Multisummability of Formal Solutions of Singular Perturbation Problems

Abstract

We investigate the summability of the unique formal power series solution of a singular perturbation problem for certain linear systems of ordinary meromorphic differential equations, using a new type of summability of power series in several (here: two) variables. # 2002 Elsevier Science (USA)

Parametric Borel summability for some semilinear system of partial differential equations

“…(b) An interesting phenomenon shown in [2] is that a certain Diophantine phenomenon appears in the summability, while it does not appear for an irregular singular equation (cf. [4]). In the case of general independent variables one can easily see that a similar multi-dimensional Diophantine condition enters in the analysis.…”

“…[3,5,8,10,11]). On the other hand, concerning the summability of formal solutions of a partial differential equation with a singular perturbation parameter we cite [2] and [4]. (See also [6,7] and [9].…”

Consistency of iterative operator‐splitting methods: Theory and applications

“…In these cases, a formal solution with respect to the asymptotic convergence of a power series, which is near the exact solution can be sought, see [19], [20].…”

Singular Perturbation of Nonlinear Systems with Regular Singularity

“…with the Poincaré rank ≥ 1, has been investigated before in [3]. Contrarily to the previous case ( = 0), the unique formal power series solution̂( , ) is always 1-summable, irrespective of whether (4), the additional condition satisfied by the eingenvalues of (0, 0), holds or not.…”

“…The case of = −1, on the other hand, has been studied in [4] for ] = 1 and the summability of the formal series can be read from the properties of the initial data of (5). The case = 0 separates the two cases and we refer to [3] for an explanation on summability properties for each of distinct cases of simple examples in which and = ( ) are a scalar and a scalar function depending only on . For recent investigations of the linear meromorphic system (5) with ≥ 1, ( = 0) ̸ = 0, and ≡ 0, about summable-resurgent of the Borel transform of its highest level's reduced formal solutions and connectionto-Stokes formulas, see [5] and references therein.…”