Singular Perturbation of Nonlinear Systems with Regular Singularity

Domingos H U Marchetti,William R P Conti

doi:10.1155/2018/5163492

Abstract

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form εzf′=Fε,z,f with F a Cν-valued function, holomorphic in a polydisc D-ρ×D-ρ×D-ρν. We show that its unique formal solution in power series of ε, whose coefficients are holomorphic functions of z, is 1-summable under a Siegel-type condition on the eigenvalues of Ff(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type.

Highlights

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form εzf󸀠 = F(ε, z, f) with F a C]-valued function, holomorphic in a polydisc Dρ × Dρ × D]ρ
We show that its unique formal solution in power series of ε, whose coefficients are holomorphic functions of z, is 1-summable under a Siegel-type condition on the eigenvalues of Ff(0, 0, 0)
When (1) is linear, i.e., F = Af − b, where b = b(ε, z) and A = A(ε, z) are, respectively, a ]-vector and a ] × ] matrix, whose entries are holomorphic in the polydisc DR × DR, R > 0, such that A(0, 0)−1 exists, Balser and Kostov [1] have established the following: (a) there exists a unique formal solution in the ring O(r)[[ε]]1 of formal power series f(ε, z) = ∑ai (z) εi i=0 in ε with coefficients ai(z) in the ring O(r) of holomorphic functions on Dr, continuous in its closure, satisfying max

Summary

Introduction

Balser-Kostov summability proof in [1] of the formal series solution fdoes not follow the usual route by which the (formal) Borel transform Bfof fis analytically continued along some sector of infinite radius (see, e.g., [2]) Their proof establishes instead Gevrey asymptotic expansion directly from (5), requiring for this an auxiliary lemma regarding an infinite system of linear equations of the same type whose coefficient matrix A = A(ε, z) is independent of z. (1) is nonlinear, the system of infinitely many equations obtained by taking derivatives of (1) with respect to ε is linear and Balser-Kostov’s method carries over to equation of the form (1) To prove these statements, suitable formulas and a simple but efficient way of estimating higher power of f are provided. The advantage of the proposed summability method is that it can be applied to nonlinear partial differential equations of evolution type [7, 15]

Power Series in z

Formal Power Series in ε

Gevrey Asymptotic

Summability of the Formal Series

Summary of Results