Cauchy-Kowalevski and polynomial ordinary differential equations

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

Electronic journal of differential equations, Tome 2012 (2012)

Résumé

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.

Détail

Classification : 34A12, 34A34, 35A10
Keywords: automatic differentiation, power series, Taylor series, polynomial ODE, majorant, error bound