×

Some properties of solutions to polynomial systems of differential equations. (English) Zbl 1076.34004

Summary: Parker and Sochacki considered iterative methods for computing the power-series solution to \({\mathbf y' = G \circ y}\) where \(G\) is a polynomial from \(\mathbb{R}^n\) to \(\mathbb{R}^n\), including truncations of Picard iteration. The authors demonstrated that many ODEs may be transformed into computationally feasible polynomial problems of this type, and the methods generalize to a broad class of initial value PDEs. In this paper, we show that the subset of the real analytic functions \(\mathcal{A}\) consisting of functions that are components of the solution to polynomial differential equations is a proper subset of \(\mathcal{A}\) and that it shares the field and near-field structure of \(\mathcal{A}\), thus making it a proper sub-algebra. Consequences of the algebraic structure are investigated. Using these results, we show that the Maclaurin or Taylor series can be generated algebraically for a large class of functions. This finding can be used to generate efficient numerical methods of arbitrary order (accuracy) for initial value problems for problems for ordinary differential equations. Examples to indicate these techniques are presented. Future advances in numerical solutions to initial value problems for ordinary differential equations are indicated.

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations

Software:

SINGULAR
PDFBibTeX XMLCite
Full Text: EuDML EMIS