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REVIEW ARTICLES

New Trends in Asymptotic Approaches: Summation and Interpolation Methods

[+] Author and Article Information
Igor V. Andrianov

Pridneprovye State Academy of Civil Engineering and Architecture, 24a Chernyshevskogo St., Dnepropetrovsk 49000, Ukraine

Jan Awrejcewicz

Technical University of Łódź, Division of Automatics and Biomechanics, 1/15 Stefanowskiego St, 90-924 Łódź, Poland

Appl. Mech. Rev 54(1), 69-92 (Jan 01, 2001) (24 pages) doi:10.1115/1.3097289 History: Online April 07, 2009

Abstract

In this review article, we present in some detail new trends in application of asymptotic techniques to mechanical problems. First we consider the various methods which allows for the possibility of extending the perturbation series application space and hence omiting their local character. While applying the asymptotic methods very often the following situation appears: an existence of the asymptotics ε → 0 implies an existence of the asymptotics ε → ∞ (or, in a more general sense, ε → a and ε → b). Therefore, an idea of constructing a single solution valid for a whole interval of parameter ε changes is very attractive. In other words, we discuss a problem of asymptotically equivalent function constructions possessing for ε → a and ε → b a known asymptotic behavior. The defined problems are very important from the point of view of both theoretical and applied sciences. In this work, we review the state-of-the-art, by presenting the existing methods and by pointing out their advantages and disadvantages, as well as the fields of their applications. In addition, some new methods are also proposed. The methods are demonstrated on a wide variety of static and dynamic solid mechanics problems and some others involving fluid mechanics. This review article contains 340 references.

Copyright © 2001 by American Society of Mechanical Engineers
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