Recent progress in the theory of formal solutions for ODE and PDE

doi:10.1016/S0096-3003(02)00325-9

Applied Mathematics and Computation

Volume 141, Issue 1, 20 August 2003, Pages 113-123

https://doi.org/10.1016/S0096-3003(02)00325-9 Get rights and content

Abstract

In this article we describe the theory of multisummability of formal power series and discuss its application to power series solutions of ordinary and partial differential equations. In particular, we give some new application to a singular perturbation of a regular singular point.

Introduction

Generally speaking, a summability method may be viewed as a linear map $S$ , defined on some vector space X of series, which we here take to be formal power series in a complex variable z. The coefficients of the power series may be complex numbers, or more generally elements of some Banach algebra, say, of functions in other variables. To each such formal power series, this linear map assigns a generalized sum, which for power series in z should be a holomorphic function in this variable. To be useful for applications to differential and other functional equations, the operator $S$ should have a number of properties:

•
The space X should contain all convergent power series, and $S$ should map each convergent series to its natural sum.
•
The linear space X should be a differential algebra; i.e., for each power series in X the formal derivative should again belong to X, and for any two power series in X, their product should also be in X.
•
The operator $S$ should be a homomorphism of differential algebras; hence it should not only be linear, but should map products to products and derivatives to derivatives.
•
For any formal power series $f ̂ ∈X$ , the function $S f ̂$ should be holomorphic in some sectorial region G, and asymptotic to $f ̂$ when the variable tends to the origin.

Section snippets

Multisummability

All the above requirements for a “good” summability method are indeed fulfilled by the following variant of Borel summability which was first introduced and studied by Ramis [27]:

Let k>0, $d∈ R$ and a power series $f ̂ (z)=∑_{0}^{∞} f_{n} z^{n}$ be given. We say that $f ̂$ is k-summable in direction d, if the following two conditions hold:

1.
The series $g(u)=∑_{0}^{∞} f_{n} u^{n} Γ(1+n/k)$ has positive radius of convergence.
2.
There exists some δ>0 so that the function g(u) defined above can be holomorphically continued into the sector |d−argu

Meromorphic ordinary differential equations

For a natural number r and constant square matrices A_j, consider a linear system of ODE of the form $zx^{′} =z^{−r} A(z)x, A(z)=∑_{0}^{∞} A_{j} z^{j},$ assuming that the series converges for |z| sufficiently small. It is well known that this system admits a formal fundamental solution of the form $X (z)= F (t)G(z)$ , where $F (t)$ is a formal matrix power series in a root t=z^1/p, while G(z) is a matrix of elementary functions such as exponents of polynomials in 1/t, integer powers of the logarithm, and general powers of z. The

Singular perturbation problems

Let us consider a linear perturbation problem of the following form: $εzx^{′} =A(z,ε)x−f(z,ε),$ with a square matrix A(z,ε) and a vector f(z,ε) whose entries are holomorphic in some polydisc, say, of radius R>0, about the origin. In addition, we shall always assume that the matrix A(0,0) is invertible. Under this assumption, one can easily see that (4.1) has a unique formal solution of the form $x ̂ (z,ε)=∑_{n=0}^{∞} x_{n} (z)ε^{n}$ . The coefficients x_n(z) are determined by $∑_{m=0}^{n} A_{n−m} (z)x_{m} (z)=zx^{′}_{n−1} (z)+f_{n} (z) ∀n⩾0,$ where A_n(z

Partial differential equations

Besides the classical theory discussing convergence of power series solutions for Cauchy problems for certain classes of PDE, there are recent results concerning the Gevrey order of such power series; e.g., see [17], [19], [20], [21], [22], [23], [25] and the literature cited there. There are also articles, e.g., by Ouchy [24], [26], showing that such formal solutions are asymptotic representations of proper solutions of the underlying equation. Only few papers exist so far about

Acknowledgements

Author Werner Balser presented this article at the workshop Advanced Special Functions and Related Topics in Differential Equation. Melfi, June 24–29, 2001. He is grateful to the organizers of the workshop, and to University of Ulm for financial support.

References (28)

W. Balser et al.
Multisummability of formal solutions of singular perturbation problems
J. Differ. Equations
(2002)
W. Balser, Einige Beiträge zur Invariantentheorie meromorpher Differentialgleichungen, Habilitationsschrift,...
W. Balser
Growth estimates for the coefficients of generalized formal solutions, and representation of solutions using Laplace integrals and factorial series
Hiroshima Math. J.
(1982)
W. Balser
Solutions of first level of meromorphic differential equations
Proc. Edinburgh Math. Soc.
(1982)
W. Balser
A constructive existence proof for first level formal solutions of meromorphic differential equations
Hiroshima Math. J.
(1985)
W. Balser
A different characterization of multisummable power series
Analysis
(1992)
W. Balser
Summation of formal power series through iterated Laplace integrals
Math. Scandinavica
(1992)
W. Balser, From Divergent Power Series to Analytic Functions, Lecture Notes in Math., vol. 1582, Springer Verlag, New...
W. Balser
Divergent solutions of the heat equation: on an article of Lutz, Miyake and Schäfke
Pacific J. Math.
(1999)
W. Balser
Formal power series and linear systems of meromorphic ordinary differential equations
(2000)

W. Balser et al.

Singular perturbation of linear systems with a regular singularity

J. Dyn. Control Syst.

(2002)

W. Balser et al.

Summability of formal solutions of certain partial differential equations

Acta Sci. Math. (Szeged)

(1999)

B.L.J. Braaksma

Multisummability of formal power series solutions of nonlinear meromorphic differential equations

Ann. Inst. Fourier (Grenoble)

(1992)

M. Canalis-Durand et al.

Gevrey solutions of singularly perturbed differential and difference equations

J. Reine und Angew. Math.

(2000)

Cited by (0)

View full text

Recent progress in the theory of formal solutions for ODE and PDE

Abstract

Introduction

Section snippets

Multisummability

Meromorphic ordinary differential equations

Singular perturbation problems

Partial differential equations

Acknowledgements

J. Differ. Equations

Growth estimates for the coefficients of generalized formal solutions, and representation of solutions using Laplace integrals and factorial series

Hiroshima Math. J.

Solutions of first level of meromorphic differential equations

Proc. Edinburgh Math. Soc.

A constructive existence proof for first level formal solutions of meromorphic differential equations

Hiroshima Math. J.

A different characterization of multisummable power series

Analysis

Summation of formal power series through iterated Laplace integrals

Math. Scandinavica

Divergent solutions of the heat equation: on an article of Lutz, Miyake and Schäfke

Pacific J. Math.

Formal power series and linear systems of meromorphic ordinary differential equations