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Mathematical Problems in Engineering
Volume 2016, Article ID 5072309, 18 pages
Research Article

Exact Boundary Derivative Formulation for Numerical Conformal Mapping Method

1Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan
2Department of Civil and Water Resources Engineering, National Chiayi University, Chiayi City 60004, Taiwan

Received 10 October 2015; Accepted 11 January 2016

Academic Editor: Ivano Benedetti

Copyright © 2016 Wei-Lin Lo et al. 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.


Conformal mapping is a useful technique for handling irregular geometries when applying the finite difference method to solve partial differential equations. When the mapping is from a hyperrectangular region onto a rectangular region, a specific length-to-width ratio of the rectangular region that fitted the Cauchy-Riemann equations must be satisfied. In this research, a numerical integral method is proposed to find the specific length-to-width ratio. It is conventional to employ the boundary integral method (BIEM) to perform the conformal mapping. However, due to the singularity produced by the BIEM in seeking the derivatives on the boundaries, the transformation Jacobian determinants on the boundaries have to be evaluated at inner points instead of directly on the boundaries. This approximation is a source of numerical error. In this study, the transformed rectangular property and the Cauchy-Riemann equations are successfully applied to derive reduced formulations of the derivatives on the boundaries for the BIEM. With these boundary derivative formulations, the Jacobian determinants can be evaluated directly on the boundaries. Furthermore, the results obtained are more accurate than those of the earlier mapping method.