Showing papers on "Method of matched asymptotic expansions published in 1973"
74 citations
TL;DR: In this article, the Fox-Goodwin technique was used for the solution of two-point boundary value problems, and it was shown that good linear independence can be obtained using a Fox-goodwin technique.
Abstract: Solutions of coupled equations for electron-atom collision theory, having known asymptotic forms, can be calculated using asymptotic expansions at some large values of r, say r=rp and rp+1. At some smaller values of r, say r=r1 and r2, it is required to match the asymptotic solutions to solutions which are bounded at the origin. When some channels are closed, inwards integrations from (rp,rp+1) to (r1,r2) can give solutions which have poor linear independence at (r1,r2). It is shown that good linear independence can be obtained using a Fox-Goodwin technique for the solution of two-point boundary value problems.
71 citations
TL;DR: Theoretical results for heat transfer from a circular cylinder oscillating in an unbounded viscous fluid which is otherwise at rest are given in this article, where the amplitude of the oscillation is assumed small compared to the radius of the cylinder, which for most of the examples considered is assumed to be at a constant temperature.
53 citations
01 Sep 1973
TL;DR: In this article, the mass transport velocity due to modal interactions is associated with a cellular structure in each fluid layer, and the order of magnitude of the velocity calculated for a general oscillatory disturbance is the same as that calculated for interfacial progressive waves by Dore(2).
Abstract: The to method of matched asymptotic expansions is employed to calculate the mass tre sport velocity due to combinations of small amplitude oscillatory waves propagatir, at a single frequency in fluid systems with density and viscosity dis-continuities. Interfacial boundary layers are considered in terms of the curvilinear coordinate system described by Longuet-Higgins(1). The order of magnitude of the mass transport velocity calculated for a general oscillatory disturbance is the same as that calculated for interfacial progressive waves by Dore(2). For standing waves, the time-averaged motion of the fluid particles forms a cellular structure in each fluid layer; the mass transport velocity due to modal interactions is associated with a similar structure.
32 citations
TL;DR: In this paper, the low-wavenumber structure of fixed-frequency cross-spectral density of turbulent pressure fluctuations at a rigid plane boundary is determined for flows with small but finite Mach number.
Abstract: The low‐wavenumber structure of fixed‐frequency cross‐spectral density of turbulent pressure fluctuations at a rigid plane boundary is determined for flows with small but finite Mach number. The method of matched asymptotic expansions is applied in the coordinate normal to the boundary. The boundary layer is an inner region, and prescribes an “effective” boundary velocity distribution for the outer region, which is governed by the acoustic wave equation. Those components of effective velocity with supersonic phase speeds account for the radiation of sound. For the inviscid infinite plate model, the wall‐pressure spectrum has a nonintegrable singularity at the acoustic critical wavenumber. Because of undamped contributions to point pressure from distant acoustic sources, in fact, the infinite model fails in an inviscid medium with any degree of compressibility. A large but finite model is considered, and the nonintegrable singularity at the critical wavenumber is removed. The spectrum coincides otherwise with the infinite plate result. The finite extent of the model can represent either a real geometrical limitation or the effect of damping over long distances. The intensity in the radiated field is shown to vary with the eighth power of velocity, but with a coefficient proportional to the logarithm of the characteristic in‐plane dimension.
30 citations
TL;DR: In this article, conditions for determining explicitly the range of parameter perturbation such that both bounded-input-bounded-output and asymptotic stabilities are insured were derived.
Abstract: This paper considers the problem of stability of a singularly perturbed system and of finding an upper bound for the parameter when the order of the system changes as a result of parameter perturbation. By means of the contraction mapping technique, conditions have been derived for determining explicitly the range of parameter perturbation such that both bounded-input-bounded-output and asymptotic stabilities are insured. In addition, bounds of the state and output of the singularly perturbed system can be found. Two examples are given to illustrate the application and significance of the results.
