Application of Gr\"obner bases for solving polynomial-nonlinear boundary problems with inexactly known boundary conditions

M. G. Dmitriev ; M. V. Nesterova ; V. P. Gerdt

Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 3, pp. 675-686.

Résumé

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 Gröbner 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.