid, and v the kinematic viscosity of the fluid). A drag formula is obtained for small values of R[sub]b and arbitrary values of R[sub]a. When R[sub]a is also small, the present drag formula reduces to the Oberbeck (1876) result for Stokes flow past a spheroid, and it gives the Oseen (1910) drag for an infinitely long cylinder when R[sub]a tends to infinity. This result thus provides a clear physical picture and explanation of the 'Stokes paradox' known in viscous flow theory.

64 citations


Journal ArticleDOI
TL;DR: In this article, a method is developed whereby the statistical characteristics of the response of non-linear stochastic systems are calculated with good accuracy based on the theory of Markov processes.
Abstract: In this paper a method is developed whereby the statistical characteristics of the response of non-linear stochastic systems are calculated with good accuracy. The method consists of the formulation and solution of the differential equations for the statistical characteristics of the processes under consideration. The development is based on the theory of Markov processes. In particular, use is made of Ito's differential rule to obtain a basic result which is used for the formulation of the differential equations. For the solution of the differential equations the method employs an approximate analytic representation of the probability density function of a random process. The representation is in the form of a finite Edgeworth (asymptotic) expansion. The method is quite general and yields accurate results. The treatment is illustrated by examples.

64 citations



Journal ArticleDOI
TL;DR: In this article, a constant-pressure axisymmetric turbulent boundary layer along a circular cylinder of radius a is studied at large values of the frictional Reynolds number a+ (based upon a) with the boundary-layer thickness δ of order a. The condition that the two expansions match requires the existence, at the lowest order, of a log region in the usual two-dimensional co-ordinates (u+, y+).
Abstract: A constant-pressure axisymmetric turbulent boundary layer along a circular cylinder of radius a is studied at large values of the frictional Reynolds number a+ (based upon a) with the boundary-layer thickness δ of order a. Using the equations of mean motion and the method of matched asymptotic expansions, it is shown that the flow can be described by the same two limit processes (inner and outer) as are used in two-dimensional flow. The condition that the two expansions match requires the existence, at the lowest order, of a log region in the usual two-dimensional co-ordinates (u+, y+). Examination of available experimental data shows that substantial log regions do in fact exist but that the intercept is possibly not a universal constant. Similarly, the solution in the outer layer leads to a defect law of the same form as in two-dimensional flow; experiment shows that the intercept in the defect law depends on δ/a. It is concluded that, except in those extreme situations where a+ is small (in which case the boundary layer may not anyway be in a fully developed turbulent state), the simplest analysis of axisymmetric flow will be to use the two-dimensional laws with parameters that now depend on a+ or δ/a as appropriate.

43 citations


Journal ArticleDOI
TL;DR: Asymptotic expansions for the distributions of latent roots of matrices in three multivariate situations are given in this paper. But they are not applicable to the noncentrality matrices.

33 citations


Journal ArticleDOI
TL;DR: In this article, a system of equations that arises in a singularly perturbed optimal control problem is studied, and conditions under which a formal asymptotic solution exists are given.
Abstract: A system of equations that arises in a singularly perturbed optimal control problem is studied. We give conditions under which a formal asymptotic solution exists. This formal asymptotic solution consists of an outer expansion and left and right boundary-layer expansions. A feature of our procedure is that we do nota priori eliminate the control function from the problem. In particular, we construct a formal asymptotic expansion for the control directly. We apply our procedure to a Mayer-type problem. The paper concludes with a worked example.

Journal ArticleDOI
TL;DR: In this paper, an analytical solution for non-equilibrium dissociating flow of an inviscid Lighthill-Freeman gas after a curved shock was obtained by dividing the flow into a thin reacting layer near the shock and a frozen region further downstream.
Abstract: Analytic solutions are obtained for non-equilibrium dissociating flow of an inviscid Lighthill-Freeman gas after a curved shock, by dividing the flow into a thin reacting layer near the shock and a frozen region further downstream. The method of matched asymptotic expansions is used, with the product of shock curvature and reaction length as the small parameter. In particular, the solution gives expressions for the reacting-layer thickness, the frozen dissociation level, effective shock values of the frozen flow and the maximum density on a stream-line as functions of free-stream, gas and shock parameters. Numerical examples are presented and the results are compared with experiments.

