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
Full-text PDF Free Access

Abstract | References | Similar Articles | 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 [Enhancements On Off] (What's this?)

  • 1. 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.
  • 2. D. K. Babu and A. S. Odeh, Productivity of a horizontal well, SPE Reservoir Eng., November (1989), 417-421.
  • 3. 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.
  • 4. G. F. D. Duff and D. Naylor, Differential equations of applied mathematics, John Wiley& Sons, Inc., New York-London-Sydney, 1966. MR 0192675
  • 5. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453
  • 6. R. D. Hazlett and D. K. Babu, Optimal well placement in heterogeneous reservoirs through semi-analytic modeling, SPE J., 10 (2005), 286-296.
  • 7. 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.
  • 8. 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).
  • 9. Michelle E. Johnston, Jan C. Myland, and Keith B. Oldham, A Green function for the equilateral triangle, Z. Angew. Math. Phys. 56 (2005), no. 1, 31–44. MR 2112839, https://doi.org/10.1007/s00033-004-3114-z
  • 10. C. M. Linton, Rapidly convergent representations for Green’s functions for Laplace’s equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1985, 1767–1797. MR 1701551, https://doi.org/10.1098/rspa.1999.0379
  • 11. Simon L. Marshall, A rapidly convergent modified Green’s function for Laplace’s equation in a rectangular region, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1985, 1739–1766. MR 1701550, https://doi.org/10.1098/rspa.1999.0378
  • 12. R. C. McCann, R. D. Hazlett, and D. K. Babu, Highly accurate approximations of Green’s and Neumann functions on rectangular domains, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2008, 767–772. MR 1875306, https://doi.org/10.1098/rspa.2000.0690
  • 13. Brian J. McCartin, Eigenstructure of the equilateral triangle. II. The Neumann problem, Math. Probl. Eng. 8 (2002), no. 6, 517–539. MR 1967479, https://doi.org/10.1080/1024123021000053664
  • 14. J. N. Newman, The Green function for potential flow in a rectangular channel, J. Engrg. Math. 26 (1992), no. 1, 51–59. MR 1154227, https://doi.org/10.1007/BF00043225
  • 15. 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.
  • 16. Milan Práger, Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle, Appl. Math. 43 (1998), no. 4, 311–320. MR 1627985, https://doi.org/10.1023/A:1023269922178
  • 17. John Strain, Fast potential theory. II. Layer potentials and discrete sums, J. Comput. Phys. 99 (1992), no. 2, 251–270. MR 1158209, https://doi.org/10.1016/0021-9991(92)90206-E

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35A20, 35B60, 35C05

Retrieve articles in all journals with MSC (2000): 35A20, 35B60, 35C05

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.

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website