29 citations
TL;DR: In this paper, the authors give conditions to ensure that all solutions of (1.1), (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 55, 56, 57
Abstract: where functions appeared in the equations are real valued. The dots indicate differentiation with respect to t and all solutions considered are assumed to be real. The problem is to give conditions to ensure that all solutions of (1.1), (1.2), (1.3) and (1.4) tend to zero as t— >oo. This problem has received a considerable amount of attention during the past twenty years, particulary when equations are autonomous. Many of these results are summarized in
28 citations
23 citations
TL;DR: In this paper, an approximate solution of the problem concerning the propagation of a planar. front of a two-stage exothermic sequential chemical reaction in a gas, by the method of matched asymptotic expansions, is presented.
22 citations
TL;DR: Following Kasminski, this paper investigated the asymptotic behavior of solutions of linear time-independent Ito equations and gave a sufficient condition for stability of the zero solution in dimension 2.
Abstract: Following Kasminski, we investigate asymptotic behavior of solutions of linear time-independent Ito equations We first give a sufficient condition for asymptotic stability of the zero solution Then in dimension 2 we determine conditions for spiraling at a linear rate Finally we give applications to the Cauchy problem for the associated parabolic equation by the use of a tauberian theorem Introduction Consider a system of linear, constant coefficients differential equations dx I (0 1) -d=Ebxj x (I 01, he gave a necessary and sufficient condition that r(t) 0 as when t --> He did not examine, however, the behavior of {X(t), t > 01 The first study of the angular behavior in the case 1 2 Received by the editors March 13, 1972, AMS (MOS) subject classifications (1970) Primary 60H10, 60J60; Secondary 34D05, 34A30, 34C05
21 citations
TL;DR: In this article, Kuzmak et al. extended the asymptotic method of G. B. Whitham to a more general class of nonlinear second order partial differential equations and obtained asymptic expansions of the solutions.
Abstract: By extending the asymptotic method of G. E. Kuzmak, J. C. Luke has shown how G. B. Whitham’s theory of nonlinear wave propagation can be derived directly from the partial differential equation without using the variational principle, in special cases. We apply the same method to a more general class of nonlinear second order partial differential equations or systems of first order equations containing a small parameter e and obtain asymptotic expansions of the solutions.
TL;DR: The low Mach number sound field induced by the motion of line vortex filaments coupled to a two-dimensional semi-infinite duct is determined by means of a singular perturbation technique as discussed by the authors.
Abstract: The low Mach number sound field induced by the motion of line vortex filaments coupled to a two-dimensional semi-infinite duct is determined by means of a singular perturbation technique. Using the method of matched asymptotic expansions, solutions for the sound field are obtained by matching with an ‘inner’ region of incompressible flow. The radiation field induced by the emergence of a single vortex from the channel exhibits the edge scattering effects typical of half-plane problems. The sound field intensity is found to have angular dependence on sin2½ θ, where θ = 0 defines the exterior axis of symmetry. When a vortex pair propagates out of the duct however, the special symmetry of the fluid motion causes cancellation of the scattered field from the duct edges. In that case the sound field is driven from sources located at the duct exit. We show that this result is consistent with the general theories of both Curle and Powell. The sound field is essentially induced by a dipole at the exit plane of the duct, part of which drives a coupled weak monopole, while the remainder corresponds to an axial ‘edge force’ originating in the r−½ velocity singularities at the duct edges.
TL;DR: In this paper, the periodic solution of a Volterra-lotka system is considered as a relaxation oscillation, and four local expansions are constructed from the implicitly given solution.
Abstract: The periodic solution of a Volterra-Lotka system is considered as a relaxation oscillation. With perturbation techniques, four local expansions are constructed from the implicitly given solution. Integration over the four regions leads to an asymptotic formula for the period.
TL;DR: In this article, a nonlinear second order differential equation with a small parameter is used as a model to discuss the validity of the method of inner and outer expansions (MIO) for treating singular perturbation problems such as those of slow viscous flow past a cylinder.
Abstract: This paper concerns singular perturbation problems such as those of slow viscous flow past a cylinder A nonlinear second order differential equation with a small parameter is used as a model to discuss the validity of the method of inner and outer expansions (MIO) for treating such problems Based on a regular perturbation procedure by Finn and Smith, it is shown that the formal asymptotic expansions constructed by MIO are indeed in some sense the asymptotic expansions for the exact solution of the problem
01 Jan 1973
TL;DR: In this paper, the amplitude of the Van der Pol equation was determined up to a higher order accuracy in v than has been done so far, by applying the method of intermediate matching for connecting the four local asymptotic solutions.
