Computing power series solutions of a nonlinear PDE system

Abstract

This paper presents a new algorithm to compute the power series solutions of a significant class of nonlinear systems of partial differential equations. The algorithm is very different from previous algorithms to perform this task. Those relie on differentiating iteratively the differential equations to get coefficients of the power series, one at a time. The algorithm presented here relies on using the linearisation of the system and the associated recurrences. At each step the order up to which the power series solution is known is doubled. The algorithm can be seen as belonging to the family of Newton iteration methods.

Properties of an optimal linearization transformation

“…If we differentiate the conjugacy relation (4.1) with respect to t and apply the differential equation that / t (x 0 ) satisfies, we obtain If one is interested in approximating w(x 0 ) using (4.2), a power series approximation can be obtained (see, e.g. [23,24]). …”

A Normal Form Algorithm for Regular Differential Chains

“…Reference texts for this section are Seidenberg (1958Seidenberg ( , 1969. See also (Rust et al, 1999;Hubert and Le Roux, 2003). The m derivations δ 1 , .…”

The Cauchy-Kowalevski Theorem

“…In recent time we can observe the renewed interest in the algorithms associated with the solution of partial differential equations using power series (see: for example [8]). This study initiated by the famous theorem of Cauchy-Kowalevski 1 (see original work [9], Theorem 2.1 and Proposition 2.1 in this article, compare [2], [3]) were later generalized by Riquier [11] for a wide class of orthonomic passive systems.…”

“…Since that time many algorithms for determining the formal solution of partial differential equations was stated. It is well known that coefficients of such a formal solution are polynomials depending on coefficients occurring in the power series expansion of right-hand side functions in partial differential equations (see: for example [1], [4], [8], see also [13]). Moreover, these polynomials have non-negative rational coefficients.…”