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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K.E. Atkinson, Introduction to Numerical Analysis, John Wiley & Sons, New York, 1978.
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.
N. Borovykh, Stability in the numerical solution of the heat equation with nonlocal boundary conditions, Appl. Numer. Math., 42:17–27, 2002.
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.
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.
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.
G. Ekolin, Finite difference methods for a nonlocal boundary value problem for heat equation, BIT, 31:245–261, 1991.
G. Fairweather and J.C. Lopez-Marcos, Galerkin methods for a semilinear parabolic problem with nonlocal conditions, Adv. Comput. Math., 6:243–262, 1996.
S.K. Godunov and V.S. Ryabenjkii, Difference Schemes Introduction to the Theory, Moscow, Nauka, 1977 (in Russian).
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.
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.
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.
A.V. Gulin and V.A. Morozova, On the stability of nonlocal finite-difference boundary value problem, Differ. Equ., 39(7):962–967, 2003.
Y. Liu, Numerical solution of the heat equation with nonlocal boundary conditions, J. Comput. Appl. Math., 110(1):115–127, 1999.
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.
R.D. Richtmyer and K.W. Marton, Difference Methods for Initial-Value Problems, Second Edition, John Wiley & Sons, 1967.
A.A. Samarskii, The Theory of Difference Schemes, Moscow, Nauka, 1977 (in Russian); Marcel Dekker, Inc., New York and Basel, 2001.
A.A. Samarskii and A.V. Gulin, Numerical Methods, Moscow, Nauka, 1989 (in Russian).
M.P. Sapagovas, The eigenvalue of some problems with a nonlocal condition, Differ. Equ., 38(7):1020–1026, 2002.
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.
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.
A. Štikonas, The Sturm-Liouville problem with a nonlocal boundary condition, Lith. Math. J., 47(3):336–351, 2007.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sapagovas, M. On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems. Lith Math J 48, 339–356 (2008). https://doi.org/10.1007/s10986-008-9017-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-008-9017-5