Showing papers on "Method of matched asymptotic expansions published in 1994"

Convergence of the Cahn-Hilliard equation to the Hele-Shaw model

Abstract: We prove that level surfaces of solutions to the Cahn-Hilliard equation tend to solutions of the Hele-Shaw problem under the assumption that classical solutions of the latter exist. The method is based on a new matched asymptotic expansion for solutions, a spectral analysis for linearizd operators, and an estimate for the difference between the true solutions and certain approximate ones.

Magnetocentrifugally driven flows from young stars and disks. 2: Formulation of the dynamical problem

Abstract: We formulate the dynamical problem of a cool wind centrifugally driven from the magnetic interface of a young star and an adjoining Keplerian disk. We examine the situation for mildly accreting T Tauri stars that rotate slowly as well as rapidly accreting protostars that rotate near break-up. In both cases a wind can be driven from a small X-region just outside the stellar magnetopause, where the field lines assume an open geometry and are rooted to material that rotates at an angular speed equal both to the local Keplerian value and to the stellar angular speed. Assuming axial symmetry for the ideal magnetohydrodynamic flow, which requires us to postpone asking how the (lightly ionized) gas is loaded onto field lines, we can formally integrate all the governing equations analytically except for a partial equation that describes how streamlines spread in the meridional plane. Apart from the difficulty of dealing with PDEs of mixed type, finding the functional forms of the conserved quantities along streamlines - the ratio beta of magnetic field to mass flux, the specific energy H of the fluid in the rotating frame, and the total specific angular momentum J carried in the matter and the field - constitutes a standard difficulty in this kind of (Grad-Shafranov) formalism. Fortunately, because the ratio of the thermal speed of the mass-loss regions to the Keplerian speed of rotation of the interface constitutes a small parameter epsilon, we can attack the overall problem by the method of matched asymptotic expansions. This procedure leads to a natural and systematic technique for obtaining the relevant functional dependences of beta, H, and J. Moreover, we are able to solve analytically for the properties of the flow emergent from the small transsonic region driven by gas pressure without having to specify the detailed form of any of the conserved functions, beta, H, and J. This analytical solution provides inner boundary conditions for the numerical computation in a companion paper by Najita & Shu of the larger region where the main acceleration to terminal speeds occurs.

Asymptotic behavior of solutions to the Coagulation-Fragmentation Equations. II. Weak Fragmentation

Abstract: The discrete coagulation-fragmentation equations are a model for the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones. The assumptions made on the fragmentation coefficients have the physical interpretation that surface effects are important. Our results on the asymptotic behavior of solutions generalize the corresponding results of Ball, Carr, and Penrose for the Becker-Doring equation.

Impulsive Differential Equations with a Small Parameter

Abstract: Preliminary notes and auxiliary assertions regularly perturbed impulsive differential equations singularly perturbed differential equations.

A Distributional Theory for Asymptotic Expansions

Abstract: The purpose of this chapter is to present the distributional theory of asymptotic expansions for functions of one variable. This chapter and the next, where the multidimensional expansions are studied, are the central part of the book.

Heavy-traffic asymptotic expansions for the asymptotic decay rates in the BMAP/G/1 queue

Abstract: In great generality, the basic steady-state distributions in theBMAP/G/l queue have asymptotically exponential tails. Here we develop asymptotic expansions for the asymptotic decay rates of these tail probabilities in powers of one minus the traffic intensity. The first term coincides with the decay rate of the exponential distribution arising in the standard heavy-traffic limit. The coefficients of these heavy-traffic expansions depend on the moments of the service-time distribution and the derivatives of the Perron-Frobenius eigenvalue δ(z) of the BMAP matrix generating function D (z) at z = 1. We give recursive formulas for the derivatives δ(κ) (1). The asymptotic expansions provide the basis for efficiently computing the asymptotic decay rates as functions of the traffic intensity, i.e., the caudal characteristic curves. The asymptotic expansions also reveal what features of the model the asymptotic decay rates primarily depend upon. In particular, δ(z) coincides with the limiting time-average of the ...

Uniform Airy-type expansions of integrals

Abstract: A new method for representing the remainder and coefficients in Airy-type expansions of integrals is given. The quantities are written in terms of Cauchy-type integrals and are natural generalizations of integral representations of Taylor coefficients and remainders of analytic functions. The new approach gives a general method for extending the domain of the saddle-point parameter to unbounded domains. As a side result the conditions under which the Airy-type asymptotic expansion has a double asymptotic property become clear. An example relating to Laguerre polynomials is worked out in detail. How to apply the method to other types of uniform expansions, for example, to an expansion with Bessel functions as approximants, is explained. In this case the domain of validity can be extended to unbounded domains and the double asymptotic property can be established as well.

Light-off behavior of catalytic converters

Abstract: This paper studies a system of partial differential equations modeling the behavior of an automotive catalytic converter. The particular phenomenon considered in detail is light-off, when the temperature of the converter changes dramatically from cold to hot somewhere within the converter. The initial position of light-off and the subsequent movement of this steep jump in temperature toward the inlet of the converter are analyzed. The method of matched asymptotic expansions is used to study light-off and to derive approximate formulas that determine its behavior in the limits of small heat diffusion and large activation energy. Numerical calculations are presented and are used to compare with the analytical formulas. These calculations reveal that the asymptotic results for small diffusion are quantitatively accurate, while those for small diffusion and large activation energy give the qualitative behavior of the solution but give poor quantitative predictions for the range of parameters encountered in pr...

