Showing papers on "Method of matched asymptotic expansions published in 1989"

Asymptotic techniques for use in statistics

Abstract: Preliminary notions.- Some basic limiting procedures.- Asymptotic expansions.- Edgeworth and allied expansions.- Miscellany on multivariate distributions.- Multivariate asymptotic expansions.- Expansions for conditional distributions.- Postscript.

Front migration in the nonlinear Cahn-Hilliard equation

Abstract: The method of matched asymptotic expansions is used to describe solutions of the nonlinear Cahn-Hilliard equation for phase separation in N > 1 space dimensions The expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature On the dominant (slowest) timescale the interface velocity is determined by the mean curvature of the interface, by a non-local relation which is identical to that in a well-known quasi-static model of solidification, which exhibits a shape instability discovered by Mullins & Sekerka (J appl Phys 34, 323-329 (1963)) On a faster timescale, the Cahn-Hilliard equation regularizes a classic two-phase Stefan problem Similarity solutions of the two-phase Stefan problem should describe boundary layers Existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

Asymptotic methods in equations of mathematical physics

Abstract: Part 1 The stationary phase method: on asymptotic expansions the stationary phase method the stationary phase method, the multidimensional case the problem of waves on the surface of a liquid the asymptotic behaviour of the Fourier transform of a function concentrated on a smooth closed surface. Part 2 The WKB method for ordinary differential equations: the asymptotic behaviour of solutions of a homogeneous equation the scattering problem the asymptotic behaviour of solutions of boundary-value problems. Part 3 Partial differential equations of the first order and characteristics for equations of higher order: quasilinear partial differential equations of the first order general partial differential equations of the first order the Hamilton-Jacobi equation example propagation of light waves in an inhomogenous medium characteristic surfaces for differential operators of high order, connection with the well-posedness of the Cauchy problem the search for characteristic surfaces. Part 4 Propagation of discontinuities, problems with rapidly oscillating data: the Leibniz formula problems with rapidly oscillating initial data discontinuous solutions of equations. Part 5 The Maslov canonical operator: the problem of scattering of a plane wave in an inhomogenous medium the Lagrange maniford the precanonical operator the canonical operator, construction of a formal asymptotic solution a field in an isotropic medium with parabolic wave front more general problems. Part 6 Elliptic problems in a bounded domain: Sobolev-Slobodetskii spaces elliptic problems elliptic problems with a parameter inversion of a finitely-meromorphic Fredholm family of operators. Part 7 Equations and systems with constant coefficients in Rn: equations with a non-zero characteristic polynomial equations and systems of the type of the Helmholtz equation radiation conditions the principle of limiting absorption. Part 8 Elliptic equations with variable coefficients and boundary-value problems in the exterior of a bounded domain: solubility and a priori estimated of solutions of exterior boundary-value problems the principle of limiting absorption for exterior problems. Part 9 Analytic properties of the resolvent of operators that depend polynomially on a parameter: equations with constant coefficients equations with variable coefficients and problems in the interior of a bounded domain the asymptotic behaviour of solutions of exterior problems for small frequencies. Part 10 Short-wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour of solutions of hyperbolic equations as t introduction short-wave asymptotic behaviour as t of solutions of mixed problems. Part 11 Quasiclasical approximations in stationary scattering problems: the asymptotic behaviour of the solution of the scattering problem and the amplitude of the scattering proof of theorems 1 and 2. Part contents...

Matched Asymptotic Expansions: Ideas and Techniques

Abstract: I. Introduction: General Concepts of Singular Perturbation Theory.- II. Layer-type Problems. Ordinary Differential Equations.- III. Layer-type Problems. Partial Differential Equations.- List of References.

