Topic
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.
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03 Dec 1962
TL;DR: In this paper, a method for deriving the asymptotic representation valid for large times for the motion of a particle under the influence of a predominantly linear restoring force and small nonlinear perturbations is presented.
Abstract: : A method is presented for deriving the asymptotic representation valid for large times for the motion of a particle under the influence of a predominantly linear restoring force and small nonlinear perturbations. It is shown that such an asymptotic representation must be a function of two time variables, in order to depict the behavior of the solution. The basic ideas are explained by the liberal use of simple examples, and the method is also applied to two idealized problems in celestial mechanics. (Author)
83 citations
TL;DR: In this paper, the authors analyzed singularly perturbed telegraph systems applying the newly developed compressed asymptotic method and showed that the diffusion equation of the singularized telegraph system is a limit of the system of equations.
Abstract: In the paper we analyze singularly perturbed telegraph systems applying
the newly developed compressed asymptotic method and show that the
diffusion equation is an asymptotic limit of singularly perturbed telegraph
system of equations. The results are applied to the random walk theory
for which the relationship between correlated and uncorrelated random
walks is explained in asymptotic terms.
83 citations
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
TL;DR: 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 as mentioned in this paper.
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.
82 citations
TL;DR: Thanks to the specific structure and the properties of Markovian generators, it is established that the solution of the system can be approximated “as close as possible” by a series expansion in terms of the small parameter $\varepsilon > 0$.
Abstract: A class of singularly perturbed time-varying systems with a small parameter $\varepsilon > 0$ is considered in this paper. The importance of the study stems from the fact that many problems arise in various applications involve a rapidly fluctuating Markov chain. To investigate the limit behavior of such systems, it is necessary to consider the corresponding singular-perturbation problems. Existing results in singular perturbation of ordinary differential equations cannot be applied since the coefficient matrix of the equation is a generator of a finite-state Markov chain, and as a result it is singular. Asymptotic properties of the aforementioned systems are developed via matched asymptotic expansion in this paper. Thanks to the specific structure and the properties of Markovian generators, it is established that the solution of the system can be approximated “as close as possible” by a series expansion in terms of the small parameter $\varepsilon > 0$.
81 citations