Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Readily computable Green's and Neumann functions for symmetry-preserving triangles

Authors: R. D. Hazlett and D. K. Babu
Journal: Quart. Appl. Math. 67 (2009), 579-592
MSC (2000): Primary 35A20; Secondary 35B60, 35C05
DOI: https://doi.org/10.1090/S0033-569X-09-01157-X
Published electronically: May 14, 2009
MathSciNet review: 2547641
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Abstract: Neumann and Green's functions of the Laplacian operator on $ 30$-$ 60$- $ 90^{\circ}$ and $ 45$-$ 45$- $ 90^{\circ}$ triangles can be generated with appropriately placed multiple sources/sinks in a rectangular domain. Highly accurate and easily computable Neumann and Green's function formulas already exist for rectangles. The extension to equilateral triangles is illustrated. In applications, closed-form expressions can be constructed for the potential, the streamfunction, or the various spatial derivatives of these properties. The derivation of analytic line integrals of these functions allows the proper handling of singularities and facilitates extended applications to problems on domains with open boundaries. Using a boundary integral method, it is demonstrated how one can construct semi-analytical solutions to problems defined on domains that exhibit spatially-dependent properties (heterogeneous media) or possess irregular boundaries.

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Additional Information

R. D. Hazlett
Affiliation: Potential Research Solutions, 1818 Shelmire Dr., Dallas, Texas 75224
Email: rdhazlett@sbcglobal.net

D. K. Babu
Affiliation: Potential Research Solutions, 1818 Shelmire Dr., Dallas, Texas 75224

DOI: https://doi.org/10.1090/S0033-569X-09-01157-X
Keywords: Neumann function, Green's function, right triangle, boundary integral method
Received by editor(s): July 2, 2008
Published electronically: May 14, 2009
Additional Notes: This work was supported by grants from the National Science Foundation’s Small Business Innovation Research Program, Contracts DMI-0128291 and DMI-0236569. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Article copyright: © Copyright 2009 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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