Nonlinear Differential Equations with Inexactly Known Boundary Conditions and Gröbner bases

Vladimir P. Gerdt Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research 141980 Dubna, Russia Email: gerdt@jinr.dubna.su and Michael G.Dmitriev and Marina V.Nesterova Program System Institute of Russian Academy of Sciences 152140 Pereslavl-Zalessky, Russia

An algorithm is presented for constructing approximate solutions of boundary problems for second-order differential equations of the form

Φ(y′′, y′, x) = 0, y(a) = y_{a}, y_{b} = y_{b} ,  (1)

where y(x) is a real function, Φ is a polynomial in its arguments over rationals, and the numbers a, b, y_{a}, y_{b} are also rational. Here y_{a} or y_{b} or both of them are assumed to be known with some reasonably small errors. 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. In particular, if equation Φ(y′′, y′, x) = 0 admits the variational formulation (ϕ_{y′})′ − ϕ_{y} = 0, and both y_{a}, y_{b} are inexactly known, then an approximate solution of (1) can be constructed from the condition

F ({c_{i}}, α, β) = ∫_{a}^{b} ϕ(ỹ′′, ỹ′, x)dx + α[ỹ(a) − y_{a}]^{2} + β[ỹ(b) − y_{b}]^{2} ⟹ min_{{c_{i}}} ,

y ≈ ỹ = ∑_{i=1}^{n} c_{i}v_{i}(x) ,

The arising system of nonlinear algebraic equations in the coefficients c_{i} for some appropriate set of basic functions v_{i}(x) is solved by 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 α and β. The related computational aspects for parametric Gröbner bases are considered. The proposed algorithm is illustrated by an example of the boundary problem with the use of computer algebra system Reduce. The obtained accuracy is analysed in comparison with some other methods used to solve that particular boundary problem.