Abstract: The purpose of this paper is to apply the method of intermediate matching for connecting the four local asymptotic solutions of the Van der Pol equation, given by Dorodnicyn [1]. It turns out that for the approximation of the periodic solution a fifth local solution is needed. The present approach results in a reduction of the computational work. The amplitude of the periodic solution is determined up to a higher order accuracy in v than has been done so far.
TL;DR: In this paper, a transonic flow solution for configurations with span-to-length ratios of order one is presented, where the angles of attack are sufficiently large to produce lift effects that are either dominant or comparable to the thickness effects.
Abstract: A transonic flow solution is presented for configurations with span-to-length ratios of order one. The angles of attack are sufficiently large to produce lift effects that are either dominant or comparable to the thickness effects. The analysis is performed with the aid of the method of matched asymptotic expansions. The results obtained are compared with data reported by Cheng and Hafez (1972).
TL;DR: In this paper, an analysis of the far field behavior of a longitudinal shock wave propagating in a one-dimensional semi-infinite elastic lattice with a velocity step applied to the first mass is made.
Abstract: An analysis is made of the far‐field behavior of a longitudinal shock wave propagating in a one‐dimensional semi‐infinite elastic lattice with a velocity step applied to the first mass. The nonlinear part of the elastic interaction force is assumed to be of parabolic form. Using the method of matched asymptotic expansions and stretched coordinates, for small nonlinearity, the original lattice equations of motion are replaced by a partial differential equation that governs the slow evolution of the wave with distance. A numerical solution of the partial differential equation shows the transition as the wave propagates from an oscillatory shock profile in the near field to a sequence of solitary waves in the far field of the lattice. The individual solitary waves at the head of the pulse eventually achieve essentially the same steady maximum amplitude and the same steady form, while continuing to spread apart from each other. The maximum particle velocity for the far‐field solitary waves is found to be approximately 1.76 times as great as the velocity applied to the first mass. For the first‐order nonlinear theory that is derived, the existence of a maximum amplitude of particle velocity that is found to be steady with distance as the wave propagates also implies that amplitude is independent of the magnitude of the nonlinearity, so long as the nonlinearity is small, but nonzero.
TL;DR: In this article, a fixed final time free end-point optimal problem with a small time delay is considered, which leads to a nonlinear boundary value problem consisting of both advanced-and retarded-type differential-difference equations.
Abstract: A fixed final time free end-point optimal problem with a small time delay is considered. This problem leads to a nonlinear boundary-value problem consisting of both advanced- and retarded-type differential-difference equations. An asymptotic power series solution to the problem is constructed in terms of time delay. This asymptotic approximation procedure is aimed at improving a "nominal design" in which the small time delay is neglected. A scalar example illustrates the construction of asymptotic expansions. Numericai results on a coupled-core nuclear-reactor power control problem clearly show the advantages of the method.
TL;DR: In this article, a study of an asymptotic theory of hyperthermal flow past a blunt object for large Knudsen number is presented, where the largest demands on such a theory are present for flow past two-dimensional bodies.
Abstract: A study of an asymptotic theory of hyperthermal flow past a blunt object for large Knudsen number is presented. The greatest demands on such a theory are present for flow past two‐dimensional bodies. This is because the large lateral extent of the emitted molecules in free molecular flow gives rise to a singularity in the integrated collision frequency of the incoming free stream molecules. This singularity is removed by accounting for the effects of collisions upon the emitted molecules which augments the usual geometric free molecular decay. The method of matched asymptotic expansions is applied to regions near the body and far from the body (one mean free path and greater). For two‐dimensional bodies, the surprising result is obtained that the first correction to free molecular flow [Kn−1 ln(Kn)−1] cannot be obtained without explicitly taking into account the collisional effects on the emitted particles in the far region. For three‐dimensional bodies, although there is no problem in the determination of the leading term (Kn−1), the evaluation of the next order term [Kn−2 ln(Kn)−1 involves precisely the same difficulties as the leading term for two‐dimensional bodies.
