Showing papers on "Method of matched asymptotic expansions published in 1981"
01 Dec 1981
TL;DR: In this paper, a quadratic-type Lyapunov function for a singularly perturbed system is obtained as a weighted sum of quadratically-type LSTM functions of two lower order systems.
Abstract: Asymptotic and exponential stability of nonlinear singularly perturbed systems are investigated via Lyapunov stability techniques. A quadratic-type Lyapunov function for a singularly perturbed system is obtained as a weighted sum of quadratic-type Lyapunov functions of two lower order systems. Estimates of domain of attraction, of upper bound on perturbation parameter, and of degree of exponential stability are obtained. The method is illustrated by studying the stability of a synchronous generator connected to an infinite bus.
257 citations
TL;DR: In this paper, a single-phase soliton-form solution of the Korteweg-de Vries equation with variable coefficients is presented, and the Dirichlet series for constructing asymptotic expansions for general equations.
Abstract: CONTENTSIntroductionChapter I. The Korteweg-de Vries equation with variable coefficients § 1. Basic definitions. A single-phase soliton-form solution of the Korteweg-de Vries equation with variable coefficients § 2. Construction of an asymptotic single-phase soliton-form solution § 3. Conservation lawsChapter II. The Kadomtsev-Petviashvili equation and the sine-Gordon equation § 1. The Kadomtsev-Petviashvili equation § 2. The sine-Gordon equation with variable coefficientsChapter III. Multi-phase asymptotic solutions of non-linear equations and Dirichlet series § 1. Multi-phase asymptotic solutions § 2. Dirichlet series for constructing asymptotic expansions for general equationsReferences
102 citations
TL;DR: In this paper, a linear regular problem involving singularly perturbed difference equations is considered, and the basic features of such problems, such as order reduction, separation of time scales, boundary layers, etc., are highlighted.
Abstract: A linear regular problem involving singularly perturbed difference equations is considered. The basic features of singularly perturbed systems-order reduction, separation of time scales, boundary layers, etc.-are highlighted. Application to the numerical treatment of control problems involving stiff differential equations is discussed.
97 citations
TL;DR: Estimates of the region of attraction and bounds on the small parameters are obtained and sufficient conditions are derived to guarantee the asymptotic stability of a class of nonlinear singularly perturbed systems with several perturbation parameters of the same order.
51 citations
TL;DR: In this article, general advantageous criteria are established for uniform asymptotic stability of singularly perturbed general and large-scale systems possibly containing non-differentiable nonlinearities, which often appear in control systems.
Abstract: New general advantageous criteria are established for uniform asymptotic stability of singularly perturbed general and large-scale systems possibly containing non-differentiable non-linearities, which often appear in control systems. For systems with multiple time scales caused by several small parameters, or for large-scale systems, an essential order reduction of aggregation matrices is achieved. The results are conceptually and numerically applied to the absolute stability analysis of singularly perturbed Lurie-Postnikov systems.
32 citations
TL;DR: In this paper, sufficient conditions are given for this situation to hold for (*, and the structure of the solution space of (* is also studied, where ρ(t), 0≦i≦n, and q(t) are continuous and positive on some half-line [a, ∞].
Abstract: The equation to be considered iswhere pi(t), 0≦i≦n, and q(t) are continuous and positive on some half-line [a, ∞). It is known that (*) always has “strictly monotone” nonoscillatory solutions defined on [a, ∞), so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of (*). In this paper sufficient conditions are given for this situation to hold for (*). The structure of the solution space of (*) is also studied.
28 citations
TL;DR: In this article, a new asymptotic expansion algorithm related to the Chapman-Enskog expansion in kinetic theory is applied to systems of linear evolution equations, and the uniform convergence of the asymPT solution to the exact one is shown.
Abstract: A new asymptotic expansion algorithm related to the Chapman-Enskog expansion in kinetic theory is applied to systems of linear evolution equations. The uniform convergence of the asymptotic solution to the exact one is shown. The algorithm is applied to the linearized Carleman model of the Boltzmann equation, to the neutron transport equation, and to the Fokker-Planck equation.
