Skip to main content
SpringerLink
Log in

Green and Neumann Functions for a Plane Degenerate Circular Domain

  • Conference paper
  • First Online:
Analysis, Probability, Applications, and Computation

Part of the book series: Trends in Mathematics ((RESPERSP))

Abstract

Harmonic Green and Neumann functions are constructed using the parqueting-reflection principle for a simply connected domain in the complex plane with two touching circles as the boundary.

This work was completed with the support of our TEX-pert.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

References

  1. M. Akel, Riemann-Hilbert problem for the Cauchy-Riemann operator in lens and lune. Complex Variables Elliptic Equ. 62, 1570–1588 (2017)

    Article  MathSciNet  Google Scholar 

  2. M. Akel, H. Begehr, Neumann function for a hyperbolic strip and a class of related plane domains. Math. Nachr. 290, 490–506 (2017). https://doi.org/10.1002/mana.201500501

    Article  MathSciNet  Google Scholar 

  3. Ü. Aksoy, A.O. Celebi, A survey on boundary value problems for complex partial differential equations. Adv. Dyn. Syst. Appl. 5, 133–158 (2010)

    MathSciNet  MATH  Google Scholar 

  4. H. Begehr, Green function for a hyperbolic strip and a class of related plane domains. Appl. Anal. 93, 2370–2385 (2014)

    Article  MathSciNet  Google Scholar 

  5. H. Begehr, Fundamental solutions to the Laplacian in plane domains bounded by ellipses, in Proceedings of Mathematics and Computing, Third International Conference, ICMC 2017, Haldia, India, January 17–21, ed. by D. Giri, R.N. Mohapatra, H. Begehr, M.S. Obaidat. Communications in Computer and Information Science, vol. 655 (Springer, Singapore, 2017), pp. 293–311

    Google Scholar 

  6. H. Begehr, Green and Neumann functions for strips, in New Trends in Analysis and Interdisciplinary Applications. Proceedings of the 10th International ISAAC Congress, Macau, 2015, ed. by P. Dang, M. Ku, T. Qiao, L. Rodino. Trends in Mathematics Research Perspectives (Birkhäuser, Cham, 2017), pp. 89–95

    Google Scholar 

  7. H. Begehr, T. Vaitekhovich, Harmonic boundary value problems in half disc and half ring. Funct. Approx. Comment. Math. 40, 251–282 (2009)

    Article  MathSciNet  Google Scholar 

  8. H. Begehr, T. Vaitekhovich, Green functions, reflections, and plane parqueting. Eur. Math. J. 1, 17–31 (2010); 2, 139–142 (2011)

    Google Scholar 

  9. H. Begehr, T. Vaitekhovich, Harmonic Dirichlet problem for some equilateral triangle. Complex Variables Elliptic Equ. 57, 185–196 (2012)

    Article  MathSciNet  Google Scholar 

  10. H. Begehr, T. Vaitekhovich, Schwarz problem in lens and lune. Complex Variables Elliptic Equ. 59, 76–84 (2014)

    Article  MathSciNet  Google Scholar 

  11. H. Begehr, T. Vaitekhovich, The parqueting-reflection principle for constructing Green function, in Analytic Methods of Analysis and Differential Equations: AMADE-2012, ed. by S.V. Rogosin, M.V. Dubatovskaya (Cambridge Scientific Publishers, Cottenham, 2014), pp. 11–20

    Google Scholar 

  12. H. Begehr, M.-R. Costache, S. Tappert, T. Vaitekhovich, Harmonic Green and Neumann representations in a triangle, quarter-disc and octo-plane, in Progress in Analysis and Its Applications. Proceedings of 7th International ISAAC Congress, ed. by M. Ruzhansky, J. Wirth (World Scientific, Singapore, 2011), pp. 74–80

    Google Scholar 

  13. H. Begehr, S. Burgumbayeva, B. Shupeyeva. Remark on Robin problem for Poisson equation. Complex Variables Elliptic Equ. 62(10), 1589–1599 (2017). http://www.tandfonline.com/eprint/W534YxAEU3RUBZDQ7Kqe/full; http://doi.org/10.1080/17476933.2017.1303052

    Article  MathSciNet  Google Scholar 

  14. H. Begehr, S. Burgumbayeva, B. Shupeyeva, Harmonic Green functions for a plane domain with two touching circles as boundary. Adv. Math. Model. Appl. 3(1), 18–29 (2018)

    Google Scholar 

  15. E. Gaertner, Basic complex boundary value problems in the upper half plane, Ph.D. thesis, FU Berlin, 2006. www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000002129

  16. V.V. Mityushev, S.V. Rogosin, Constructive methods for linear and nonlinear boundary value problems for analytic functions: theory and applications. Chapman and Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 108 (Chapman and Hall/CRC, Boca Raton, 2000)

    Google Scholar 

  17. B. Shupeyeva, Harmonic boundary value problems in a quarter ring domain. Adv. Pure Appl. Math. 3, 393–419 (2012); 4, 103–105 (2013)

    Google Scholar 

  18. B. Shupeyeva, Some basic boundary value problems for complex partial differential equations in quarter ring and half hexagon, Ph.D. thesis, FU Berlin, 2013. www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000094596

  19. N. Taghizadeh, V.S. Mohammadi, M.N. Foumani, Schwarz boundary value problem in half lens and half lune. Complex Variables Elliptic Equ. 61, 484–495 (2016)

    Article  MathSciNet  Google Scholar 

  20. T.S. Vaitekhovich, Boundary value problems to second order complex partial differential equations in a ring domain. Šiauliai Math. Sem. 2(10), 117–146 (2007)

    MathSciNet  MATH  Google Scholar 

  21. T.S. Vaitekhovich, Boundary value problems to first order complex partial differential equations in a ring domain. Integral Transform. Spec. Funct. 19, 211–233 (2008)

    Article  MathSciNet  Google Scholar 

  22. T. Vaitsiakhovich, Boundary value problems for complex partial differential equations in a ring domain, Ph.D. thesis, FU Berlin, 2008. www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000003859

  23. Y. Wang, Boundary value problems for complex partial differential equations in fan-shaped domains, Ph.D. thesis, FU Berlin, 2011. www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000021359

  24. Y.-F. Wang, Y.-J. Wang, Schwarz-type problem of nonhomogeneous Cauchy-Riemann equation on a triangle. J. Math. Anal. Appl. 377, 557–570 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Freie Universität Berlin, Berlin, Germany

    H. Begehr

  2. Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Astana, Kazakhstan

    S. Burgumbayeva

  3. Nazarbayev University, Astana, Kazakhstan

    B. Shupeyeva

Authors
  1. H. Begehr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Burgumbayeva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Shupeyeva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Begehr .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Karl‐Olof Lindahl

  2. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Torsten Lindström

  3. Department of Mathematics, University of Torino, Torino, Italy

    Luigi G. Rodino

  4. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Joachim Toft

  5. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Patrik Wahlberg

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Begehr, H., Burgumbayeva, S., Shupeyeva, B. (2019). Green and Neumann Functions for a Plane Degenerate Circular Domain. In: Lindahl, K., Lindström, T., Rodino, L., Toft, J., Wahlberg, P. (eds) Analysis, Probability, Applications, and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04459-6_13

Download citation

Publish with us

Policies and ethics

Search

Navigation