Lithuanian Mathematical Journal

, Volume 48, Issue 3, pp 339–356

On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems

Article

Abstract

We deal with the stability analysis of difference schemes for a one-dimensional parabolic equation subject to integral conditions. It is based on the spectral structure of the transition matrix of a difference scheme. The stability domain is defined by using the hyperbola which is the locus of points where the transition matrix has trivial eigenvalues. The stability conditions obtained are much more general compared with those known in the literature. We analyze three separate cases of nonlocal integral conditions and solve an example illustrating the efficiency of the technique.

Keywords

nonlocal integral conditions parabolic equations finite-difference schemes stability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K.E. Atkinson, Introduction to Numerical Analysis, John Wiley & Sons, New York, 1978.MATHGoogle Scholar
  2. 2.
    B.I. Bandyrskii, I. Lazurchak, V.L. Makarov, and M. Sapagovas, Eigenvalue problem for the second order differential equation with nonlocal conditions, Nonlin. Anal. Model. Control, 11(1):13–32, 2006.MATHMathSciNetGoogle Scholar
  3. 3.
    N. Borovykh, Stability in the numerical solution of the heat equation with nonlocal boundary conditions, Appl. Numer. Math., 42:17–27, 2002.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    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.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R. Čiegis, A. Štikonas, O. Štikonienė, and O. Suboč, A monotonic finite-diference scheme for a parabolic problem with nonlocal conditions, Differ. Equ., 38(7):1027–1037, 2002.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    R. Čiupaila, Ž. Jesevičiūtė, and M. Sapagovas, On the eigenvalue problem for one-dimensional differential operator with nonlocal integral condition, Nonlinear Anal. Model. Control, 9(2):109–116, 2004.MATHMathSciNetGoogle Scholar
  7. 7.
    G. Ekolin, Finite difference methods for a nonlocal boundary value problem for heat equation, BIT, 31:245–261, 1991.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    G. Fairweather and J.C. Lopez-Marcos, Galerkin methods for a semilinear parabolic problem with nonlocal conditions, Adv. Comput. Math., 6:243–262, 1996.CrossRefMathSciNetGoogle Scholar
  9. 9.
    S.K. Godunov and V.S. Ryabenjkii, Difference Schemes Introduction to the Theory, Moscow, Nauka, 1977 (in Russian).Google Scholar
  10. 10.
    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.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A.V. Gulin, N.I. Ionkin, and V.A. Morozova, Stability criterion of difference schemes for the heat conduction equation with nonlocal boundary conditions, Comput. Methods Appl. Math., 6(1):31–55, 2006.MATHMathSciNetGoogle Scholar
  12. 12.
    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.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    A.V. Gulin and V.A. Morozova, On the stability of nonlocal finite-difference boundary value problem, Differ. Equ., 39(7):962–967, 2003.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Y. Liu, Numerical solution of the heat equation with nonlocal boundary conditions, J. Comput. Appl. Math., 110(1):115–127, 1999.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    S. Pečiulytė, O. Štikonienė, and A. Štikonas, Sturm-Liouville problem for stationary differential operator with nonlocal integral boundary conditions, Nonlin. Anal. Model. Control, 11(1):47–78, 2006.MATHGoogle Scholar
  16. 16.
    R.D. Richtmyer and K.W. Marton, Difference Methods for Initial-Value Problems, Second Edition, John Wiley & Sons, 1967.Google Scholar
  17. 17.
    A.A. Samarskii, The Theory of Difference Schemes, Moscow, Nauka, 1977 (in Russian); Marcel Dekker, Inc., New York and Basel, 2001.Google Scholar
  18. 18.
    A.A. Samarskii and A.V. Gulin, Numerical Methods, Moscow, Nauka, 1989 (in Russian).MATHGoogle Scholar
  19. 19.
    M.P. Sapagovas, The eigenvalue of some problems with a nonlocal condition, Differ. Equ., 38(7):1020–1026, 2002.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    M.P. Sapagovas, On stability of finite-difference schemes for one-dimensional parabolic equations subject to integral condition, Zh. Obchysl. Prykl. Mat., 92:70–90, 2005.Google Scholar
  21. 21.
    M.P. Sapagovas and A.D. Štikonas, On the structure of the spectrum of a differential operator with a nonlocal condition, Differ. Equ., 41(7):1010–1018, 2005.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    A. Štikonas, The Sturm-Liouville problem with a nonlocal boundary condition, Lith. Math. J., 47(3):336–351, 2007.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilniusLithuania

Personalised recommendations