Skip to main content

Calculation of potential in a sector

  • 375 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1121)

Abstract

About 25 years ago Wasow and Lehman got an asymptotic series for a harmonic function in a sector. This showed that, in various important cases, the first derivative of the function becomes infinite as one approaches the vertex of the sector. For this reason, finite difference or finite element methods of a fixed mesh give poor accuracy near the vertex. The asymptotic series of Wasow and Lehman cannot be used for calculations near the vertex since the derivations of the series gave no means to compute the numerical values of the coefficients of the series.

In this talk means are provided for computing the numerical values of the coefficients. Moreover, by suitably pairing some terms of the series, the resulting series of terms and pairs turns out to be convergent. It is therefore quite suitable for calculating values of the harmonic function near the vertex.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. D. Bergland, "A radix-eight Fast Fourier Transform subroutine for real-valued series," IEEE Trans. on Audio and Electroacoustics, vol. AU-17 (1969), pp. 138–144.

    CrossRef  Google Scholar 

  2. G. C. Evans, "The logarithmic potential," A.M.S. Colloquium Publications, vol. VI, 1927.

    Google Scholar 

  3. G. H. Hardy and E. M. Wright, "An introduction to the theory of numbers," third edition, Clarendon Press, Oxford, 1954.

    MATH  Google Scholar 

  4. O. D. Kellogg, "Foundations of potential theory," Dover Publications, Inc., New York, 1953.

    MATH  Google Scholar 

  5. R. Sherman Lehman, "Developments in the neighborhood of the beach of surface waves over an inclined bottom," Communications on Pure and Applied Mathematics, vol. 7 (1954), pp. 393–439.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. R. Sherman Lehman, "Developments at an analytic corner of solutions of elliptic partial differential equations," Journal of Mathematics and Mechanics, vol. 8 (1959), pp. 727–760.

    MathSciNet  MATH  Google Scholar 

  7. W. E. Milne, "Numerical solutions of differential equations," John Wiley and Sons, Inc., 1953.

    Google Scholar 

  8. Wolfgang Wasow, "Asymptotic development of the solution of Dirichlet's problem at analytic corners," Duke Mathematical Journal, vol. 24 (1957), pp. 47–56.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. E. T. Whittaker and G. N. Watson, "A course of modern analysis," American Edition, The Macmillan Company, New York, 1946.

    MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1985 Springer-Verlag

About this paper

Cite this paper

Rosser, J.B. (1985). Calculation of potential in a sector. In: Grisvard, P., Wendland, W.L., Whiteman, J.R. (eds) Singularities and Constructive Methods for Their Treatment. Lecture Notes in Mathematics, vol 1121. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076274

Download citation

  • DOI: https://doi.org/10.1007/BFb0076274

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15219-4

  • Online ISBN: 978-3-540-39377-1

  • eBook Packages: Springer Book Archive