Application of the Gevrey Theory to the Parameterization of Nonlinear Problems
by
Sandrine Bargiacchi
MIP Laboratory. 118 route de Narobonne - 31 062 Toulouse Cedex
Coauthors: Mohamed MASMOUDI (MIP Laboratory - Toulouse)

Application of the Gevrey Theory to the Parameterization of Nonlinear Problems

Sandrine BARGICCHI and Mohamed MASMOUDI
CNRS - Universite Paul Sabatier - UMR MIP 5640
118 route de narbonne - F31062 Toulouse Cedex

Let's consider a family of linear systems depending on a parameter p

A(p) U=B.

If A is invertible and depends analytically on p then the solution U is meromorphic (i.e. U(p) = P(p)/Q(p) where P and Q are holomorphic). Pade approximation is then a natural way to approximate U.

If U(p) is the solution to a nonlinear system F(p, U)=0, then the map p --> U(p) is not generally meromorphic. The most simple example is the scalar equation U²-p=0. In this case Pade is not appropriate for the approximation of U. The Gevrey theory gives an interesting alternative to Pade approximation.

Pade tries to extend the extrapolation domain over the poles. The basic idea of the Gevrey theory is to avoid the singularities by the mean of a variable change. The singularities are rejected out of the convergence domain. The approximation is then calculated by a Laplace transform.

http://mip.ups-tlse.fr/~bargiac/AD2000_ext_abstract.ps

Date received: December 21, 1999