Method of matched asymptotic expansions

About: Method of matched asymptotic expansions is a research topic. Over the lifetime, 4233 publications have been published within this topic receiving 73311 citations.

The Asymptotic Properties of Estimates of the Parameters of a Single Equation in a Complete System of Stochastic Equations

Abstract: In a previous paper [2] the authors have given a method for estimating the coefficients of a single equation in a complete system of linear stochastic equations. In the present paper the consistency of the estimates and the asymptotic distributions of the estimates and the test criteria are studied under conditions more general than those used in the derivation of these estimates and criteria. The point estimates, which can be obtained as maximum likelihood estimates under certain assumptions including that of normality of disturbances, are consistent even if the disturbances are not normally distributed and (a) some predetermined variables are neglected (Theorem 1) or (b) the single equation is in a non-linear system with certain properties (Theorem 2). Under certain general conditions (normality of the disturbances not being required) the estimates are asymptotically normally distributed (Theorems 3 and 4). The asymptotic covariance matrix is given for several cases. The criteria derived in [2] for testing the hypothesis of over-identification have, asymptotically, $\chi^2$-distributions (Theorem 5). The exact confidence regions developed in [2] for the case that all predetermined variables are exogenous (that is, that the difference equations are of zero order) are shown to be consistent and to hold asymptotically even when this assumption is not true (Theorem 6).

The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

Abstract: Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter e which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.

Quadratic-type Lyapunov functions for singularly perturbed 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.