An asymptotic description of transient settling and ultrafiltration of colloidal dispersions

Abstract: Sedimentation and ultrafiltration are important processes for removing solids from suspensions. The Kynch theory describes the transient settling of noncolloidal particles forming an incompressible sediment by providing a solution to the convective conservation equation. This solution predicts the existence of several different regions as settling progresses. Subsequent treatments have accounted for compressibility within the sediment. These modifications focus largely on stretching the Kynch theory to fit the problem at hand rather than on formulating a new model to include the relevant physics. In this paper a model of sedimentation for colloidal systems is presented by including a diffusion term in the governing equation. In the regions above the sediment, this term acts as a small perturbation to the Kynch theory. Within the sediment, owing to the high solid volume fraction, diffusion is comparable to convection. Slow compression to the maximum sediment volume fraction contrasts the incompressibility of the Kynch theory. Application of the method of matched asymptotic expansions to the conservation equation enables the formulation of a complete description of the settling process, and, in particular, the volume fraction evolution in the sediment. This method is also applied to the related ultrafiltration process. Where the properties of the sediment, or filtercake, are important, such as in ceramics manufacturing, a quantitative understanding of its formation is of obvious value.

Asymptotic behavior of the solutions of the dead-core problem

Abstract: u, u, + Pup, = 0. (1.1) This equation originates from the study of a class of models for the reaction-diffusion processes of a gas inside a chemical reactor [2, 81. From the mathematical point of view the major interest of equation (1.1) lies in the competition of the linear diffusion and the nonlinear “fast” (p 0, t > 0, with initial and boundary conditions

Shadowing lemma and singularly perturbed boundary value problems

Abstract: A complete procedure is given to determine the outer and inner expansions of a singularly perturbed boundary value problem in $\mathbb{R}^n $. The validity of such expansions is deduced from a gene...

Effects of bubbly layers on wave propagation

Abstract: The transmission and reflection of an acoustic wave through a bubbly layer are investigated. Nonlinear model equations for the bubbly liquid are used. These equations first linearized and solved exactly for a time‐harmonic incident wave. Then, numerical solutions of the nonlinear system are found. It is found that even for a small amplitude incident pressure wave, it is possible to have nonlinear transmitted and reflected waves. The limit where the bubbly layer is thin relative to the incident wavelength is also considered. By using the method of matched asymptotic expansions, it is found that the bubbly layer can be replaced by an interface subject to the continuity of pressure and an effective nonlinear jump condition. The latter involves the internal effects of the layer. Solutions of this limiting case are compared with the numerical results and good agreement is found even when the ratio of the bubbly layer thickness to the incident wavelength is of order 1.

Mixmaster cosmological model in theories of gravity with a quadratic Lagrangian

Abstract: We use the method of matched asymptotic expansions to examine the behavior of the vacuum Bianchi type-IX mixmaster universe in a gravity theory derived from a purely quadratic gravitational Lagrangian. The chaotic behavior characteristic of the general-relativistic mixmaster model disappears and the asymptotic behavior is of the monotonic, nonchaotic form found in the exactly soluble Bianchi type-I models of the quadratic theory. The asymptotic behavior far from the singularity is also found to be of monotonic nonchaotic type.

New initial-value method for singularly perturbed boundary-value problems

Abstract: An initial-value method is given for second-order singularly perturbed boundary-value problems with a boundary layer at one endpoint. The idea is to replace the original two-point boundary value problem by two suitable initial-value problems. The method is very easy to use and to implement. Nontrivial text problems are used to show the feasibility of the given method, its versatility, and its performance in solving linear and nonlinear singularly perturbed problems.

The mathematics of ship slamming

Abstract: Motivated by the motion of a ship in a heavy sea, a mathematical model for the vertical impact of a two-dimensional solid body onto a half-space of quiescent, inviscid, incompressible fluid is formulated. No solutions to the full problem are known, but in the case when the impacting body has small deadrise angle (meaning that the angle between the tangent to the profile and the horizontal is everywhere small) a uniformly valid solution is obtained by using the method of matched asymptotic expansions. The pressure on the body is calculated and is in fair agreement with experimental results. The model is generalised for more complicated impacts and the justifications for the model are discussed. The method is extended to three-dimensional bodies with small deadrise angle and solutions are obtained in some special cases. A variations! formulation of the leading order outer problem is derived, which gives information about the solution and leads to an fixed domain scheme for calculating solutions numerically. A partial linear stability analysis of the outer problem is given which indicates that entry problems are stable but exit problems are unstable to small perturbations. A mathematical model for the effect of a cushioning air layer between the body and the fluid is presented and analysed both numerically and in appropriate asymptotic limits. Finally, the limitations of the models are discussed and directions for future work indicated.

Asymptotic solution of some singularly perturbed Fredholm integral equations

Abstract: A formal approach is described for obtaining the asymptotic solution to a class of singularly perturbed Fredholm integral equations. The approach is illustrated through application to some example problems which arise in heat transfer, diffraction theory, crack mechanics, Markov processes and low order eigenvalue problems.

Three-dimensional atmospheric entry problem using method of matched asymptotic expansions

Abstract: A composite solution consisting of an outer solution, an inner solution, and a common solution is obtained. The outer solution arises from the gravitationally dominant region, whereas the aerodynamically dominant region contributes to the inner solution. The common solution accounts for the overlap between the outer and inner regions. In contrast to previous work, this simplified methodology yields explicit analytical expressions for various compounds of the composite solution without resorting to any type of transcendental equation that can be solved only by numerical methods. The present method is application for obtained autonomous guides and control strategies for a variety of aerospace vehicles. >

Singularly perturbed integral equations with endpoint boundary layers

Abstract: A methodology is developed for certain singularly perturbed integral equations whose solution exhibits a boundary layer at either one or both endpoints of its domain. The scheme indicates how the width and magnitude of the boundary layer(s) are determined, and gives a construction of an asymptotic solution as well. Application to both linear and nonlinear equations is discussed and illustrated with examples.

The stretching of a thin viscous inclusion and the drawing of glass sheets

Abstract: A thin, two‐dimensional inclusion of a highly viscous fluid is deformed by an external Stokes flow—a suggested mechanism by which the Earth’s oceanic crust is entrained and thinned by the mantle. The method of matched asymptotic expansions is applied with the small expansion parameter being the inverse aspect ratio of the inclusion. The kinematic condition and continuity of shear and normal stresses lead to boundary conditions for the biharmonic equation governing the outer flow in the two cases of stretching by a pure shear and by a simple shear flow. The evolution of the inclusion can then, in principle, be determined. The equations governing the drawing of glass sheets may be derived as a limiting case, and any asymmetry in the centerline of the sheet is deformed over a short time scale before stretching occurs. Several examples are computed, showing how the straightening of the sheet can be slowed down by constraints on the gradients at the two ends. It is shown that an inclusion with smooth initial data cannot break up in a finite time.

Effective Roughness Length for Turbulent Flow over a Wavy Surface

Abstract: A two-equation turbulence model is used to calculate the effective roughness length for two-dimensional turbulent flow over small amplitude, wavy surface topography. The governing equations are solved using the method of matched asymptotic expansions for the case ϵ ≪ 1, δ ≪ 1, where ϵ is the square root of a characteristic drag coefficient for flow over a plane surface and δ is the wave slope. Analytical expressions are derived for the effective roughness length and drag coefficient for flow over stationary topography and over a progressive wave. The results are used to determine the drag coefficient and the form drag for flow over a progressive water wave, and are found to be consistent with observations.

The Asymptotic Solution of a Connection Problem of a Second Order Ordinary Differential Equation

Abstract: The solutions of the equation are discussed in the limit ρ 0. The solutions which oscillate about − |t| as t ∞ have asymptotic expansions whose leading terms are where A+, , A−, and are constants. The connection problem is to determine the asymptotic expansion at + ∞. In other words, we wish to find (A+, ) as functions of A− and The nonlinear solutions with A± not small are analyzed by using the method of averaging. It is shown that this method breaks down for small amplitudes. In this case a solution can be obtained on [0, ∞) as a small amplitude perturbation about the exact nonoscillating solution W(t) whose asymptotic expansion is This is a solution of (1) which corresponds to A+ ≡ 0 in (2). A quantity which determines the scale of the small amplitude response is −W'(0). This quantity is found to be exponentially small. The determination of this constant is shown to reduce to a solution of the equation for the first Painleve transcendent. The asymptotic behavior of the required solution is determined by solving an integral equation.