22 citations
18 citations
16 citations
TL;DR: The Lagrangian analysis as mentioned in this paper is a special case of the Lagrangians, which was introduced by the W.K.B. method for solving the Dirac equation.
Abstract: H. Poincare defined asymptotic expansions. Their use by the W. K. B. method introduced a new kind of solution of linear differential equations. Maslov showed their singularities to be merely apparent. The clarification of those results leads to the introduction of "Lagrangian functions", of their scalar product and of "Lagrangian operators", which constitutes a new structure: the "Lagrangian analysis". The last step of its definition requires the choice of a constant. That constant has to be Planck's constant, when the equation is the Schrodinger or the Dirac equation describing the hydrogen atom-the study of atoms with several electrons is very incomplete. 1. Henri Poincare's main field, more precisely the one where the number of his publications is the highest, happens to be celestial mechanics. For instance, he tried to establish the convergence of the series by means of which the motion of the solar system is computed; it was a failure. He proved indeed the opposite: the divergence of those series, whose numerical values furnished the most impressive, precise and famous predictions in science during the last century! Henri Poincare explained that paradox: those series give a very good approximation of the wanted result, provided only their first terms, namely, a reasonable number of them, are taken into account. Of course, demanding mathematicians to be reasonable is dubious but Henri Poincare [4] made it clear by defining the asymptotic expansion ^mi0anx n of a function of x at the origin: it is a, formal series such that for each natural number N there exists a positive number cN such that I N I ƒ(•*) 2 <*n* < cN\x\ * for x near 0. (1.1) I "° I Thus an asymptotic expansion of ƒ is a formal series able to give a very good approximation of ƒ(*), when x is small, but unable to supply the exact value of/(x). 2. The W.K.B. method constructs asymptotic solutions of a linear differential equation n(x, ~ -j£ W *) 0 ( x G l = R ^ 6 i [0, oo[), (2.1) whose unknown is the function u and whose parameter v tends to /oo. Presented at the Symposium on the Mathematical Heritage of Henry Poincare in April, 1980; received by the editors June 15, 1980. 1980 Mathematics Subject Classification. Primary 47B99, 81C99; Secondary 35S99, 42B99.
TL;DR: In this paper, the defect of the traditional boundary layer methods (including the matched asymptotic expansions and the method of Visik-Lyusternik) is noted, from those methods, from which we can not construct the expansion of boundary layer term substantially.
Abstract: In this paper, the defect of the traditionary boundary layer methods (including the method of matched asymptotic expansions and the method of Visik-Lyusternik) is noted, from those methods we can not construct the asymptotic expansion of boundary layer term substantially. So the method of multiple scales is proposed for constructing the asymptotic expansion of boundary layer term, the reasonable result is obtained. Furthermore, we compare this method with the method used by Levinson, and find that both methods give the same asymptotic expansion of boundary layer term, but our method is simpler.
TL;DR: In this article, an approximate solution to the wave equation in a time-dependent domain which is moving slowly compared with the speed of the waves is found using the two timing method.
Abstract: In this note we find an approximate solution to the wave equation in a time-dependent domain which is moving slowly compared with the speed of the waves. The solution is found using the two timing method. The validity of the approximate solution is verified finding an appropriate energy inequality with the method of multipliers. Finally the problem is solved using an appropriate averaged Lagrangian.
TL;DR: In this article, the problem of classifying an observation X into one of k multivariate normal distributions is considered, and asymptotic expansions for the expected values and variances of these random variables, in terms of the inverses of the sample sizes, are found.
Abstract: SUMMARY The problem of classifying an observation X into one of k multivariate normal distributions is considered. When samples are used to estimate the population parameters, the probabilities of correct classification, and the associated error rates, are random variables. Asymptotic expansions for the expected values and variances of these random variables, in terms of the inverses of the sample sizes, are found. Simulations have been performed to evaluate the expansions.
