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Showing papers on "Method of matched asymptotic expansions published in 1991"


Journal ArticleDOI
TL;DR: In this article, the Thouless-Anderson-Palmer equations at low-order were derived for spin glasses with general couplings between spins, and these expansions can be converted into 1/d expansions around mean-field theory.
Abstract: High-temperature expansions performed at a fixed-order parameter provide a simple and systematic way to derive and correct mean-field theories for statistical mechanical models. For models like spin glasses which have general couplings between spins, the authors show that these expansions generate the Thouless-Anderson-Palmer equations at low order. They explicitly calculate the corrections to TAP theory for these models. For ferromagnetic models, they show that their expansions can easily be converted into 1/d expansions around mean-field theory, where d is the number of spatial dimensions. Only a small finite number of graphs need to be calculated to generate each order in 1/d for thermodynamic quantities like free energy or magnetization. Unlike previous 1/d expansions, the expansions are valid in the low-temperature phases of the models considered. They consider alternative ways to expand around mean-field theory besides 1/d expansions. In contrast to the 1/d expansion for the critical temperature, which is presumably asymptotic, these schemes can be used to devise convergent expansions for the critical temperature. They also appear to give convergent series for thermodynamic quantities and critical exponents. They test the schemes using the spherical model, where their properties can be studied using exact expressions.

158 citations


Journal ArticleDOI
TL;DR: In this paper, it is shown how to improve exponentially the well-known Poincare expansions for the generalized exponential integral (or incomplete Gamma function) of large argument by allowing the number of terms in an expansion to depend on the asymptotic variable.
Abstract: By allowing the number of terms in an asymptotic expansion to depend on the asymptotic variable, it is possible to obtain an error term that is exponentially small as the asymptotic variable tends to its limit. This procedure is called “exponential improvement.” It is shown how to improve exponentially the well-known Poincare expansions for the generalized exponential integral (or incomplete Gamma function) of large argument. New uniform expansions are derived in terms of elementary functions, and also in terms of the error function.Inter alia, the results supply a rigorous foundation for some of the recent work of M. V. Berry on a smooth interpretation of the Stokes phenomenon.

82 citations


Journal ArticleDOI
TL;DR: In this article, the authors derived asymptotic approximations for the complex order parameter which are valid in the core region and in the far field, and showed that the line moves in the binormal direction with a curvature-dependent velocity.

76 citations


Journal ArticleDOI
TL;DR: In this paper, a new asymptotic equation for the motion of thin vortex filaments in an incompressible fluid at high Reynolds numbers is derived, which includes self-stretching of the filament in a nontrivial but to some extent analytically tractable, fashion.

73 citations


Journal ArticleDOI
Stephen O'Brien1
TL;DR: In this article, the problem of obtaining asymptotic expressions describing the shape of small sessile and pendant drops is revisited, and the method of matched asymmptotic expansions is used to obtain solutions.
Abstract: The problem of obtaining asymptotic expressions describing the shape of small sessile and pendant drops is revisited. Both cases display boundary-layer behaviour and the method of matched asymptotic expansions is used to obtain solutions. These give good agreement when compared with numerical results. The sessile solutions are relatively straightforward, while the pendant drop displays a behaviour which is both rich and interesting.

