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A Hybrid Perturbation-Galerkin Method for Differential Equations Containing a Parameter

[+] Author and Article Information
James F. Geer

Department of Systems Science, Watson School of Engineering, Applied Science and Technology, S.U.N.Y., Binghamton, NY 13901

Carl M. Andersen

Department of Mathematics, College of William and Mary, Williamsburg, VA 23185

Appl. Mech. Rev 42(11S), S69-S77 (Nov 01, 1989) doi:10.1115/1.3152410 History: Online June 03, 2009

Abstract

A two-step hybrid perturbation-Galerkin method to solve a variety of differential equations which involve a parameter is presented and discussed. The method consists of: (1) the use of a perturbation method to determine the asymptotic expansion of the solution about one or more values of the parameter; and (2) the use of some of the perturbation coefficient functions as trial functions in the classical Bubnov-Galerkin method. This hybrid method has the potential of overcoming some of the drawbacks of the perturbation method and the Bubnov-Galerkin method when they are applied by themselves, while combining some of the good features of both. The proposed method is illustrated first with a simple linear two-point boundary value problem and is then applied to a nonlinear two-point boundary value problem in lubrication theory. The results obtained from the hybrid method are compared with approximate solutions obtained by purely numerical methods. Some general features of the method, as well as some special tips for its implementation, are discussed. A survey of some current research application areas is presented and its degree of applicability to broader problem areas is discussed.

Copyright © 1989 by The American Society of Mechanical Engineers
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