A Picard-Maclaurin theorem for initial value PDEs. (English) Zbl 0987.35006

Parker, G. Edgar; Sochacki, James S.

Abstr. Appl. Anal. 5, No. 1, 47-63 (2000).

The authors extend their result for the ODE \(y'=F\circ y; \quad y(0)=x\) To PDEs. It was proved for the ODE that if \(F\) is a polynomial from \(\mathbb{R}^n \to \mathbb{R}^n \), then the \(k\)-th Picard iteration defined by \[ p_k(t)=x+\int_0^tF\circ p_{k-1}, \quad p_1(t)=x \] is the \((k-1)\)-st degree Maclaurin polynomial plus a polynomial all of those terms have degree greater than \(k-1\). In the paper for PDE one component is designated by analogy to ODE. The generator, supposed to be polynomial and autonomous with respect to this component and no partial derivatives with respect to this component, can appear in the domain of the generator. The initial conditions are given in the designated component at zero and are analytic in the nondesignated components. It is shown that the power series solution of the Cauchy problem for a such PDE (existed by the Cauchy theorem) can be generated to arbitrary degree by Picard iteration. Some examples are given.

Reviewer: Irina V.Melnikova (Ekaterinburg)

MSC:

35A07	Local existence and uniqueness theorems (PDE) (MSC2000)
35A35	Theoretical approximation in context of PDEs
35C10	Series solutions to PDEs
35F25	Initial value problems for nonlinear first-order PDEs
35G10	Initial value problems for linear higher-order PDEs
35G25	Initial value problems for nonlinear higher-order PDEs

Keywords:

Picard iteration; polynomial generator