TL;DR: In this article, matched asymptotic expansions are used to obtain simple analytical expressions for the time-dependent effectiveness factor of variously shaped porous catalyst particles undergoing slow first-order self-poisoning at large Thiele modulus.
Abstract: The method of matched asymptotic expansions is used to obtain simple analytical expressions for the time-dependent effectiveness factor of variously shaped porous catalyst particles undergoing slow first-order self-poisoning at large Thiele modulus. These asymptotic solutions are barely distinguishable from their exact numerical counterparts for Thiele moduli above 100. The position of the activity wave as a function of time is given by compact analytical expressions and the structure of the wave front and the way in which it is formed are described in detail.
TL;DR: In this paper, the structure of the shear Alfven waves in a collisional plasma with shear was investigated and the fourth-order equation obtained by combining Ampere's law and the quasineutrality condition was solved by the method of matched asymptotic expansions for k⊥ρs≫≪1 to obtain the dispersion relation.
Abstract: The structure of the shear Alfven waves is investigated in a collisional plasma with shear. The fourth‐order equation obtained by combining Ampere’s law and the quasi‐neutrality condition is solved by the method of matched asymptotic expansions for k⊥ ρs≫≪1 to obtain the dispersion relation. A hierarchy of damped, localized modes are found which can have either even or odd parity. The solutions basically have the structure of kinetic Alfven modes trapped between the two Alfven cutoffs on either side of the rational surface. The mode damping arises from Ohmic dissipation by electrons near the rational surface. The relation between these modes, microtearing modes, and the Alfven continuum is discussed.
TL;DR: In this paper, an asymptotic method due to Achenbach was used to analyze the torsional modes of wave propagation in a solid circular cylinder of piezoelectric material of (622) crystal class.
TL;DR: In this article, a composite expansion solution to the Troesch problem using the method of matched asymptotic expansions is presented. But their solution is uniformly valid for all n>0 and y⩾0.
TL;DR: In this paper, a variational matched asymptotic expansion is employed to develop shape functions which are particularly useful when convection effects dominate diffusion effects in these problems, in conjunction with the standard Galerkin method, to solve convection-diffusion equations, increased solution accuracy is obtained.
Abstract: Approximation procedures for the solution of convection-diffusion equations, occurring in various physical problems, are considered. Several finite-element algorithms based on singular-perturbation methods are proposed for the solution of these equations. A method of variational matched asymptotic expansions is employed to develop shape functions which are particularly useful when convection effects dominate diffusion effects in these problems. When these shape functions are used, in conjunction with the standard Galerkin method, to solve convection-diffusion equations, increased solution accuracy is obtained. Numerical results for various one-dimensional problems are presented to establish the workability of the developed methods.
TL;DR: In this article, the nonlinearity of the gravity sea flow past a three-dimensional flat blunt ship with a length-based Froude number of order unity was studied using the method of matched asymptotic expansions.
Abstract: The nonlinearity of the gravity sea flow past a three-dimensional flat blunt ship with a length-based Froude number of order unity is studied using the method of matched asymptotic expansions. It is shown that the nonlinearity is important in an inner domain near the ship, whereas the flow in the rest of the fluid domain is the solution of a Neumann-Kelvin problem. Two possible inner solutions – a jet and a wave – are obtained and discussed.
TL;DR: In this paper, a-posteriori error estimates for the finite element solution of the given parabolic problem are derived on every discrete time level which provide a basis for an adaptive mesh refinement.
Abstract: For the Crank-Nicolson-Galerkin-method of solving a singularly perturbed parabolic initial value problem, upper and lower a-posteriori error estimates are established. As a principal tool a-posteriori error estimates in [14] for singularly perturbed boundary value problems in ordinary differential equations are extended to approximately given inhomogeneous right-hand sides containing boundary layer contributions. Herewith, a-posteriori error estimates for the finite element solution of the given parabolic problem are derived on every discrete time level which provide a basis for an adaptive mesh refinement. The efficiency of our method is demonstrated by a numerical example.
