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 article, the authors consider large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives and obtain sharp and realistic error bounds.
Abstract: In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise from an application of the method of steepest descents.
17 citations
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TL;DR: In this paper, a method for improvement of the numerical solution of differential equations by incorporation of asymptotic approximations is investigated for a class of singular perturbation problems, and uniform error estimates are derived for this method when implemented in known difference schemes and applied to linear second order O.D.E.'s.
Abstract: A method for improvement of the numerical solution of differential equations by incorporation of asymptotic approximations is investigated for a class of singular perturbation problems.
Uniform error estimates are derived for this method when implemented in known difference schemes and applied to linear second order O.D.E.'s. An improvement by a factor of?n+1 can be obtained (where ? is the "small" parameter andn is the order of the asymptotic approximation) for a small amount of extra work. Numerical experiments are presented.
17 citations
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01 Mar 2015TL;DR: In this paper, the equation of motion of a small, compact body in an external vacuum spacetime through second order in the body's mass was derived using a rigorous method of matched asymptotic expansions.
Abstract: Using a rigorous method of matched asymptotic expansions, I derive the equation of motion of a small, compact body in an external vacuum spacetime through second order in the body's mass (neglecting effects of internal structure). The motion is found to be geodesic in a certain locally defined regular geometry satisfying Einstein's equation at second order. I outline a method of numerically obtaining both the metric of that regular geometry and the complete second-order metric perturbation produced by the body.
17 citations
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17 citations
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TL;DR: In this paper, the evolution of long nonlinear Rossby waves in a sheared zonal current in the regime where a competition sets in between weak nonlinearity and weak dispersion is considered.
Abstract: This study considers the evolution of long nonlinear Rossby waves in a sheared zonal current in the regime where a competition sets in between weak nonlinearity and weak dispersion. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean-flow velocity at a certain latitude, due to the appearance of a singularity in the leading order equation, which strongly modifies the flow in the critical layer. Here, nonlinear effects are invoked to resolve this singularity, since the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear critical-layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg-de Vries equation, modified by new nonlinear terms, depending on the critical-layer shape. These lead to depression or elevation solitary waves.
17 citations