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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 613082, 11 pages
http://dx.doi.org/10.1155/2013/613082
Research Article

Analytical Particular Solutions of Multiquadrics Associated with Polyharmonic Operators

Department of Marine Environmental Engineering, National Kaohsiung Marine University, Kaohsiung 811, Taiwan

Received 9 January 2013; Accepted 29 March 2013

Academic Editor: Kue-Hong Chen

Copyright © 2013 Chia-Cheng Tsai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We derive two- and three-dimensional analytical particular solutions of multiquadrics (MQ) associated with the polyharmonic operators, named as the polyharmonic multiquadrics (PMQs). The methods of undetermined coefficients are constructed by observing the first few orders of the PMQs which are obtained by the symbolic software, Mathematica. By expanding the PMQs into the Laurent series, the unknown coefficients of the PMQs can be determined. The homogeneous parts of the PMQs are suitably arranged so that the PMQs are hierarchically unique and infinitely differentiable. Mathematica codes are provided for obtaining the PMQs of arbitrary orders. The derived PMQs are validated by numerical solutions for Poisson’s equation. Numerical results indicate that the solutions obtained by the PMQs are more accurate than those by the MQ.