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
Full-text PDF Free Access
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Additional Information
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.
References
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References
- H. Ahmedov and I. H. Duru, Regularized Green’s function and group of reflections in a cavity with triangular cross section, Phys. Atom. Nucl., 68 (2005), 1621–1624.
- D. K. Babu and A. S. Odeh, Productivity of a horizontal well, SPE Reservoir Eng., November (1989), 417–421.
- R. Chadha and K. C. Gupta, Green’s functions for triangular segments in planar microwave circuits, IEEE Trans. Microwave Theory Tech., 28 (1980), 1139–1143.
- G. F. D. Duff and D. Naylor, Differential equations of applied mathematics, Wiley, New York, NY, 1966. MR 0192675 (33:900)
- I. M. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Corrected and enlarged edition, Academic Press, San Diego, CA, 1980, pp. 39–40. MR 0582453 (81g:33001)
- R. D. Hazlett and D. K. Babu, Optimal well placement in heterogeneous reservoirs through semi-analytic modeling, SPE J., 10 (2005), 286–296.
- R. D. Hazlett, D. K. Babu, and L. W. Lake, Semianalytical stream-function solutions on unstructured grids for flow in heterogeneous media, SPE J., 12 (2007), 179–187.
- R. D. Hazlett, D. K. Babu, V. Chodur, S. Cook, and L. W. Lake, A mirror on the world: Undergraduate researchers advance physics in triangular spaces, submitted to Journal of Engineering Education (2009).
- M. E. Johnston, J. C. Myland, and K. B. Oldham, A Green function for the equilateral triangle, Z. Angew. Math. Phys., 56 (2005), 31–44. MR 2112839 (2005h:35004)
- C. M. Linton, Rapidly convergent modified Green’s function for Laplace’s equation, Proc. R. Soc. Lond. A, 455 (1999), 1767–1797. MR 1701551 (2000c:65097)
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- J. N. Newman, The Green function for potential flow in a rectangular channel, J. Engrg. Math., 26 (1992), 51–59. MR 1154227 (92j:76009)
- A. S. Odeh and D. K. Babu, Transient flow behavior of horizontal wells: Pressure drawdown and buildup analysis, SPE Formation Eval., March (1990), 7–15.
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- J. Strain, Fast potential theory. II. Layer potentials and discrete sums, J. Comput. Phys., 99 (1992), 251–270. MR 1158209 (93b:65197)
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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
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.