Showing papers on "Method of matched asymptotic expansions published in 2000"

Balance and the Slow Quasimanifold: Some Explicit Results

Abstract: The ultimate limitations of the balance, slow-manifold, and potential vorticity inversion concepts are investigated. These limitations are associated with the weak but nonvanishing spontaneous-adjustment emission, or Lighthill radiation, of inertia–gravity waves by unsteady, two-dimensional or layerwise-two-dimensional vortical flow (the wave emission mechanism sometimes being called “geostrophic” adjustment even though it need not take the flow toward geostrophic balance). Spontaneous-adjustment emission is studied in detail for the case of unbounded f-plane shallow-water flow, in which the potential vorticity anomalies are confined to a finite-sized region, but whose distribution within the region is otherwise completely general. The approach assumes that the Froude number F and Rossby number R satisfy F ≪ 1 and R ≳ 1 (implying, incidentally, that any balance would have to include gradient wind and other ageostrophic contributions). The method of matched asymptotic expansions is used to obtain ...

Asymptotic Solution of a Boundary-Value Problem for Linear Singularly-Perturbed Functional Differential Equations Arising in Optimal Control Theory

Abstract: The Hamiltonian boundary-value problem, associated with a singularly-perturbed linear-quadratic optimal control problem with delay in the state variables, is considered. A formal asymptotic solution of this boundary-value problem is constructed by application of the boundary function method. The justification of this asymptotic solution is done. The asymptotic solution of the Hamiltonian boundary-value problem is constructed and justified assuming boundary-layer stabilizability and detectability.

Approximate potential symmetries for partial differential equations

Abstract: The method of approximate potential symmetries for partial differential equations with a small parameter is introduced. By writing a given perturbed partial differential equation R in a conserved form, an associated system S with potential variables as additional variables is obtained. Approximate Lie point symmetries admitted by S induce approximate potential symmetries of R. As applications of the theory, approximate potential symmetries for a perturbed wave equation with variable wave speed and a nonlinear diffusion equation with perturbed convection terms are obtained. The corresponding approximate group-invariant solutions are also derived.

Singularly Perturbed Discrete-Time Markov Chains

Abstract: Originating from a wide range of applications in optimization and control of large-scale systems (such as telecommunications, queueing networks, and manufacturing systems), this work is devoted to a class of singularly perturbed discrete-time Markov chains. The states of the Markov chain are naturally decomposable into recurrent and transient classes such that within each class the interactions are strong and among different classes the interactions are weak. Our study focuses on the difference equations representing the probabilityvector, and aims at deriving matched asymptotic expansions of the solutions. Justification and error analysis are also provided. Both time-homogeneous and time-inhomogeneous models are examined.

Numerical and asymptotic aspects of parabolic cylinder functions

Abstract: Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered.

Quasi-continuum approximations to lattice equations arising from the discrete nonlinear telegraph equation

Abstract: We show how quasi-continuum methods can be used to construct approximate solutions of nonlinear differential delay equations derived from symmetry reductions of the discrete nonlinear telegraph equation. Travelling wave solutions are proven to exist and the existence of solutions to two other symmetry reductions are studied. Two of the less familiar reductions are studied; the first supports both a one-parameter family of single-pulse solitary-wave-type solutions and a one-parameter family of periodic waves. The size and shape of these waves are examined using the quasi-continuum technique; this approximates the differential-difference equation with a higher- order differential equation, which is integrated and analysed using phase plane techniques. In the large-amplitude limit, the shape of the pulse approaches a limiting form which has a corner at its peak. The manner of this approach is elucidated using matched asymptotic expansions. The second reduction, though differing only by the addition of a single term, appears not to support the solitary-wave type of solution—even in the limit where the additional term is premultiplied by an asymptotically small constant.

Numerical and asymptotic aspects of parabolic cylinder functions

Abstract: Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered.

