[go: up one dir, main page]

Skip to main content
Log in

On One Integral Representation of Solutions of Polyharmonic Equation

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

In the paper, an elementary solution of polyharmonic equation is determined and, with the help of it, an integral representation of solutions of polyharmonic equation in a bounded domain is presented.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

REFERENCES

  1. H. Begehr, ‘‘Biharmonic Green functions,’’ Matematiche LXI, 395–405 (2006).

    MathSciNet  MATH  Google Scholar 

  2. H. Begehr and T. Vaitekhovich, ‘‘Modified harmonic Robin function,’’ Complex Variab. Ellipt. Equat. 58, 483–496 (2013).

    MATH  Google Scholar 

  3. M. A. Sadybekov, B. T. Torebek, and B. Kh. Turmetov, ‘‘On an explicit form of the Green function of the Robin problem for the Laplace operator in a circle,’’ Adv. Pure Appl. Math. 6, 163–172 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  4. Wang Ying and Ye. Liuqing, ‘‘Biharmonic Green function and biharmonic Neumann function in a sector,’’ Complex Variab. Ellipt. Equat. 58, 7–22 (2013).

    MATH  Google Scholar 

  5. Wang Ying, ‘‘Tri-harmonic boundary value problems in a sector,’’ Complex Variab. Ellipt. Equat. 59, 732–749 (2014).

    Google Scholar 

  6. V. V. Karachik, ‘‘The Green function of the Dirichlet problem for the triharmonic equation in the ball,’’ Math. Notes 107 (1), 105–120 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Boggio, ‘‘Sulle funzioni di Green d’ordine \(m\),’’ Palermo Rend. 20, 97–135 (1905).

    Article  MATH  Google Scholar 

  8. T. Sh. Kalmenov, B. D. Koshanov. and M. Y. Nemchenko, ‘‘Green function representation for the Dirichlet problem of the polyharmonic equation in a sphere,’’ Complex Variab. Ellipt. Equat. 53, 177–183 (2008).

    MATH  Google Scholar 

  9. H. Begehr, J. Du, and Y. Wang, ‘‘A Dirichlet problem for polyharmonic functions,’’ Ann. Mat. Pura Appl. 187, 435–457 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  10. V. V. Karachik and B. Kh. Turmetov, ‘‘On Green’s function of the Robin problem for the Poisson equation,’’ Adv. Pure Appl. Math. 10, 203–214 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  11. M. A. Sadybekov, B. T. Torebek, and B. K. Turmetov, ‘‘Representation of Green’s function of the Neumann problem for a multi-dimensional ball,’’ Complex Variab. Ellipt. Equat. 61, 104–123 (2016).

    MATH  Google Scholar 

  12. V. V. Karachik and B. T. Torebek, ‘‘On the Dirichlet–Riquier problem for biharmonic equations,’’ Math. Notes 102 (1), 31–42 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. P. Soldatov, ‘‘On the Fredholm property and index of the generalized Neumann problem,’’ Differ. Equat. 56, 212–220 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. V. Karachik, ‘‘Green’s functions of the Navier and Riquier–Neumann problems for the biharmonic equation in the ball,’’ Differ. Equat. 57, 654–668 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  15. G. Sweers, ‘‘A survey on boundary conditions for the biharmonic,’’ Complex Variab. Ellipt. Equat. 54, 79–93 (2009).

    MATH  Google Scholar 

  16. V. V. Karachik, B. Turmetov, and H. Yuan, ‘‘Four boundary value problems for a nonlocal biharmonic equation in the unit ball,’’ Mathematics 10 (7), 1–21 (2022).

    Article  Google Scholar 

  17. A. Cabada and F. A. F. Tojo, ‘‘On linear differential equations and systems with reflection,’’ Appl. Math. Comput. 305, 84–102 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  18. F. A. F. Tojo, ‘‘Computation of Green’s functions through algebraic decomposition of operators,’’ Boundary Value Probl. 167, 1–15 (2016).

    MathSciNet  MATH  Google Scholar 

  19. A. V. Bitsadze, Equations of Mathematical Physics (Nauka, Moscow, 1982) [in Russian].

    MATH  Google Scholar 

  20. V. V. Karachik, ‘‘Green’s function of Dirichlet problem for biharmonic equation in the ball,’’ Complex Variab. Ellipt. Equat. 64, 1500–1521 (2019).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. V. Karachik.

Additional information

(Submitted by T. K. Yuldashev)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karachik, V.V. On One Integral Representation of Solutions of Polyharmonic Equation. Lobachevskii J Math 44, 2749–2756 (2023). https://doi.org/10.1134/S1995080223070247

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080223070247

Keywords:

Navigation