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