Boundary conditions for the Child-Langmuir sheath model

Abstract: A collision-free space-charge sheath formed by cold ions at a negative surface is considered. The method of matched asymptotic expansions is applied, small parameter being the ratio of the electron temperature to the sheath voltage. Two expansions are considered, one describing bulk of the sheath where the electron density is exponentially small as compared to the ion density, and another describing the outer section of the sheath in which the densities are comparable. Boundary conditions for equations describing the bulk are found such that the model have exponential accuracy. A physical meaning of these conditions is that the ions are accelerated in the outer section from the Bohm velocity to twice the Bohm velocity and that the voltage drop, in the outer section equals (3/2)(kT/sub e//e). The model predicts the electric field and ion velocity at the surface and the thickness of the ion layer to the accuracy of several percent for sheath voltages exceeding 3(kT/sub e//e).

Direct asymptotic integration of the equations of transversely isotropic elasticity for a plate near cut-off frequencies

Abstract: Summary Direct asymptotic integration of the equations of transversely isotropic elasticity, for a layer with in-plane axis of transverse isotropy and zero surface traction, is carried out in the vicinity of the cut-off frequencies. In direct contrast to the corresponding isotropic case, in which there is only one family of thickness shear resonance frequencies, two such families are observed to exist. Consequently, the two-dimensional equations for the associated long-wave amplitudes are each scalar equations, rather than the single vector equation which arises in the isotropic case. These equations, together with that corresponding to thickness stretch resonance, are obtained. The exact dispersion relation is also derived and asymptotic expansions, giving frequency as a function of scaled wave number, are obtained in the neighbourhood of the cut-off frequencies. This both reveals the appropriate asymptotic orders of stress and displacement components to help facilitate the direct asymptotic integration and retrospectively acts as a check on the coefficients of the two-dimensional equations. The mathematical equivalence of approximations derived from the exact solutions and exact solutions derived from asymptotically approximate equations is therefore verified.

Asymptotic expansions of symmetric standard elliptic integrals

Abstract: Symmetric standard elliptic integrals are considered when one of their parameters is larger than the others. The distributional approach is used for deriving five convergent expansions of these int...

Noise generation by high-frequency gusts interacting with an airfoil in transonic flow

Abstract: The method of matched asymptotic expansions is used to describe the sound generated by the interaction between a short-wavelength gust (reduced frequency k, with k [Gt ] 1) and an airfoil with small but non-zero thickness, camber and angle of attack (which are all assumed to be of typical size O(δ), with δ [Lt ] 1) in transonic flow. The mean-flow Mach number is taken to differ from unity by O(δ2/3), so that the steady flow past the airfoil is determined using the transonic small-disturbance equation. The unsteady analysis is based on a linearization of the Euler equations about the mean flow. High-frequency incident vortical and entropic disturbances are considered, and analogous to the subsonic counterpart of this problem, asymptotic regions around the airfoil highlight the mechanisms that produce sound. Notably, the inner region round the leading edge is of size O(k−1), and describes the interaction between the mean-flow gradients and the incident gust and the resulting acoustic waves. We consider the preferred limit in which kδ2/3 = O(1), and calculate the first two terms in the phase of the far-field radiation, while for the directivity we determine the first term (δ = 0), together with all higher-order terms which are at most O(δ2/3) smaller – in fact, this involves no fewer than ten terms, due to the slowly-decaying form of the power series expansion of the steady flow about the leading edge. Particular to transonic flow is the locally subsonic or supersonic region that accounts for the transition between the acoustic field downstream of a source and the possible acoustic field upstream of the source. In the outer region the sound propagation has a geometric-acoustics form and the primary influence of the mean-flow distortion appears in our preferred limit as an O(1) phase term, which is particularly significant in view of the complicated interference between leading- and trailing-edge fields. It is argued that weak mean- flow shocks have an influence on the sound generation that is small relative to the effects of the leading-edge singularity.