71 citations


Journal ArticleDOI
TL;DR: In this article, the effect of fully coupled poroelasticity on an impulsively loaded crack in plane strain is investigated; the solution is obtained using Laplace and Fourier transforms in time and space respectively and then using the Wiener-Hopf technique to solve the resulting functional equations.
Abstract: The effect of fully coupled poroelasticity on an impulsively loaded crack in plane strain is investigated. A formally exact solution for a semi-infinite crack in a linear, isotropic, poroelastic medium with a prescribed internal stress is considered; the solution is obtained using Laplace and Fourier transforms in time and space respectively and then using the Wiener-Hopf technique to solve the resulting functional equations. The stress intensity factor is found as a function of the Laplace variable s and is evaluated explicitly for small times and numerically for all times. The problem of a finite length crack embedded in a poroelastic medium under uniform impulsively applied tension at infinity is solved using the method of matched asymptotic expansions for small times. The formal solution for a steadily propagating semi-infinite crack under tension is outlined, the crack-tip fields are examined and the crack-tip stress intensity factors are found as functions of the crack velocity. Analytical solutions for the pore pressure and stress ahead of the crack are obtained and their relevance to the retardation of fracture discussed. The results extend the range of possible solutions of the fully coupled poroelastic equations to mixed boundary-value problems in fracture mechanics. These are fundamental to the study of the interaction between a diffusing pore fluid and the solid elastic skeleton. In particular, time dependent solutions to the symmetric problems of impulsive loadings and explicit solutions to the steady problems are considered.

67 citations


Journal ArticleDOI
TL;DR: Degond and Raviart as mentioned in this paper provided a mathematical framework to this physical theory, by successively investigating the reduced problem (when the perturbation parameter e is set equal to zero) and the boundary layer problem.
Abstract: 187 Degond, P. and P.A. Raviart, An asymptotic analysis of the one-dimensional Vlasov-Poisson system: the Child-Langmuir law, Asymptotic Analysis 4 (1991) 187-214. We perform the asymptotic analysis of the one-dimensional Vlasov-Poisson system when singular boundary data are prescribed. Such a singular perturbation problem arises in the modelling of vacuum diodes under very large applied bias, and gives rise to the well-known "Child-Langmuir law". In this paper, we provide a mathematical framework to this physical theory, by successively investigating the reduced problem (when the perturbation parameter e is set equal to zero) and the boundary layer problem, which gives a sharp qualitative information.

50 citations


Journal ArticleDOI
TL;DR: In this article, the global asymptotic stability of the linear difference equation was studied and sufficient conditions for the global stability of linear difference equations were obtained. But the conditions were not defined.
Abstract: Consider the linear difference equation where by investigating the asymptotic behavior first of the nonoscillatory solutions and then of the oscillatory solutions of Eq.(1) we obtain sufficient conditions for the global asymptotic stability of Eq.(1)

46 citations


Journal ArticleDOI
TL;DR: In this article, a new analysis of Petrov-Galerkin finite element methods for solving linear singularly perturbed two-point boundary value problems without turning points is given.
Abstract: We give a new analysis of Petrov-Galerkin finite element methods for solving linear singularly perturbed two-point boundary value problems without turning points. No use is made of finite difference methodology such as discrete maximum principles, nor of asymptotic expansions. On meshes which are either arbitrary or slightly restricted, we derive energy norm and L norm error bounds. These bounds are uniform in the perturbation parameter. Our proof uses a variation on the classical Aubin-Nitsche argument, which is novel insofar as the L bound is obtained independently of the energy norm bound.

32 citations


Journal ArticleDOI
TL;DR: In this paper, matched asymptotic expansions for surface water waves trapped by fixed bodies in a deep channel with vertical walls were constructed by using simple properties of the bodies, and approximate relations between the frequency of trapped waves and the channel width were obtained in terms of simple properties.
Abstract: Solutions are constructed by the method of matched asymptotic expansions for surface water waves trapped by fixed bodies in a deep channel with vertical walls. In particular, approximate relations between the frequency of trapped waves and the channel width are obtained in terms of simple properties of the bodies. A number of geometries are considered

30 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of finding the basin of attraction of an equilibrium point of a Q?s planar system of differential equations (Q = f(x), where f is the linear part of the system at the equilibrium point).

