Skip to main content
SpringerLink
Log in

On the stability of a difference scheme for a two-dimensional parabolic equation with an integral condition*

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract

We deal with the stability of a locally one-dimensional method for two-dimensional parabolic equations subject to nonlocal boundary condition. To analyze the stability, we study the structure of the spectrum of the transition matrix of the difference scheme. The results of a numerical experiment are presented.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.P. Agarwal, Focal Boundary-Value Problems for Differential and Difference Equations, Kluwer Academic, Dordrecht, 1998.

    Book  MATH  Google Scholar 

  2. B.I. Bandyrskii, I. Lazurchak, V.L. Makarov, and M. Sapagovas, Eigenvalue problem for the second-order differential equation with nonlocal conditions, Nonlinear Anal., Model. Control, 11(1):13–32, 2006.

    MathSciNet  MATH  Google Scholar 

  3. B. Cahlon, D.M. Kulkarni, and P. Shi, Stepwise stability for the heat equation with a nonlocal constrain, SIAM J. Numer. Anal., 32(2):571–593, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.R. Cannon, Y. Lin, and A.L. Matheson, The solution of the diffusion equation in two-space variables subject to specification of mass, Appl. Anal, 50(1):1–15, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Čiegis, Economical difference schemes for the solution of a two-dimensional parabolic problem with an integral condition, Differ. Equ., 41(7):1025–1029, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Čiegis and N. Tumanova, Numerical solution of parabolic problems with nonlocal boundary conditions, Numer. Funct. Anal. Optim., 31(12):1318–1329, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Čiegis, A. Štikonas, O. Štikonienė, and O. Suboč, A monotonic finite-difference scheme for a parabolic problem with nonlocal conditions, Differ. Equ., 38(7):1027–1037, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Dehghan, Locally explicit schemes for three-dimensional diffusion with nonlocal boundary specifications, Appl. Math. Comput., 138(2):489–501, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  9. G. Ekolin, Finite-difference methods for a nonlocal boundary-value problem for heat equation, BIT, 31:245–261, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Fairweather and J.C. Lopez-Marcos, Galerkin methods for a semilinear parabolic problem with nonlocal conditions, Adv. Comput. Math., 6:243–262, 1996.

    Article  MathSciNet  Google Scholar 

  11. J. Gao, D. Sun, and M. Zhang, Structure of eigenvalues of multipoint boundary-value problems, Adv. Difference Equ., 2010, Article ID 381932, 24 pp., 2010.

  12. A.V. Gulin, N.I. Ionkin, and V.A. Morozova, Stability of a nonlocal two-dimensional finite-difference problem, Differ. Equ., 37(7):970–978, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  13. A.V. Gulin, N.I. Ionkin, and V.A. Morozova, Study of the norm in stability problems for nonlocal difference schemes, Differ. Equ., 42(7):974–984, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  14. A.V. Gulin, V.A. Morozova, and N.S. Udovichenko, Stability of a nonlocal difference problem with a complex parameter, Differ. Equ., 47(8):1116–1129, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  15. G. Infante, Eigenvalues of some nonlocal boundary-value problems, Proc. Edinb. Math. Soc., II. Ser., 46:75–86, 2003.

    MathSciNet  MATH  Google Scholar 

  16. F. Ivanauskas, T. Meškauskas, and M. Sapagovas, Stability of difference schemes for two-dimensional parabolic equations with nonlocal boundary conditions, Appl. Math. Comput., 215(7):2716–2732, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Jachimavičienė, Ž. Jesevičiutė, and M. Sapagovas, The stability of finite-difference schemes for a pseudoparabolic equations with nonlocal conditions, Numer. Funct. Anal. Optim., 30(9):988–1001, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  18. Y. Lin, S. Xu, and H.-M. Yin, Finite-difference approximations for a class of nonlocal parabolic equation, Int. J. Math. Math. Sci., 20(1):147–164, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  19. Y. Liu, Numerical solution of the heat equation with nonlocal boundary conditions, J. Comput. Appl. Math., 110:115–127, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Sapagovas, On the stability of finite-difference schemes to integral conditions, Zh. Obchysl. Prykl. Mat., 92:77–90, 2005.

    Google Scholar 

  21. M. Sapagovas, Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions, Differ. Equ., 44(7):1018–1028, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Sapagovas, On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems, Lith. Math. J., 48(3):339–356, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Sapagovas and K. Jakubėlienė, Alternating direction method for two-dimensional parabolic equation with nonlocal integral condition, Nonlinear Anal., Model. Control, 17(1):91–98, 2012.

    MathSciNet  Google Scholar 

  24. M. Sapagovas, G. Kairytė, O. Štikonienė, and A. Štikonas, Alternating direction method for a two-dimensional parabolic equation with a nonlocal boundary condition, Math. Model. Anal., 12(1):131–142, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Sapagovas, T. Meškauskas, and F. Ivanauskas, Numerical spectral analysis of a difference operator with nonlocal boundary conditions, Appl. Math. Comput., 218(14):7515–7527, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Sapagovas, A. Štikonas, and O. Štikonienė, Alternating direction method for the Poisson equation with variable weight coefficients in an integral condition, Diff. Equ., 47(8):1176–1187, 2011.

    Article  MATH  Google Scholar 

  27. M. Sapagovas and O. Štikonienė, Alternating-direction method for a mildly nonlinear elliptic equation with nonlocal integral conditions, Nonlinear Anal., Model. Control, 16(2):220–230, 2011.

    MathSciNet  MATH  Google Scholar 

  28. M.P. Sapagovas and A.D. Štikonas, On the structure of the spectrum of a difference operator with a nonlocal condition, Differ. Equ., 41(7):1010–1018, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Štikonas, Investigation of characteristic curve for Sturm–Liouville problem with nonlocal boundary conditions on torus, Math. Model. Anal., 16(1):1–22, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Wang, Solutions to nonlinear elliptic equations with a nonlocal boundary condition, Electron. J. Differ. Equ., 2002(05):1–16, 2002.

    Google Scholar 

  31. H.-M. Yin, On class of parabolic equations with nonlocal boundary conditions, J. Math. Anal. Appl., 294:712–728, 2004.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, LT-08663, Vilnius, Lithuania

    Kristina Jakubėlienė & Mifodijus Sapagovas

Authors
  1. Kristina Jakubėlienė
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mifodijus Sapagovas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mifodijus Sapagovas.

Additional information

*This research was funded by grant (No. MIP-051/2011) from the Research Council of Lithuania.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jakubėlienė, K., Sapagovas, M. On the stability of a difference scheme for a two-dimensional parabolic equation with an integral condition* . Lith Math J 53, 311–323 (2013). https://doi.org/10.1007/s10986-013-9210-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-013-9210-z

MSC

Keywords

Search

Navigation