Nineteenth-century roots of the boundary-layer idea

Abstract: Ludwig Prandtl is properly credited with the development of the boundary-layer idea in viscous flow, which was generalized to the method of matched asymptotic expansions. However, the idea of matching a global and a local approximation was previously applied to specific problems in the nineteenth century by a number of well-known natural philosophers, starting with Laplace in 1805. The author summarizes their applications in hydrostatics, hydrodynamics, elasticity, electrostatics, and acoustics, with particular attention to the process of matching.

Exponentially Improved Asymptotic Solutions of Ordinary Differential Equations. II Irregular Singularities of Rank One

Abstract: Re-expansions are found for the optimal remainder terms in the well-known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one. The re-expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions, as well being considerably more accurate and providing a smooth interpretation of the Stokes phenomenon. They are also of strikingly simple form. In addition, explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained.

An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation

Abstract: We examine traveling-wave solutions for a generalized nonlinear-diffusion Fisher equation studied by Hayes [J. Math. Biol. 29, 531–537 (1991)]. The density-dependent diffusion coefficient used is motivated by certain polymer diffusion and population dispersal problems. Approximate solutions are constructed using asymptotic expansions. We find that the solution will have a corner layer (a shock in the derivative) as the diffusion coefficient approaches a step function. The corner layer at z = 0 is matched to an outer solution for z 0 to produce a complete solution. We show that this model also admits a new class of nonphysical solutions and obtain conditions that restrict the set of valid traveling-wave solutions.

Eulerian Number Asymptotics

Abstract: Asymptotic formulae for the eulerian numbers A(n, k) for n ≫ 1 are obtained directly from their recursion relation by the ray method and the method of matched asymptotic expansions. These are formal methods, so they do not prove that the formulae are asymptotic, although they suggest it. The formulae agree with the previously known results where those results are valid. They also agree very well with the exact values of A(n, k) for 1 ≼ n ≼ 170, and the agreement improves as n increases.

Asymptotic modelling for separating boundary layers

Abstract: The method of matched asymptotic expansions is used to explain bow the triple deck structure in a boundary layer can be formed. In the context of a laminar steady flow of an incompressible fluid over a flat plate, a theory is developed to explain the separation over significant wall disturbances. In particular, we show that the triple deck structure is the first perturbation that can both displace the classical boundary layer and cause separation of the flow. Above this exist a serie of perturbations, smaller but “stronger”, that cause a separation of the boundary layer without displacing it. This serie is limited by the smallest perturbation compatible with the hypothesis of the theory, thus leading to a theory in double deck.

New approach to block-diagonalization of singularly perturbed systems by Taylor expansion

Abstract: Block-diagonalization of a singularly perturbed system requires the solution of the Riccati equation and the Lyapunov equation. A new approach is suggested for both equations, using Taylor expansions. The convergence is studied in detail; it is shown that it is ensured under restrictions which are less restrictive than the case of previously suggested methods. >

Asymptotic approximation of convex curves

Abstract: L. Fejes Toth gave asymptotic formulae as n → ∞ for the distance between a smooth convex disc and its best approximating inscribed or circumscribed polygons with at most n vertices, where the distance is in the sense of the symmetric difference metric. In this paper these formulae are extended by specifying the second terms of the asymptotic expansions. Tools are from affine differential geometry.

On the connection between nonlinear differential-algebraic equations and singularly perturbed control systems in nonstandard form

Abstract: Considers a class of control systems represented by nonlinear differential equations depending on a small parameter. The systems are not in the standard singularly perturbed form, and therefore, one of the challenges is to show that the control systems do represent singularly perturbed two-time-scale systems. Assumptions are introduced which guarantee that an equivalent representation for the systems can be obtained in the standard singularly perturbed form, thereby justifying the two-time-scale property. The equations for the slow dynamics are characterized by a set of differential-algebraic equations which have been studied previosly in the literature. The fast dynamics are characterized by differential equations. Both the slow and the fast dynamics are easily derived and are defined in terms of variables that define the original control system. Control design for the class of systems being considered is studied using the composite control approach. >

Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations

Abstract: In the neighborhood of an irregular singularity of rank one at infinity, a differential equation of the form dw „, x dw , N has well-known asymptotic solutions of the form oo oo n=0 n—Q in which Ai, A2, Ml? i ^2 are constants. It is proved that for large n, the coefficients an,i, and an>2 can be expanded in asymptotic series of inverse factorials with explicit coefficients.

An asymptotic theory for dynamic response of anisotropic inhomogeneous and laminated cylindrical shells

Abstract: An asymptotic theory is developed for dynamic response of anisotropic inhomogeneous and laminated cylindrical shells. The formulation is based on three-dimensional elasticity without a priori assumption. By means of asymptotic expansion with multiple time scales, the basic equations are decomposed into differential equations of various orders which can be integrated successively. It is shown that the three-dimensional asymptotic solution to the problem can be determined hierarchically by solving the two-dimensional equations in the classical laminated shell theory. Modifications to the solution are obtained in a systematic way by treating the higher-order equations and eliminating the secular terms. Various thickness effects, such as transverse shear and rotary inertia, can be accounted for in a natural and consistent manner. It is demonstrated along the way that a uniform expansion that yields asymptotic solution valid regardless of the time span is obtained with the use of multiple time scales, whereas a straightforward expansion fails to produce valid results for long times. The theory is illustrated by determining the vibration characteristics of a multilayered cross-ply cylindrical shell.

Use of central-difference operators for solution of singularly perturbed problems

Abstract: Singularly perturbed second-order elliptic equations with boundary layers are considered. These may be considered as model problems for the advection of some quantity such as heat or a pollutant in a flow field or as linear approximations to the Navier-Stokes equations for fluid flow. Numerical methods composed of central-difference operators on special piece-wise-uniform meshes are constructed for the above problems. Numerical results are obtained which show that these methods give approximate solutions with error estimates that are independent of the singular perturbation parameter. An open theoretical problem is posed.