Journal ArticleDOI
TL;DR: In this paper, a technique for computing asymptotic expansions of combinatorial quantities from their recursion relations is presented, which is applied to the Stirling numbers of the first and second kinds.
Abstract: A technique for computing asymptotic expansions of combinatorial quantities from their recursion relations is presented. It is applied to the Stirling numbers of the first and second kinds, s(n,k) and S(n,k), for n>1 and three ranges of k:(i) k=O(1), (ii) n−k=O(1), (iii) k>1, 0

Journal ArticleDOI
TL;DR: In this article, a new algorithm for the asymptotic expansion of a solution to an initial value problem for systems of ordinary differential equatioms is presented, which leads to the reduction of numerical effort needed to achieve a given accuracy M compm-ed with the st~mdard uymptotlc expansion method.
Abstract: Atmtract--Pre~mt ed in this paper is a new algorithm for the asymptotic expansion of a solution to an initial value problem for ~ngularly perturbed (stiff) systems of ordinary dlfferemtial eqtmtlons. This algorithm is related to the Chapman-Enskog asymptotic expamdon method mind in the kinetic theory to derive the equatiorm of hydrodynamics, whcxeas the standard algorithm pertains to the Hilbert approach known to give inferior results. In cases of systems of ordinary differential equatioms the new algorithm leads to the reduction of numerical effort needed to achieve a given accuracy M compm-ed with the st~mdard uymptotlc expansion method. The proof of the asymptotic c~vc=lpmce is given. The numerical example demonstrates the feasibility of the new approach.

Journal ArticleDOI
TL;DR: In this paper, matched asymptotic expansions are used to show that the added mass limits for the two cases differ by an amount that depends very simply on the geometry of the bodies.

Journal ArticleDOI
TL;DR: In this article, effective sufficient and necessary conditions for asymptotic stability of the trivial solution of the differential equation with "maxima" were found, as well as necessary and sufficient conditions for a trivial solution with "minima".
Abstract: Differential equations with “maxima” are considered. Effective sufficient as well as necessary and sufficient conditions for asymptotic stability of the trivial solution are found.



Journal ArticleDOI
TL;DR: Canonical polynomials are constructed as new basis for collocation solution in the smooth region which is superposed with an exponential function in the boundary layer region for the linear problem.
Abstract: This paper concerns the numerical solutions of two-point singularly perturbed boundary value problems for second order ordinary differential equations. For the linear problem, Canonical polynomials are constructed as new basis for collocation solution in the smooth region which is superposed with an exponential function in the boundary layer region. Numerical examples are given which show that the exponential fitting leads to a numerical asymptotic procedure. Extension to nonlinear problems is demonstrated by an example.


Journal ArticleDOI
TL;DR: In this paper, matched asymptotic expansions are applied to the flow analysis of a three-dimensional thin wing, moving uniformly in very close proximity to a curved ground surface.
Abstract: The method of matched asymptotic expansions is applied to the flow analysis of a three-dimensional thin wing, moving uniformly in very close proximity to a curved ground surface. Four flow regions, i.e. exterior, bow, gap, and wake, are analysed and matchcd in an appropriate sequence. The solutions in expansions up to third order are given both in nonlinear and linear cases. It is shown that the flow above the wing is reduced to a direct problem, and the flow beneath it appears to be a twodimensional channel flow. The wake assumes a vortex-sheet structure close to the curved ground, undulating with the amplitude of the ground curvature, and the flow beneath it is also two-dimensional channel flow. As a consequence, an equivalence is found between the extreme curved-ground effect and the corresponding flat-ground effect, which can be treated by the image method.

Journal ArticleDOI
TL;DR: In this article, an algorithm for constructing a formal asymptotic expansion of solutions of an equation of divergence form of arbitrary order is presented, where the solvability conditions for these problems lead to an averaged equation (system) with constant coefficients.
Abstract: Elliptic equations of arbitrary order with smooth, rapidly oscillating coefficients are considered. An algorithm is set forth for constructing a formal asymptotic expansion of solutions of such equations. The algorithm consists in the successive solution of a number of periodic problems. The solvability conditions for these problems lead to an averaged equation (system) with constant coefficients. It is proved that if the solution of the equation is bounded and converges to some limit in a suitable sense, then the limit function (vector) satisfies the averaged equation (system).An asymptotic expansion of solutions of an equation of divergence form of arbitrary order is constructed. This makes it possible to obtain for such equations estimates of the form where is the order of the equation, is a solution of the equation, and comprises terms of the asymptotic expansion.Bibliography: 22 titles.

