FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
1999, VOLUME 5, NUMBER 3, PAGES 675-686
M. G. Dmitriev
M. V. Nesterova
V. P. Gerdt
Abstract
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In the present paper an algorithm is presented for constructing
approximate solutions of boundary problems for second-order
polynomial-nonlinear ordinary differential equations such that one of
the boundary conditions or both of them are inexactly known. The algorithm
is based on the use of the quadratic penalty functions for the approximately
given boundary conditions and solving the corresponding unconditional
extremum problem. The arising system of nonlinear algebraic equations in
the coefficients of expansion of the solution for some appropriate basic
functions set is solved by the construction of a lexicographical
Gr\"obner basis. It is shown that the construction
of such a basis allows one
to develop a perturbation scheme in the inverse degrees of the penalty
parameters. The proposed algorithm is illustrated by an example of
the boundary problem with the use of computer algebra system Reduce.
The accuracy obtained is analyzed in comparison with some
other methods used to solve that particular boundary problem.
All articles are published in Russian.
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Last modified: November 11, 1999