Showing papers on "Method of matched asymptotic expansions published in 1990"
TL;DR: Asymptotic expansions of renormalized Feynman amplitudes in limits of large momenta and/or masses are proved in this article, where the coefficients of these expansions are homogeneous within a regularization of dimensional or analytic type.
Abstract: Asymptotic expansions of renormalized Feynman amplitudes in limits of large momenta and/or masses are proved. The corresponding asymptotic operator expansions for theS-matrix, composite operators and their time-ordered products are presented. Coefficient functions of these expansions are homogeneous within a regularization of dimensional or analytic type. Furthermore, they are explicitly expressed in terms of renormalized Feynman amplitudes (at the diagrammatic level) and certain Green functions (at the operator level).
134 citations
60 citations
TL;DR: A simple and relatively obscure asymPTotic expansion derived by Raudys is found to yield better approximation than the well-known asymptotic expansions.
51 citations
TL;DR: A survey of some methods for finding formal solutions of differential equations, including methods forFinding power series solutions, elementary and liouvillian solutions, first integrals, Lie theoretic methods, transform methods, asymptotic methods.
51 citations
TL;DR: In this paper, a numerical method for solving linear and nonlinear second-order singularly perturbed boundary-value problems is presented, where the required approximate solution is obtained by solving the reduced problem and one or two suitable initial value problems directly deduced from the given problem.
Abstract: In this paper, we present a numerical method for solving linear and nonlinear second-order singularly perturbed boundary-value-problems. For linear problems, the method comes from the well-known WKB method. The required approximate solution is obtained by solving the reduced problem and one or two suitable initial-value problems, directly deduced from the given problem. For nonlinear problems, the quasilinearization method is applied. Numerical results are given showing the accuracy and feasibility of the proposed method.
39 citations
TL;DR: In this article, the stability problem for core-annular flow of very viscous crude oil and water is treated by the method of matched asymptotic expansions using e=m/Rα as a small parameter.
Abstract: It is known that the stability problem for core‐annular flow of very viscous crude oil and water is singular, the water annulus appears to be inviscid with boundary layers at the pipe wall and at the interface. In the present paper, this singular problem is treated by the method of matched asymptotic expansions using e=m/Rα as a small parameter. There are two cases of instability corresponding to different positions of the critical point in the annulus. One case is when the critical point is far away from the interface, the other is when the critical point is close to the interface within a distance of order e1/3. In both cases, we derive the equations for the eigenvalues, and give the explicit forms for the neutral curves. The stability problem is also treated by the modified finite element code used by Hu and Joseph [J. Fluid Mech. 205, 359 (1989); Phys. Fluids A 1, 1659 (1989)], taking into account the boundary layers at the pipe wall and at the interface. The results of the two methods agree where the...
37 citations
TL;DR: In this article, a new method is presented for obtaining the asymptotic behavior of the number of partitions of an integer n into s parts of various kinds, for both n and s large.
Abstract: A new method is presented for obtaining the asymptotic behavior, for both n and s large, of the number of partitions of an integer n into s parts of various kinds. It involves solving a recursion equation satisfied by the number of partitions, using asymptotic methods of applied mathematics such as the WKB method, the ray method, and the method of matched asymptotic expansions. It is applied to partitions of n into s parts, into s distinct parts, into srth powers, and into s parts which differ by at least d. The method can be applied to many other problems in partition theory and in combinatorics.
31 citations
TL;DR: In this article, the first boundary value problem for a parabolic-type equation in an π -dimensional band domain is considered, where the coefficients of the equation and the free term have discontinuities of the first kind on surfaces parallel to the sides of the band domain.
