Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 40, pp. 1-17.
Title: Some properties of solutions to polynomial systems of
differential equations
Authors: David C. Carothers (James Madison Univ., Harrisonburg, USA)
G. Edgar Parker (James Madison Univ., Harrisonburg, USA)
James S. Sochacki (James Madison Univ., Harrisonburg, USA)
Paul G. Warne (James Madison Univ., Harrisonburg, USA)
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.