Singular perturbation method applied to the open-loop discrete optimal control problem with two small parameters

Abstract: The open-loop optimal control of a linear, shift-invariant, singularly perturbed system with two small parameters is considered. The two small parameters are interrelated and approach zero simultaneously. The resulting two-point boundary value problem is put in the singularly perturbed form. A singular perturbation method is developed to obtain approximate solutions composed of an outer series, two initial correction series and two final correction series to recover the lost boundary conditions in the process of degeneration. An example is provided to illustrate the proposed method.

Asymptotic Behaviour of Solutions to Laplace's Tidal Equations At Low Frequencies

Abstract: SUMMARY The asymptotic behaviour of solutions to Laplace’s tidal equations at low frequencies is considered. The method used is based on perturbation in small parameters, these being the ratios of tidal frequency and the coefficient of bottom friction to the angular frequency of the Earth’s rotation. It is shown that the resulting solutions are unstable in that the functions involved in the zero-order approximation are not uniquely determined by the zero-order equations, but depend on first-order terms as well. Because of this instability, direct methods of numerical integration are inefficient. We propose a different procedure, replacing the original set of equations in partial derivatives by ordinary differential equations that have a stable solution. The equations are examined qualitatively. It is shown, in particular, that for the case of an ocean of uniform depth over the whole Earth, they coincide with the well-known Lamb’s equations. The asymptotic behaviour of the solutions is examined as modified by basin shape, bottom topography and bottom friction.

On the method of matched asymptotic expansions

Abstract: The present evaluation of the method of asymptotic expansions (MAE) indicates that the various terms of the common solution of MAE can be generated as polynomials in stretched variables, without actually solving them from the outer solution, as is currently the practice. It is also noted that the common solution of the MAE and the intermediate solution of the singular-perturbation method are the same; these methods therefore yield identical results for a certain class of problems. Two illustrative problems are treated.

On the asymptotic behaviour at infinity of solutions of the traction boundary value problem

Abstract: Korn's inequalities are proved for star-shaped domains and it is shown how the constants in these inequalities depend on the dimensions of the domain. These inequalities are then used to prove a generalisation of Saint-Venant's Principle for nonlinear elasticity and additionally to establish the asymptotic behaviour of solutions to the traction boundary value problem for a non-prismatic cylinder.

Balanced realizations of singularly perturbed systems

Abstract: Balanced realizations of linear time-invariant singularly perturbed systems are studied. The reduced model of a singularly perturbed system that is obtained, based on the balanced realization, is compared to that obtained by time-scale considerations. An approximate balancing transformation for singularly perturbed systems is constructed based on balancing transformations of the approximating slow and fast subsystems.

The averaging method for weakly nonlinear operator equations

Abstract: A method asymptotic with respect to a small parameter is presented for solving Cauchy problems for the evolution equations where is a linear operator and is a nonlinear operator. It is assumed that the method of regular expansion in powers of leads to secular terms. Such terms can be removed by suitably defining how the terms of the asymptotic solution depend on the slow variable . The proposed method is modified for equations of second order in . The possibility of getting rid of the terms secular with respect to , and of applying the asymptotic methods in the case of problems with strong forced resonance, is indicated. Examples are given which illustrate possibilities for the proposed methods.Bibliography: 16 titles.

Vortex filament calculations by Analytical/Numerical Matching with comparison to other methods

Abstract: The calculation of fluid velocity from the Biot-Savart law integrated over vortex filaments has traditional1 been computationally expensive Discretizing the filaments into N vortex elements results l in order N elemental velocity evaluations per time step Further, the elemental resolution has been governed by the need to resolve the large velocity gradients in the near field of the filaments, resulting in unnecessarily high element densities in the far field, where the velocities are slowly varying The method of Analytical/Numerical Matching (ANM) improves the efficiency of the filament velocity calculation without loss of near-field accuracy This is done by using a far field comprised of computationally inexpensive vortex particles with a large core size for smoothing The near field is done by an analytical correction which uses a thin physically correct core size to predict the large rapidly varying near-field velocities, and a second correction with the large core size to cancel the local vortex particle error and match to the far-field solution As such, the ANM method is similar to the method of matched asymptotic expansions The entire approach has been analytically linearized, which provides additional efficiency and allows unique solution opportunities Examples are given which illustrate the efficiency and accuracy of the ANM method in vortex dynamics calculations