Abstract: The first boundary value problem is considered for a parabolic-type equation in an π -dimensional band domain. Highest-order derivatives of the equation contain a parameter taking arbitrary values in the half-open interval (0,1]. With the parameter value equal to zero, the equation degenerates into a transport equation which is a first-order equation containing derivatives with respect to spatial variables (this leads to an appearance of regular boundary layers). The coefficients of the equation and the free term have discontinuities of the first kind on surfaces parallel to the sides of the band domain. On these surfaces there operate lumped sources. To solve the problem by using grids condensing in boundary and internal layers, a difference scheme is constructed which converges uniformly in the parameter everywhere in the domain. An investigation of processes of complex heat and mass transfer in moving non-homogenious media leads to boundary value problems for singularly perturbed parabolic equations with convective terms [3,5,10]. In such problems, when the parameter tends to zero there may appear boundary layers in neighbourhoods of side boundaries of the domain depending on the mutual location of characteristics of the singular equation and the internal normal to the boundary of the domain. In neighbourhoods of surfaces of the discontinuity of the coefficients of the equation and the surfaces on which the lumped sources operate there arise internal layers. To solve the problem, an approach is suggested in which classical difference approximations of boundary value problems are used on specially condensing grids (such approach was first proposed in [2]). For the one-dimensional parabolic equation in case of smooth coefficients with lumped sources absent, in [1,12] the authors constructed a difference scheme uniformly converging in the parameter. When constructing the scheme they used the adaptation method [4,7]. The singularly perturbed boundary value problem for an equation with discontinuous coefficients in the case where the singular equation does not contain derivatives in spatial variables was considered in [17]. Note that the terms of the differential equation analysed in this paper unlike the equation contained in [17] are not uniformly bounded in the parameter. 1. FORMULATION OF PROBLEM IN CLASS OF DISCONTINUOUS COEFFICIENTS Let us consider in an η-dimensional domain D with the boundary Γ (Γ = D\\D) the Dirichlet problem for the elliptic equation with discontinuous coefficients and a free term. Let the ^discontinuity surfaces partition the domain D into subdomains D^\\ j = l,...,/; D = \\JjD®9 £ nl)0') = 0, ι */. In each subdomain G® =D& χ (0,7] the following equation is given: L ωφ,Ο = {eL^ + L^}u(X,t) =/(jc,i) , (r,f) € G& . (1.
30 citations
TL;DR: In this article, the problem of mass or heat transfer into steady shear flow from a conducting film, mounted flush in an insulating wall, is considered and the transfer rate depends only on the Peclet number and on the geometry of the film.
Abstract: The problem of mass or heat transfer into steady shear flow from a conducting film, mounted flush in an insulating wall, is considered. Under suitable conditions, the transfer rate depends only on the Peclet number (a dimensionless measure of shear rate) and on the geometry of the film. We present a calculation of the transfer rate for a disc-shaped film at low Peclet number by the method of matched asymptotic expansions
30 citations
TL;DR: In this paper, the first boundary value problem for the elliptic equation sAu-x^u=f is considered on the rectangle D = (Ο,^χίΞ,^); the parameter ε takes arbitrary values in the half-open segment (0,1).
Abstract: This paper deals for the first time with grid approximation for the singular equation degenerating at the boundary. The problem under consideration is a model one for processes of complex heat and mass transfer. The distinctive feature of the problem is a rapid change of the width of the boundary layer along certain segments of the boundary. Such a behaviour of the solution does not permit us to indicate the rectangular grid on which a variation of the solution to the differential problem at neighbouring grid nodes tends to zero, with the number of grid nodes rising (Lemma 4.1). This fact makes it impossible to construct a scheme (converging uniformly in the parameter) of the condensing grid method if only singular rectangular grids are used. The paper suggests for such singular problem a difference scheme converging uniformly in the parameter. When constructing the scheme we use condensing grids and special coordinate systems in a neighbourhood of the rapid change of the width of the boundary layer. The first boundary value problem for the elliptic equation sAu-x^u=f is considered on the rectangle D = (Ο,^χίΟ,^); the parameter ε takes arbitrary values in the half-open segment (0,1]. The limit equation (the elliptic equation for ε = 0) degenerates at the side xl = 0. Unlike the singularly perturbed boundary value problems considered earlier, in this problem the width of the boundary layer in a neighbourhood of the sides x2 = 0,^ defined by the quantity ε\"*^/ rapidly changes along the axis *r It is shown that there is no classical difference scheme converging uniformly in the parameter on rectangular grids in the original variables x^ and jc2. To solve the boundary value problem, we construct a difference scheme converging uniformly in the parameter. When constructing the scheme we use new coordinates in the neighbourhood of the sides x2 = Q,d2 and grids condensing in boundary layers.