Journal ArticleDOI
TL;DR: An analytical solution of the Possio integral equation is obtained for low reduced-frequency to, correct to or- der wM/(l-M2) = ir/4 in this article.
Abstract: An analytical solution of the Possio integral equation is obtained for low reduced-frequency to, correct to or- der wM/(l-M2). The resulting loading, lift, and moment differ from that derived from GASP theory by Osborne, in agreement with recent work of Amiet, who has shown GASP theory to be inapplicable to two- dimensional flow with shed vorticity. The solution is applied to a generalized gust, a power law upwash, plunging motion, pitching motion, and a sinusoidal gust. Comparisons with numerical solutions are given for lift in the latter three cases. They show that one of the forms of the solution, the Osborne lift times a phase correction, is remarkably accurate up to coM/(l —M2) = ir/4. This approximation should prove convenient and useful in applications. HE nonsteady load distribution on two-dimension al thin airfoils oscillating in subsonic flow is governed by the Possio integral equation, which has no known exact analytical solution. There have been many numerical approaches to its solution over the years, and also a number of approximate analytical approaches. For a given airfoil motion, the forces depend on two parameters, the stream Mach number M and the reduced frequency of oscillation co. The task of any ap- proximate analytical solution is to give a satisfactory ap- proximation in as large a region of the M, co plane (M< 1) as possible. Some years ago, Miles1 briefly discussed the com- pressibility correction to the incompressible problem (which has an exact solution). He transformed the governing dif- ferential equation and boundary conditions by a combined Galilean and Lorentz transformation, expanded to first order in coM//3 2, where /32 = 1 — M2, and derived a compressibilit y correction rule. In another paper, Miles2 derived a quasisteady theory by expanding the Possio integral equation to first order in co. Recently, Amiet and Sears3 applied the method of matched asymptotic expansions to small- perturbation subsonic flows, and, again, obtained Miles' correction rule1 in the course of their work, which has been called GASP (Glauert-Amiet-Sears-Prandtl) theory. Osborne4 applied this correction rule to obtain analytical formulas for the lift and moment of oscillating, thin two- dimensional airfoils flying in a generalized gust, a sinusoidal gust, and in plunging motion, although his results were in finite form only for the sinusoidal gust. Kemp5 showed that the results for the generalized gust and the plunging motion also could be put in finite form, and added the other in- teresting case of pitching motion. (He also showed, sub- sequently6, that the lift and moment for any integer power law upwash distribution could be derived by purely algebraic

Journal ArticleDOI
TL;DR: Using Heaviside's exponential series, a power series solution of the differential equation is split into formal solutions with known asymptotic expansions as discussed by the authors, where the expansion is defined by a set of constants.
Abstract: Using Heaviside’s exponential series, a power series solution of the differential equation is split into formal solutions with known asymptotic expansions.


Journal ArticleDOI
TL;DR: In this article, the effect of gap width on flow through homogeneous porous soil into a row of drain pipes is considered, by solving Darcy's law in a model configuration with axial symmetry.

Journal ArticleDOI
TL;DR: In this article, the problem of generating approximations to functions, specified by asymptotic expansions about one or more points, is considered, and the development is limited to expansions of the power series type.
Abstract: The problem of generating approximations to functions, specified by asymptotic expansions about one or more points, is considered. The development is limited to expansions of the power series type ...

Journal ArticleDOI
TL;DR: In this article, the authors considered the plane vibrations of a slab composed of material which displays also viscoelastic and nonlinear elastic constitutive behaviour, in the case when the faces of the slab are held fixed.
Abstract: The plane vibrations are considered of a slab composed of material which, although predominantly linearly elastic, displays also viscoelastic and nonlinear elastic constitutive behaviour, in the case when the faces of the slab are held fixed. An averaging method is used to simplify the problem, treating the nonlinear elastic and viscoelastic terms as perturbations. When the viscoelastic decay time is sufficiently short, the viscoelasticity may be treated by a Voigt approximation, and the averaging method reduces the problem to Burgers' equation. If, however, the decay time is of the same order as the shock thickness, the viscoelastic terms must be left in functional form, and Burgers' equation is replaced by a certain integro-differential equation. A method of matched asymptotic expansions is used to analyse the structure of the solution in the neighbourhood of the shock waves which form, and in particular it is shown that the paths of these shocks may be found from a version of Whitham's equal areas rule, for a general relaxation function. When the relaxation function is of exponential form the solution in the shock layer is obtained explicitly, the shock profile showing a distinct asymmetry about its centre.



01 Mar 1976
TL;DR: In this paper, the exact equations of motion for three-dimensional, aerodynamically affected flight are derived and the equations are transformed into a set suitable for analytic integration using asymptotic expansions.
Abstract: The problem of determining the trajectories, partially or wholly contained in the atmosphere of a spherical, nonrotating planet, is considered. The exact equations of motion for three-dimensional, aerodynamically affected flight are derived. Modified Chapman variables are introduced and the equations are transformed into a set suitable for analytic integration using asymptotic expansions. The trajectory is solved in two regions: the outer region, where the force may be considered a gravitational field with aerodynamic perturbations, and the inner region, where the force is predominantly aerodynamic, with gravity as a perturbation. The two solutions are matched directly. A composite solution, valid everywhere, is constructed by additive composition. This approach of directly matched asymptotic expansions applied to the exact equations of motion couched in terms of modified Chapman variables yields an analytical solution which should prove to be a powerful tool for aerodynamic orbit calculations.

