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Mathematical Problems in Engineering
Volume 2014, Article ID 853252, 9 pages
Research Article

Analysis of the Boundary Knot Method for 3D Helmholtz-Type Equation

1School of Mathematical Sciences, Huaibei Normal University, Dongshan Road 100, Huaibei 235000, China
2College of Water Conservancy and Ecological Engineering, Nanchang Institute of Technology, Tianxiang Road 289, Nanchang 330099, China

Received 12 January 2014; Accepted 10 February 2014; Published 13 March 2014

Academic Editor: Yumin Cheng

Copyright © 2014 F. Z. Wang and K. H. Zheng. 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.


Numerical solutions of the boundary knot method (BKM) always perform oscillatory convergence when using a large number of boundary points in solving the Helmholtz-type problems. The main reason for this phenomenon may contribute to the severely ill-conditioned full coefficient matrix. In order to obtain admissible stable convergence results, regularization techniques and the effective condition number are employed in the process of simulating 3D Helmholtz-type problems. Numerical results are tested for the 3D Helmholtz-type equation with noisy and non-noisy boundary conditions. It is shown that the BKM in combination with the regularization techniques is able to produce stable numerical solutions.