27 citations
TL;DR: In this paper, a class of coupled equations for composite problems in elasticity theory for non-uniform bodies was studied, where the approximate solutions, constructed using the method described in /1/ by reducing the problem to a finite system of algebraic equations, are two-sided asymptotically exact in terms of the characteristic geometric parameter.
TL;DR: In this article, the Navier-Stokes Langevin equations for a particle of an arbitrary shape and the correlation functions for the fluctuating forces, torques or force-torque acting on the particle in a rotating flow are derived from the semimicroscopic level of coarse graining by using fluctuating hydrodynamics.
Abstract: The Langevin equations for a particle of an arbitrary shape and the correlation functions for the fluctuating forces, torques, or force-torque acting on the particle in a rotating flow are derived from the semimicroscopic level of coarse graining by using fluctuating hydrodynamics. In order to obtain the solution of the Navier-Stokes Langevin equation valid over the entire flow region, use is made of the method of matched asymptotic expansions in (Ω
f
a2/v)1/2≪ 1. The cases of slow and rapid rotation are analyzed. It is shown that the fluctuation-dissipation theorems hold up to the order of (Ω
f
a2/v)1/2 in both slow and rapid rotation, and that the diffusivity tensor depends on the angular velocity of the fluid and becomes anisotropic.
TL;DR: In this article, the behavior of metal oxide semiconductor field effect transistors (MOSFETs) with small aspect ratio and large doping levels is analyzed using formal perturbation techniques.
Abstract: The behavior of metal oxide semiconductor field effect transistors (MOSFETs) with small aspect ratio and large doping levels is analyzed using formal perturbation techniques. Specifically, the influence of interface layers in the potential on the averaged channel conductivity is closely examined. The interface and internal layers that occur in the potential are resolved in the limit of large doping using the method of matched asymptotic expansions. This approach, together with other asymptotic techniques, provides both a pointwise description of the state variables as well as lumped current-voltage relations that vary uniformly across the various bias regimes. These current-voltage relations are derived for a variable doping model respresenting a particular class of devices.
01 Jan 1990
TL;DR: In this paper, a symmetric positive definite matrix (B$ ) is used as a Liapunov function to investigate the asymptotic behaviors of solutions of (1.2) and (3) problems.
Abstract: where $A$ is a constant $n¥times n$ matrix, $C(t, s)$ is an $n¥times n$ matrix continuous for $ 0¥leqq s¥leqq t<¥infty$ and $D(t)$ is an $n¥times n$ matrix continuous for $t$ $¥geqq 0$ . In case $A$ is a stable matrix, there exists a symmetric positive definite matrix $B$ such that (1.4) $A^{T}B+BA=-I$ , and we can use the function $V=x^{T}Bx$ as a Liapunov function to investigate asymptotic behaviors of solutions of (1.2) $(¥mathrm{c}.¥mathrm{f}. [6, 12])$. For the equation (1.3) there is another method, that is, we can use a nice resolvent $Z(t)$ for (1.3)
TL;DR: A new asymptotic method is developed for analyzing closed BCMP queuing networks with a single class (chain) consisting of a large number of customers, a single infinite server queue, and aLarge number of single server queues with fixed (state-independent) service rates.
Abstract: In this paper, a new asymptotic method is developed for analyzing closed BCMP queuing networks with a single class (chain) consisting of a large number of customers, a single infinite server queue, and a large number of single server queues with fixed (state-independent) service rates. Asymptotic approximations are computed for the normalization constant (partition function) starting directly from a recursion relation of Buzen. The approach of the authors employs the ray method of geometrical optics and the method of matched asymptotic expansions. The method is applicable when the servers have nearly equal relative utilizations or can be divided into classes with nearly equal relative utilizations. Numerical comparisons are given that illustrate the accuracy of the asymptotic approximations.
TL;DR: In this paper, an asymptotic method of solution for scattering of acoustic waves from solid elastic targets is presented, where the perturbations to the background rigid scattered field are regular for frequencies away from the frequencies of free vibration of the target in vacuo.