Journal ArticleDOI
TL;DR: A singularly perturbed model problem with multiple distinct regular singular points is studied in this paper, where the uniform approximation of Wazwaz and Hanson (1986) is effectively extended to many singular points, thus establishing a generalized version of that theorem where the classic inner and outer expansions are not employed.

Proceedings ArticleDOI
14 May 1991
TL;DR: In this paper, the in-bound problem in singularly perturbed discrete-time systems is considered and a stability criteria based on the frequency domain representation is derived, and two examples are used to demonstrate the proposed scheme.
Abstract: The in -bound problem in singularly perturbed discrete-time system is considered. A stability criteria based on the frequency domain representation is derived. Two examples are used to demonstrate the proposed scheme. >

Journal ArticleDOI
Martin Corless1
TL;DR: In this article, a class of linear singularly perturbed systems with singular perturbation parameter µ > 0 is considered and it is shown that the full-order system is asymptotically stable for sufficiently small µ.
Abstract: A class of linear singularly perturbed systems with singular perturbation parameter µ > 0 is considered. To assure asymptotic stability of the full-order system for sm > 0 sufficiently small, it is customary to require that both the reduced-order system (µ = 0) and the boundary-layer system are asymptotically stable. Here we relax the requirement on the boundary-layer system to stability (i.e., stability, but not necessarily asymptotic stability) and show that, subject to one additional condition, the full-order system is asymptotically stable for sufficiently small µ. The result is illustrated by an application in which we consider the stability robustness of a feedback-controlled mechanical system with respect to an unmodeled flexibility.

Journal ArticleDOI
TL;DR: In this paper, the problem of the Basymptotic representation of solutions of a regularly perturbed system of differential equations with impulse action on surfaces and of differential equation with discontinuous righthand side is considered.
Abstract: The problem of the B-asymptotic representation, with respect to a small parameter, of solutions of a regularly perturbed system of differential equations with impulse action on surfaces and of differential equations with discontinuous righthand side is considered. Theorems concerning the B-analytic dependence of solutions on the small parameter are proved. Algorithms for calculating the coefficients of the expansion are developed.

Journal ArticleDOI
Jet Wimp1
TL;DR: In this paper, the authors discuss five topics of current interest in asymptotic analysis: the use of probabilistic methods to estimate the growth of combinatorial sequences, asymPTotic methods in the theory of random walks, the estimation of solutions of difference equations, asmptotic expansions in generalized scales, and the computation by asymptonotic methods of distributions whose moments are known.

Journal ArticleDOI
TL;DR: In this paper, a hybrid frictional-kinetic equations are used to predict the velocity, grain temperature, and stress fields in hoppers, and an approximate semi-analytical solution is constructed using perturbation methods.

Journal ArticleDOI
TL;DR: In this article, the boundary value problems for a class of linear ODEs with turning points are studied under suitable assumptions, and the author proves the existence and uniqueness of solutions, and obtains the uniformly valid asymptotic expansions of the solutions.


Journal Article
TL;DR: In this article, the forced flow induced by a continuously moving semi-infinite horizontal flat plate has been studied and the Navier-Stokes equations are solved by the method of matched asymptotic expansions and by repeated use of this method several flow regions are identified.
Abstract: We are concerned with the forced flow induced by a continuously moving semi-infinite horizontal flat plate which has a velocity that is proportional to a power law variation with the distance along the surface of the plate. The flow domain is taken to be the region bounded by the horizontal surface and a second vertical wall which forms a right-angled corner. The full Navier-Stokes equations are solved by the method of matched asymptotic expansions and by repeated use of this method several flow regions are identified