TL;DR: In this article, the Keller box method was modified to include an asymptotic outer solution for the case of the self-similar solution for compressible flow in a boundary layer.
TL;DR: In this paper, nonlinear differential delay equations are investigated by means of their associated semigroups and conditions are found for which solutions have nonexponential decay, but nevertheless behave asymptotically as an inverse power of t.
01 Oct 1981
TL;DR: In this article, a nonlinear quasi-one dimensional theory of sound transmitted through a converging-diverging duct section is extended to the case where the acoustical source is located well downstream of the throat, at a point where the flow Mach number is low.
Abstract: A nonlinear quasi-one dimensional theory of sound transmitted through a converging-diverging duct section is extended to the case where the acoustical source is located well downstream of the throat, at a point where the flow Mach number is low. The development and subsequent effects of shocks in the acoustic quantities are of primary consideration. The analysis uses a method of matched asymptotic expansions, yielding a set of inner equations of motion and shock conditions valid in the near-sonic throat region. The analysis leads to a generalization of the 'equal area' relation of weak shock theory. The manner in which nonlinear effects increase with source strength, frequency, and throat Mach number is illustrated by the numerical results and corresponding graphs. The shock waves are shown to cause significant dissipation in acoustic power.
TL;DR: In this article, it was shown that dominant-recessive approximations of the Orr-Sommerfeld equation can be computed to higher order linear ODEs.
Abstract: Reid (1974) derived “first approximations” to solutions of the Orr–Sommerfeld equation, which are uniformly valid in a full neighbourhood of a critical point. This paper shows that such approximations may be calculated to higher order, and makes a first step towards placing the theory on a rigorous basis by providing error bounds for the dominant-recessive approximations. These are obtained by generalizing methods discussed by Olver (1974) for second order linear ordinary differential equations.
TL;DR: In this article, a detailed description of reactor performance from start-up to final catalyst death is obtained for parallel and series self-poisoning mechanisms for a CSTR with one-half and integer-order kinetics, as well as Michaelis-Menten kinetics are considered.
TL;DR: In this article, the singularly perturbed linear evolution equations of resonance type are considered in a Banach space and the Hilbert and Chapman-Enskog algorithms for generating asymptotic solutions are presented and shown to lead to different results at each finite order of approximation.
01 Apr 1981
TL;DR: In this paper, a low frequency unsteady lifting line theory for a harmonically oscillating wing of large aspect ratio is developed by use of the method of matched asymptotic expansions which reduces the problem from a singular integral equation to quadrature.
Abstract: A low frequency unsteady lifting-line theory is developed for a harmonically oscillating wing of large aspect ratio. The wing is assumed to be chordwise rigid but completely flexible in the span direction. The theory is developed by use of the method of matched asymptotic expansions which reduces the problem from a singular integral equation to quadrature. The wing displacements are prescribed and the pressure field, airloads, and unsteady induced downwash are obtained in closed form. The influence of reduced frequency, aspect ratio, planform shape, and mode of oscillation on wing aerodynamics is demonstrated through numerical examples. Compared with lifting-surface theory, computation time is reduced significantly. Using the present theory, the energetic quantities associated with the propulsive performance of a finite wing oscillating in combined pitch and heave are obtained in closed form. Numerical examples are presented for an elliptic wing.
01 Jan 1981
TL;DR: In this article, a series of simple examples are considered, some model and some physical, in order to demonstrate the application of various techniques concerning limit process expansions, and the main tasks in the various problems is to discover the nature of this dependence by working with suitable approximate differential equations.
Abstract: In this chapter a series of simple examples are considered, some model and some physical, in order to demonstrate the application of various techniques concerning limit process expansions. In general we expect analytic dependence of the exact solution on the small parameter e, but one of the main tasks in the various problems is to discover the nature of this dependence by working with suitable approximate differential equations. Another problem is to systematize as much as possible the procedures for discovering these expansions.