Electron. J. Diff. Equ., Vol. 2012 (2012), No. 11, pp. 1-8.

Cauchy-Kowalevski and polynomial ordinary differential equations

Roger J. Thelwell, Paul G. Warne, Debra A. Warne

Abstract:
The Cauchy-Kowalevski Theorem is the foremost result guaranteeing existence and uniqueness of local solutions for analytic quasilinear partial differential equations with Cauchy initial data. The techniques of Cauchy-Kowalevski may also be applied to initial-value ordinary differential equations. These techniques, when applied in the polynomial ordinary differential equation setting, lead one naturally to a method in which coefficients of the series solution are easily computed in a recursive manner, and an explicit majorization admits a clear a priori error bound. The error bound depends only on immediately observable quantities of the polynomial system; coefficients, initial conditions, and polynomial degree. The numerous benefits of the polynomial system are shown for a specific example.

Submitted October 11, 2010. Published January 17, 2012.
Math Subject Classifications: 34A12, 34A34, 35A10.
Key Words: Automatic differentiation; power series; Taylor series; polynomial ODE; majorant; error bound.

Show me the PDF file (226 KB), TEX file, and other files for this article.

Roger J. Thelwell
Department of Mathematics and Statistics
James Madison University
Harrisonburg, VA 22807, USA
email: thelwerj@jmu.edu
Paul G. Warne
Department of Mathematics and Statistics
James Madison University
Harrisonburg, VA 22807, USA
email: warnepg@jmu.edu
Debra A. Warne
Department of Mathematics and Statistics
James Madison University
Harrisonburg, VA 22807, USA
email: warneda@jmu.edu

Return to the EJDE web page