Abstract: An asymptotic method of solution is presented for scattering of acoustic waves from solid elastic targets. The asymptotic parameter is the ratio of the fluid density to that of the solid, and the solution is developed using the method of matched asymptotic expansions in this small quantity. The perturbations to the background rigid scattered field are regular for frequencies away from the frequencies of free vibration of the target in vacuo, but, near these frequencies, the perturbation is singular in that an asymptotically small value of the density ratio produces a change in the scattered field of order unity. By combining the regularly and singularly perturbed expansions, a solution is obtained that is uniformly correct at all frequencies. The elements in the uniform solution depend only upon the in vacuo modes and frequencies, and the Green’s function for the equivalent rigid target. At no stage is it necessary to solve the fully coupled system. An analysis of the asymptotic approximation for a spheri...
TL;DR: In this paper, the first term in such a series describes a field in an infinite-dimensional square-well potential, and the method of matched asymptotic expansions to determine the first five terms in the series is used.
Abstract: The possibility of expressing the solution to a φ2P quantum field theory as a series in powers of 1/P is proposed. Such a series would be nonperturbative in its dependence on the fundamental parameters of the theory such as the mass and the coupling constant. The first term in such a series describes a field in an infinite‐dimensional square‐well potential. In this paper, the quantum‐mechanical Hamiltonian H=p2+q2P is studied as a model calculation and the expansion of the energy levels as series in powers of 1/P is examined. The method of matched asymptotic expansions to determine the first five terms in the series for all energy levels is used. The results are compared with extensive numerical calculations of the ground‐state energy and it is found that the series is extremely accurate: When P=2, the five‐term series has a relative error of 6%, when P=10 the relative error is 0.009%, and when P=200 the relative error is 3.4×10−9%.
TL;DR: In this article, the authors derived an analytic solution to the diffraction problem for two parallel semi-infinite plates arranged to form a duct and used it to determine several terms in the asymptotic expansion of the velocity potential.
TL;DR: In this paper, second-order linear differential equations with a turning point and a double pole with complex exponent are examined, where the turning point is assumed to be a real continuous function of a parameter and coalesces with the pole at the origin.
Abstract: Second-order linear differential equations having a turning point and double pole with complex exponent are examined. The turning point is assumed to be a real continuous function of a parameter $\alpha $, and coalesces with the pole at the origin when $\alpha \to 0$. Asymptotic expansions for solutions, as a second parameter $u \to \infty$, are constructed in terms of Bessel functions of purely imaginary order. The asymptotic solutions are uniformly valid for the argument lying in both real and complex regions that include both the coalescing turning point and the pole. The theory is then applied to obtain uniform asymptotic expansions for Legendre functions of large real degree and purely imaginary order.
05 Dec 1990
TL;DR: In this article, a class of control systems that are represented by equations which are not in the standard singularly perturbed form is studied, and it is shown that the equations for the slow dynamics can be characterized by a set of differential-algebraic equations which can be easily derived from the original equations by setting the parameter to zero.
Abstract: A class of control systems is studied that are represented by equations which are not in the standard singularly perturbed form. Assumptions are introduced which guarantee that an equivalent representation can be obtained which is in the standard singularly perturbed form, thereby justifying the two time scale property. It is then possible to show that the equations for the slow dynamics can be characterized by a set of differential-algebraic equations which are easily derived from the original equations by setting the parameter to zero; the original assumptions guarantee the existence of solutions of the obtained differential-algebraic equations. In addition, the equations for the fast dynamics can be expressed in terms of matrices that define the original control system. Control design for the system being considered is studied using the composite control approach. As an example, the problem of contact force and position regulation in a robot with its end effector in contact with a stiff surface is considered. >
TL;DR: In this article, it was shown that the inner and outer expansions must necessarily be supplemented by a third transition expansion in order to obtain a uniformly valid approximation beyond O(1)$ on a complete half-period.
Abstract: The analysis of the relaxation oscillations of Van der Pol’s equation presents an especially challenging test of the formal techniques of the method of matched asymptotic expansions for solving singular perturbation problems. The formal analysis is described in Kevorkian and Cole’s monograph [J. Kevorkian and J. D. Cole, “Perturbation Methods in Applied Mathematics”, Springer-Verlag, Berlin, New York, 1981], which explains why the inner and outer expansions must necessarily be supplemented by a third “transition” expansion in order to obtain a uniformly valid approximation beyond $O(1)$ on a complete half-period. Kevorkian and Cole carry out the construction and delicate matching of several terms in the expansions. The present paper mathematically justifies their formal results to $O(\varepsilon ^{{1 / 3}} )$, and is the first such proof for any transition expansion. Partly for this reason, but also because the idea underlying the proof has been and will be applied to other singular perturbation problems,...
