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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TL;DR: In this paper, the stability of the in-phase and out-of-phase modes of vibration is shown to be governed by the behavior of a linear variational equation with periodic coefficients.
Abstract: This paper concerns the dynamics of a pair of identical, linearly coupled van der Pol relaxation oscillators. We study the stability of the in-phase and out-of-phase modes of vibration. The stability of both modes is shown to be governed by the behavior of a linear Variational equation with periodic coefficients. Approximate analytical solutions are obtained by the method of matched asymptotic expansions. These analytical results are supplemented by numerical integrations based on Floquet theory.It is shown that previous work based on the sinusoidal (nonrelaxation) limit fails to predict a significant region of instability for both modes.
45 citations
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TL;DR: In this paper, it was shown that a small body with possibly strong internal gravity moves through an empty region of a curved, and not necessarily asymptotically flat, external space-time on an approximate geodesic.
Abstract: This paper shows that a small body with possibly strong internal gravity moves through an empty region of a curved, and not necessarily asymptotically flat, external space-time on an approximate geodesic. By "approximate geodesic," one means the following: Suppose the ratio $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{m}{L}$, where $m$ is the body's mass and $L$ is a curvature reference length of the unperturbed external field, is a small parameter. Then $O(L)$ deviations from geodesic motion in the unperturbed external field vanish over times of $O(L)$, with possible $O(L)$ corrections occurring only over times of order $\frac{L}{\ensuremath{\epsilon}}$ or longer. The world line is here calculated directly from the Einstein field equation using a generalized method of matched asymptotic expansions based on a previous paper concerning singular perturbations on manifolds and related to a technique used by D'Eath. Aside from D'Eath's work, previous results on the motion of realistic bodies have assumed weak internal gravity, in some cases incorporating additional assumptions such as perfect fluids or high symmetry. This calculation makes no assumptions about the details of the body, such as weak fields, symmetry, the equations of state for matter, or even the presence of matter. Most previous treatments assumed asymptotic flatness of the external field. Here, it is only assumed that, in the region of interest, the external spacetime is empty and free of singularities. The results extend the work of D'Eath to a more general class of objects that includes nonstationary black holes, naked singularities, and neutron stars, as well as ordinary astrophysical objects. This method can be applied to related problems, such as the motion of a charged black hole through an external gravitational and electromagnetic field. A future paper will combine this method with Burke's method of obtaining radiation reaction to calculate the orbital-period shortening of gravitationally bound, slow-motion systems, such as the binary pulsar PSR 1913 + 16, containing objects with strong internal gravity.
45 citations
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TL;DR: In this paper, the authors developed asymptotic expansions for solutions of integro-differential equations arising from transition densities of singularly perturbed switching-diffusion processes.
45 citations
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TL;DR: It is shown that the initial layer terms in the expansion decay at an exponential rate, and error bounds on the remainder terms also are obtained, indicating the validity of the expansion is rigorously justified.
Abstract: We derive limit theorems for the transition densities of diffusion processes and develop asymptotic expansions for solutions of a class of singularly perturbed Kolmogorov–Fokker–Planck equations. The model under consideration can be viewed as a Markov process having two time scales. One of them is a rapidly changing scale, and the other is a slowly varying one. The study is motivated by a wide range of applications involving singularly perturbed Markov processes in manufacturing systems, reliability analysis, queueing networks, statistical physics, population biology, financial economics, and many other related fields. In this work, the asymptotic expansion is constructed explicitly. It is shown that the initial layer terms in the expansion decay at an exponential rate. Error bounds on the remainder terms also are obtained. The validity of the expansion is rigorously justified.
45 citations
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TL;DR: In this article, asymptotic expansions for the density functions of the TSLS and LIML estimates of coefficients in a simultaneous equation system when the sample size increases and the effect of the exogenous variables increases along the sample length is derived.
45 citations