TL;DR: In this paper, the authors give sufficient conditions for oscillatory solutions of variants of the nonlinear delay differential equation, and show that these conditions can be obtained for any variant of the problem.
TL;DR: In this article, the authors proposed a method of matched asymptotic expansions for approximating small ϵi, for the system where G is a general function of the N-dependent variables fi.
TL;DR: In this paper, the second-order effects due to surface curvature and displacement on the velocity components, pressure, shear stress, heat transfer, and mass transfer are determined.
TL;DR: In this article, a relatively simple asymptotic solution is presented which allows one to determine the effects due to ionic currents, providing at the same time a valid approximation to many problems which in the past proved very difficult to solve.
Abstract: Solutions of the combined electrodiffusion equations have been considered for a homogeneous medium from which fixed charges are absent. An exact solution becomes possible provided only positive or negative ions are present. For the general case a relatively simple asymptotic solution is presented which allows one to determine the effects due to the ionic currents, providing at the same time a valid approximation to many problems which in the past proved very difficult to solve. A comparison between the results of numerical calculations and the predictions from the asymptotic solution provides an estimate as to the validity range of the asymptotic method. An analysis of the dynamic changes within the ion concentration shows that the steady state solution for many problems is only applicable to rather limited space regions of biological membranes. An approximate nonsteady state solution is therefore given which is valid for large space regions.
01 Sep 1973
TL;DR: In this paper, the Szaniawski power series is derived using matched asymptotic expansions, and the inner throat region is governed by the nonlinear transonic equations which admit as one class of solutions, similarity solutions.
Abstract: : Many solutions have been presented for two dimensional transonic nozzle flows, with several different methods being represented Two of the more interesting of these solutions are those presented by Tomotika and Tamada (1950) and Szaniawski (1965) However, it has not been made clear under what conditions either solution is valid It is the purpose of the paper using the methods of matched asymptotic expansions, to derive the Szaniawski power series systematically and to show that this solution should be considered as an outer solution which may not be uniformly valid as the throat is approached The inner throat region is governed by the nonlinear transonic equations which admit as one class of solutions, similarity solutions The analysis is performed using the general non-steady inviscid equations of motion, with the steady flow results being derivable as a special case (Modified author abstract)
01 Jan 1973
TL;DR: In this article, it is shown that for f bounded, locally integrable and positive, it is known that a unique positive solution u of (i.i.I) exists and this solution has been studied in various asymptotic limits.
Abstract: Here ¢ > 0 is a constant and f is a given function. Some background and references related to this physical problem are given in [4]° For f bounded, locally integrable and positive, it is known that a unique positive solution u of (i.I) exists° This solution has been studied in various asymptotic limits [3], [4], [6]° Of particular interest here are the limits t ~ for s fixed and ¢ ~ 0 for all t ~ O. We shall discuss both of these cases.
01 May 1973
TL;DR: In this paper, the two-dimensional flow in a jet, falling under gravity from a slot in a horizontal plane, is studied and the method of matched asymptotic expansions is applied.
Abstract: The two-dimensional flow in a jet, falling under gravity from a slot in a horizontal plane, is studied. The fluid is considered to be incompressible and inviscid; the flow is taken to be irrotational; and the reciprocal e of the Froude number is considered to be small. By taking the complex potential as the independent variable we overcome the difficulty that the boundary geometry is not known in advance. The method of matched asymptotic expansions is applied. The first two terms of an inner asymptotic expansion and the first three of an outer one are found: the inner expansion is valid above and near the slot, but is inappropriate far downstream, while the outer expansion is valid far downstream, but fails to satisfy the conditions upstream. The two expansions are matched and ‘composite’ approximations, covering the whole flow field, are derived.
TL;DR: In this paper, it was shown that the multiplicative rule of the method of matched asymptotic expansions fails under certain conditions, such as the velocity distribution in front of an elliptic airfoil.
Abstract: It is shown that the multiplicative rule of the method of matched asymptotic expansions fails under certain conditions. The velocity distribution in front of an elliptic airfoil is given as an example. The reason for the breakdown is explained by inspecting the usual formal justification of multiplicative composition.