Interface Development and Local Solutions to Reaction-Diffusion Equations

Abstract: The evolution of interfaces and the local behavior of solutions near the interface in problems for one-dimensional reaction-diffusion equations are studied. In all cases explicit formulae for the i...

Asymptotic Properties of the Solutions to Stochastic KPP Equations

Abstract: A reduction method is used to prove the existence and uniqueness of strong solutions to stochastic Kolmogorov–Petrovskii–Piskunov (KPP) equations, where the initial condition may be anticipating. The asymptotic behaviour of the solution for large time and space and the random travelling waves are then studied under two different basic assumptions.

New insight into the generation of ship bow waves

Abstract: The generation of ship bow waves is studied within the framework of potential flow theory. Assuming the ship bow to be slender, or thin, a pattern of the flow is derived using the method of matched asymptotic expansions. This method leads to the determination of three different zones in which three asymptotic expansions are performed and matched. To first order with respect to the slenderness parameter, the near-field flow appears to be two-dimensional in each transverse plane along the bow. However, it is demonstrated that three-dimensional effects are important in front of the ship and must be taken into account in the composite solution. This leads to a three-dimensional correction to be added to the two-dimensional solution along the ship. The asymptotic approach is then applied to explain the structure of the bow flow in connection with experimental observations and numerical simulations.

Multivariable circle criteria for multiparameter singularly perturbed systems

Abstract: A direct approach to the Lur'e problem for singularly perturbed systems (SPS) is proposed. In contrast to previous results, the feedback connection between the linear and nonlinear parts of SPS is allowed to depend essentially on both the slow and the fast variables. The Lur'e problem for multiparameter SPS is studied by the same framework.

Asymptotic solution of zero-sum linear-quadratic differential game with cheap control for minimizer

Abstract: A two-players zero-sum linear-quadratic differential game with the cheap control for the minimizer (the player which minimizes the cost functional) is considered. An asymptotic solution of the singularly perturbed matrix differential Riccati equation with singular terminal conditions for the "fast" variables, associated with this problem, is constructed. Based on this result, some asymptotic properties of suboptimal feedback strategies of the players are investigated. In particular, an asymptotic equilibrium of the suboptimal strategies is established.

Matched asymptotic expansions for bent and twisted rods : applications for cable and pipeline laying

Abstract: The geometrically exact theory of linear elastic rods is used to formulate the general three-dimensional problem of a twisted, clamped rod hanging under gravity and subject to buoyancy forces from a fluid. The resulting boundary-value problem is solved by the method of matched asymptotic expansions. The truncated analytical solution is compared with results obtained from a numerical scheme and shows good agreement. The method is used to consider the near-catenary application of a clamped pipeline.

Matched asymptotic expansion for the low-frequency scattering by a semi-circular trough in a ground plane

Abstract: Plane wave scattering by an electrically small circular trough cut in an infinite ground plane is solved analytically for both the TM and TE polarizations. A quasi-static solution for the inner field based upon a transformation to bipolar coordinates exploits the failure of the narrow trough to react to the detailed wave nature of the incident field and forms the starting point for the method of matched asymptotic expansions. The distant behavior of the inner field must agree with the near behavior of the outer field, which is a radiative solution of the Helmholtz equation. In addition to yielding several analytic terms of the solution in low-order powers and the logarithm of the trough wave size k/spl alpha/ the matching process provides an account of the interplay between all of the physical parameters.

Exponentially improved asymptotic expansions for resonances of an elliptic cylinder

Abstract: Scattering by an elliptic cylinder is considered. Asymptotic expansions for Regge poles and resonances are derived from the uniform asymptotic expansions of Mathieu functions and modified Mathieu functions constructed by applying the Langer-Olver method. In addition, asymptotic expansions for resonances are exponentially improved by emphasizing the role of the symmetries of the scatterer. The splitting up of resonances is then explained in terms of the symmetry breaking O(2)2v.