TL;DR: This work develops a procedure for obtaining the full asymptotic series of the stationary distribution of unfinished work in powers of one minus the traffic intensity, and shows that the correction terms have different forms in different regions of the state space.
Abstract: We consider single server M/G/1 and GI/G/1 queues in the limit of heavy traffic. We develop a procedure for obtaining the full asymptotic series of the stationary distribution of unfinished work in powers of one minus the traffic intensity. The leading term in this series is the exponential density obtained from the heavy traffic limit theorem. We show that the correction terms have different forms in different regions of the state space. These corrections are constructed using the method of matched asymptotic expansions. We assume that the method of matched asymptotic expansions is valid.
TL;DR: In this paper, a finite element method for linear and non-linear singularly perturbed boundary-value problems is considered, and it is proved that the approximate solutions converge to the exact solution in the norm of the space of continuous functions, uniformly in the small parameter.
Abstract: A finite element method for linear and non-linear singularly perturbed boundary-value problems is considered. It is proved that the approximate solutions converge to the exact solution in the norm of the space of continuous functions, uniformly in the small parameter. The proposed scheme is suitable for solving a wider class of problems than can be handled by the popular “hinged element method”, and also produces a higher order of approximation.
TL;DR: In this paper, asymptotic expansions of the wave functions of three identical particles are constructed as series in powers of the hyperradius, its logarithm powers and unknown functions of the hyperspherical angles.
Abstract: In the framework of the integro-differential approach asymptotic expansions of the wave functions of three identical particles are constructed as series in powers of the hyperradius, its logarithm powers and unknown functions of the hyperspherical angles. For calculations of these functions a recurrence chain of ordinary second-order differential equations is obtained. The dependence of the asymptotic expansions on the total angular momentum and the behaviour of the potentials at small distances are investigated.
TL;DR: In this article, the Sturm type separation theorem of Morse [9] was used to show that any solution of (1.1) can be expressed by a linear matrix function A + tB satisfying
TL;DR: In this article, the problem of the scattering of acoustic waves by a large, sound soft (hard), corrugated surface in two dimensions is addressed and the method of matched asymptotic expansions is applied to analyze the problem in the limit as e=a/L→0.
Abstract: The problem of the scattering of acoustic waves by a large, sound soft (hard), corrugated surface in two dimensions is addressed. The surface undulates periodically up to a characteristic length L beyond which it becomes planar. The height of the corrugation is measured by a characteristic length a and its period by Λ. The ordering of these scales is taken to be λ∼Λ∼a≪L, where λ is the wavelength of the incident plane acoustic wave. The method of matched asymptotic expansions is applied to analyze the problem in the limit as e=a/L→0. This approach is both mathematically systematic and physically intuitive. The results that are obtained in the far field are identical to those obtained by using a finite beam approximation for a sound hard surface in two dimensions and almost the same for a sound soft case; the only difference being a sine factor that yields correct boundary behavior. Results for three‐dimensional scattering problems are also derived and are compared similarly.
01 Jan 1990
TL;DR: In this paper, the established necessary conditions for optimality in nonlinear control problems that involve state-variable inequality constraints are applied to a class of singularly perturbed systems and the distinguishing feature of this class of two-time-scale systems is a transformation of the state variable inequality constraint, present in the full order problem, to a constraint involving states and controls in the reduced problem.
Abstract: The established necessary conditions for optimality in nonlinear control problems that involve state-variable inequality constraints are applied to a class of singularly perturbed systems. The distinguishing feature of this class of two-time-scale systems is a transformation of the state-variable inequality constraint, present in the full order problem, to a constraint involving states and controls in the reduced problem. It is shown that, when a state constraint is active in the reduced problem, the boundary layer problem can be of finite time in the stretched time variable. Thus, the usual requirement for asymptotic stability of the boundary layer system is not applicable, and cannot be used to construct approximate boundary layer solutions. Several alternative solution methods are explored and illustrated with simple examples.