01 Apr 1976
TL;DR: In this paper, the three-dimensional transonic flow about lifting wing-body configurations is reduced to a two-variable computational problem with the method of matched asymptotic expansions.
Abstract: The three-dimensional problem of transonic flow about lifting wing-body configurations is reduced to a two-variable computational problem with the method of matched asymptotic expansions The computational problem is solved with the method of relaxation The method accounts for leading-edge separation, the presence of shock waves, and the presence of solid, slotted, or porous tunnel walls The Mach number range of the method extends from zero to the supersonic value at which the wing leading edge becomes sonic A modified form of the transonic area rule which accounts for the effect of lift is developed This effect is explained from simple physical considerations

Journal ArticleDOI
Syozo Kubo1
TL;DR: In this article, the effects of anisotropic scattering on temperature distribution of a steady one-dimensional radiative heat transfer through an absorbing, emitting gray medium in a semi-infinite space are analyzed by method of matched asymptotic expansions.
Abstract: Effects of anisotropic scattering on temperature distribution of a steady one-dimensional radiative heat transfer through an absorbing, emitting gray medium in a semi-infinite space are analyzed by method of matched asymptotic expansions. An external radiation beam is imposed through a transparent boundary. Rayleigh scattering is considered as a typical model of an anisotropic scattering. The medium is assumed to be non-heat conducting, not to move and to have constant physical properties, The results show that no qualitative difference is shown between the temperature distribution in a scattering medium and that in a non-scattering medium. Only a little quantitative difference is shown for various values of the beam angle. The difference is in 10 percent of the non-scattering value at most.

Journal ArticleDOI
TL;DR: In this paper, the optimal lift and bank modulation of a reentry vehicle using the method of matched asymptotic expansions is presented. But it is only for the inner region, where gravity is predominant, and only four of the five adjoint equations for the free time problem have been integrated in closed form.


Journal ArticleDOI
TL;DR: In this paper, multiple-scale perturbation solutions are obtained for a class of nonlinear second order elliptic Dirichlet boundary value problems in two independent variables, on a semi-infinite strip and on a square.
Abstract: Multiple-scale perturbation solutions are obtained for a class of nonlinear second order elliptic Dirichlet boundary value problems in two independent variables, on a semi-infinite strip and on a square. Complex characteristics are used, and only nonlinearities that involve functions of a first derivative of the dependent variable are covered. The order one approximation is obtained from the solution of a pair of nonlinear ordinary differential equations. The method is especially of interest when boundary conditions are specified on an unbounded domain.

Journal ArticleDOI
TL;DR: In this paper, Liapunov-like conditions for asymptotic stability of singularly perturbed dynamic systems are established, which are used as a basis to link singular perturbation approach and vector LFO approach to the stability analysis of large-scale systems.

Journal ArticleDOI
TL;DR: In this article, the rotational diffusion of a polar cylinder about its long axis in an electric field has been solved and the results are presented as a function of a reduced time and the variable κ, which is a measure of the ratio of mean electrical energy to thermal energy.
Abstract: The problem of rotational diffusion of a polar cylinder about its long axis in an electric field has been solved. The solutions apply to the high as well as the low field limits. The results are presented as a function of a reduced time and the variable κ, which is a measure of the ratio of mean electrical energy to thermal energy. Asymptotic expansions are obtained for the distribution function and the fractional polarization. The eigenvalues and even solution of a Whittaker–Hill–Ince differential equation have been calculated as functions of κ. Asymptotic expansions of the eigenvalue and solution of this equation are given.

Journal ArticleDOI
TL;DR: In this article, the vertical force and pitching moment coefficients are calculated as functions of the reduced frequency and Froude number with a view towards possible application to ship safety considerations, and the method of matched asymptotic expansions is used to take advantage of the simplified governing equations in the near and far fields.
Abstract: The problem under investigation is the unsteady subcritical potential flow generated by a slender ship translating over a wavy wall in shallow water. The method of matched asymptotic expansions is used to take advantage of the simplified governing equations in the near and far fields. The vertical force and pitching moment coefficients are calculated as functions of the reduced frequency and Froude number with a view towards possible application to ship safety considerations.

Journal ArticleDOI
TL;DR: In this article, the authors introduce comparison techniques for studying the oscillatory and asymptotic behavior of solutions of third order differential equations with retarded argument, allowing the use of Kneser-type, as well as integral, criteria for deciding the behavior of the solutions.
Abstract: In this paper we introduce comparison techniques for studying the oscillatory and asymptotic behavior of solutions of third order differential equations with retarded argument. This allows the use of “Kneser-type”, as well as integral, criteria for deciding the behavior of the solutions.