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Hybrid Padé-Galerkin Technique for Differential Equations

[+] Author and Article Information
James F. Geer

Department of Systems Science, Watson School of Engineering and Applied Science, State University of New York, Binghamton NY 13902

Carl M. Andersen

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

Appl. Mech. Rev 46(11S), S255-S265 (Nov 01, 1993) doi:10.1115/1.3122644 History: Online April 29, 2009

Abstract

A three-step hybrid analysis technique, which successively uses the regular perturbation expansion method, the Padé expansion method, and then a Galerkin approximation, is presented and applied to some model boundary value problems. In the first step of the method, the regular perturbation method is used to construct an approximation to the solution in the form of a finite power series in a small parameter ε associated with the problem. In the second step of the method, the series approximation obtained in step one is used to construct a Padé approximation in the form of a rational function in the parameter ε. In the third step, the various powers of ε which appear in the Padé approximation are replaced by new (unknown) parameters {δj }. These new parameters are determined by requiring that the residual formed by substituting the new approximation into the governing differential equation is orthogonal to each of the perturbation coordinate functions used in step one. The technique is applied to model problems involving ordinary or partial differential equations. In general, the technique appears to provide good approximations to the solution even when the perturbation and Padé approximations fail to do so. The method is discussed and topics for future investigations are indicated.

Copyright © 1993 by American Society of Mechanical Engineers
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