Abstract
A multipoint scheme of the method of lines is applied to the boundary-value problem
The second derivative
replaced by a
-point (ρ is any positive integer) central-difference approximation with an error of order
where
is the step of the net of lines. An approximating system of ordinary differential equations, associated with problem (1), (2), is transformed into a reducing one. The uniform convergence of the approximate solution of the method of lines to the solution of the original boundary-value problem with order
is established. For this the solution of the reducing system with zero boundary conditions is examined for the difference between the exact solution of problem (1), (2) and the approximate solution obtained by the method of lines. The behavior of this solution as
is studied at point
and, next, at any point
by transforming the independent variable
that transfers point z to the origin.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis [in Russian], Moscow-Leningrad (1962).
A. P. Kubanskaya, “One matrix equality,” J. Sov. Math.,13, No. 2, 225–230 (1980).
A. P. Kubanskaya, “On the application of a multipoint differential-difference scheme to one boundary-value problem,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,70, 76–88 (1977).
V. N. Faddeeva, “The method of lines applied to some boundary-value problems,” Tr. Mat. Inst. Steklova, Akad. Nauk SSSR,28, 73–103 (1949).
D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra, W. H. Freeman, San Francisco-London (1963).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 39–45, 1979.
Rights and permissions
About this article
Cite this article
Kubanskaya, A.P. Convergence with orderh 2p-1 of a 2p+1-point scheme of the method of lines for one boundary-value problem. J Math Sci 20, 1923–1928 (1982). https://doi.org/10.1007/BF01680563
Issue Date:
DOI: https://doi.org/10.1007/BF01680563