Advertisement

On a Generalised Samarskii-Ionkin Type Problem for the Poisson Equation

  • Aishabibi A. Dukenbayeva
  • Makhmud A. Sadybekov
  • Nurgissa A. Yessirkegenov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 264)

Abstract

In this paper, we consider a generalised form of the Samarskii-Ionkin type boundary value problem for the Poisson equation in the disk and show its well-posedness. The possibility of separation of variables is justified. We obtain an explicit form of the Green function for this problem and an integral representation of the solution.

Keywords

Poisson equation Periodic boundary conditions Samarskii-Ionkin type boundary value problem Green function 

Notes

Acknowledgements

The authors were supported in parts by the MES RK grant AP051333271 as well as by the MES RK target grant BR05236656.

References

  1. 1.
    Ammari, H., Kang, H.: Polarization and Moment Tensors. Applied Mathematical Sciences, vol. 162. Springer, New York (2007)zbMATHGoogle Scholar
  2. 2.
    Ionkin, N.I.: Solution of a boundary-value problem in heat conduction theory with a nonclassical boundary condition. Differentsial’nye Uravneniya [Differ. Equ.] 13(2), 294–304 (1977)zbMATHGoogle Scholar
  3. 3.
    Ionkin, N.I., Moiseev, E.I.: A problem for a heat equation with two-point boundary conditions. Differentsial’nye Uravneniya [Differ. Equ.] 15(7), 1284–1295 (1979)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Kapanadze, D., Mishuris, G., Pesetskaya, E.: Exact solution of a nonlinear heat conduction problem in a doubly periodic 2D composite material. Arch. Mech. (Arch. Mech. Stos.) 67, 157–178 (2015)Google Scholar
  5. 5.
    Kapanadze, D., Mishuris, G., Pesetskaya, E.: Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials. Complex Var. Elliptic Equ. 60, 1–23 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Lanza de Cristoforis, M., Musolino, P.: A singularly perturbed nonlinear Robin problem in a periodically perforated domain: a functional analytic approach. Complex Var. Elliptic Equ. 58, 511–536 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Musolino, P.: A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain: a functional analytic approach. Math. Methods Appl. Sci. 35, 334–349 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Sadybekov, M.A., Turmetov, B.Kh.: On analogues of periodic boundary value problems for the Laplace operator in a ball. Eurasian Math. J. 3(1), 143–146 (2012)Google Scholar
  9. 9.
    Sadybekov, M.A., Turmetov, B.Kh.: On an analog of periodic boundary value problems for the Poisson equation in the disk. Differ. Equ. 50(2), 268–273 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sadybekov, M.A., Torebek, B.T., Yessirkegenov, N.A.: On an analog of Samarskii-Ionkin type boundary value problem for the Poisson equation in the disk. AIP Conf. Proc. 1676, 020035 (2015)CrossRefGoogle Scholar
  11. 11.
    Sadybekov, M.A., Yessirkegenov, N.A.: Spectral properties of a Laplace operator with Samarskii-Ionkin type boundary conditions in a disk. AIP Conf. Proc. 1759, 020139 (2016)CrossRefGoogle Scholar
  12. 12.
    Sadybekov, M.A., Turmetov, B.Kh., Torebek, B.T.: Solvability of nonlocal boundary-value problems for the Laplace equation in the ball. Electron. J. Differ. Equ. 2014, 1–14 (2014)Google Scholar
  13. 13.
    Sadybekov, M.A., Yessirkegenov, N.A.: On a generalised Samarskii-Ionkin type problem for the Poisson equation. Kazakh Math. J. 17(1), 115–116 (2017)Google Scholar
  14. 14.
    Sadybekov, M.A., Torebek, B.T., Turmetov, B.Kh.: Representation of Green’s function of the Neumann problem for a multi-dimensional ball. Complex Var. Elliptic Equ. 61(1), 104–123 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Sadybekov, M.A., Torebek, B.T., Turmetov, B.Kh.: On an explicit form of the Green function of the third boundary value problem for the Poisson equation in a circle. AIP Conf. Proc. 1611, 255–260 (2014)Google Scholar
  16. 16.
    Sadybekov, M.A., Torebek, B.T., Turmetov, B.Kh.: Representation of the Green’s function of the exterior Neumann problem for the Laplace operator. Sib. Math. J. 58(1), 153–158 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Tihonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics, Courier Corporation (2013)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Aishabibi A. Dukenbayeva
    • 1
  • Makhmud A. Sadybekov
    • 1
  • Nurgissa A. Yessirkegenov
    • 2
  1. 1.Institute of Mathematics and Mathematical ModelingAlmatyKazakhstan
  2. 2.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations