Electron. J. Diff. Eqns., Vol. 2005(2005), No. 40, pp. 1-17.

Some properties of solutions to polynomial systems of differential equations

David C. Carothers, G. Edgar Parker, James S. Sochacki, Paul G. Warne

Abstract:
In [7] and [8], Parker and Sochacki considered iterative methods for computing the power series solution to ${\bf y' = G \circ y}$ where ${\bf G}$ is a polynomial from $\mathbb{R}^n$ to $\mathbb{R}^n$, including truncations of Picard iteration. The authors demonstrated that many ODE's may be transformed into computationally feasible polynomial problems of this type, and the methods generalize to a broad class of initial value PDE's. 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 ordinary differential equations. Examples to indicate these techniques are presented. Future advances in numerical solutions to initial value ordinary differential equations are indicated.

Submitted August 26, 2004. Published April 5, 2005.
Math Subject Classifications: 34A05, 34A25, 34A34, 65L05.
Key Words: Analytic functions; inverse functions; Maclaurin polynomials; Pade expansions; Grobner bases.

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David C. Carothers
Department of Mathematics and Statistics
James Madison University
Harrisonburg, VA 22807, USA
email: carothdc@jmu.edu
G. Edgar Parker
Department of Mathematics and Statistics
James Madison University
Harrisonburg, VA 22807, USA
email: parkerge@jmu.edu
James S. Sochacki
Department of Mathematics and Statistics
James Madison University
Harrisonburg, VA 22807, USA
email: jim@math.jmu.edu
Paul G. Warne
Department of Mathematics and Statistics
James Madison University
Harrisonburg, VA 22807, USA
email: warnepg@math.jmu.edu

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