## Optimal uniform estimates and rigorous asymptotics beyond all orders for a class of ordinary differential equations (1996)

Venue: | Proceedings of the Royal Society. London. Series A |

Citations: | 7 - 2 self |

### BibTeX

@ARTICLE{Costin96optimaluniform,

author = {Ovidiu Costin and Martin D. Kruskal},

title = {Optimal uniform estimates and rigorous asymptotics beyond all orders for a class of ordinary differential equations},

journal = {Proceedings of the Royal Society. London. Series A},

year = {1996}

}

### OpenURL

### Abstract

For first order differential equations of the form y ′ = ∑ P p=0 Fp(x)y p and second order homogeneous linear differential equations y ′ ′ +a(x)y ′ +b(x)y = 0 with locally integrable coefficients having asymptotic (possibly divergent) power series when |x | → ∞ on a ray arg(x) =const, under some further assumptions, it is shown that, on the given ray, there is a one-to-one correspondence between true solutions and (complete) formal solutions. The correspondence is based on asymptotic inequalities which are required to be uniform in x and optimal with respect to certain weights. 1 Introduction and Main Results The main purpose of the present paper is to give, in terms of uniform asymptotic estimates, a precise meaning to complete asymptotic expansions (e.g., as power series followed by exponentially small terms) of solutions of a class of differential equations in a neighborhood of an irregular singular point (chosen to be infinity).

### Citations

693 |
N.: Theory of Ordinary Differential Equations
- Coddington, Levinson
- 1955
(Show Context)
Citation Context ...solution of the differential equation (1.23) of the form y(x) = V (x) + ǫ(x) with V (x) the solution given in Lemma 3.2 and ǫ(x) ∼ Ce−αx (xr+1 + O(xr )) (|x| → ∞). This is a standard result (see [8], =-=[9]-=-) ; in our case ǫ(x) could be also directly obtained using the contractive mapping arguments above. The uniqueness proof is very similar to the one in the linear case since no two solutions can differ... |

151 |
Asymptotic expansions for ordinary differential equations
- Wasow
- 1987
(Show Context)
Citation Context ...any large enough B. We also want to stress that uniqueness is relative to the space of solutions of the differential equations treated here. 3.) As a manifestation of the Stokes phenomenon ([2], [3], =-=[8]-=-, [14]), the constants Ci in the asymptotic representation of a given solution depends on the direction θ of the ray considered (provided, of course, that the differential equation satisfies our hypot... |

110 |
Differential algebra
- Ritt
- 1950
(Show Context)
Citation Context ...y determined. The expression (1.12) containing two arbitrary constants is the general formal solution of our equation, in the differential algebra generated by power series and exponentials (see [6], =-=[7]-=-, [12]). The notion of optimal asymptoticity can be extended in a natural way to asymptotic structures of the form (1.12). We say that f is uniformly asymptotic to ˜ S for a weight (w1, w2) along the ... |

76 | Asymptotic expansions: their derivation and interpretation - Dingle - 1973 |

50 |
Global theory of a second order linear ordinary differential equation with a polynomial coefficient
- Sibuya
- 1975
(Show Context)
Citation Context ... for any large enough B. We also want to stress that uniqueness is relative to the space of solutions of the differential equations treated here. 3.) As a manifestation of the Stokes phenomenon ([2], =-=[3]-=-, [8], [14]), the constants Ci in the asymptotic representation of a given solution depends on the direction θ of the ray considered (provided, of course, that the differential equation satisfies our ... |

28 | Les fonctions résurgentes - Écalle - 1981 |

18 | Uniform asymptotic smoothing of Stokes’s discontinuities - Berry - 1989 |

11 | Asymptotics beyond all orders in a model of crystal growth - Kruskal, Segur - 1991 |

7 | 1990] “Finitude des cycles limites et accélérosommation de L’application de retor - Écalle |

6 |
Finitude des cycles limites.., Preprint 90-36 of Universite de Paris-Sud
- Écalle
- 1990
(Show Context)
Citation Context ...ow Theorem 1.1, with obvious adaptations, also apply in this case. We could actually continue the construction of the formal solution and consider the complete formal solutions or “transseries”([12] ,=-=[13]-=-) which in this case have the form Y0 + x r e −α x Y1 + x r2 e −2 α x Y2 + ... 11in which Yi are formal power series. Y1 is determined up to an arbitrary constant which, once given, determines comple... |

3 | Theory of ordinary dioeerential equations - Coddington, Levinson - 1955 |

3 | Asymptotics Beyond all Orders - Segur, Tanveer, et al. - 1991 |

3 |
Ecalle in Bifurcations and periodic orbits of vector fields
- unknown authors
- 1993
(Show Context)
Citation Context ...ermined. The expression (1.12) containing two arbitrary constants is the general formal solution of our equation, in the differential algebra generated by power series and exponentials (see [6], [7], =-=[12]-=-). The notion of optimal asymptoticity can be extended in a natural way to asymptotic structures of the form (1.12). We say that f is uniformly asymptotic to ˜ S for a weight (w1, w2) along the ray Rθ... |

1 | Formal solutions of irregular dioeerential equations - Cope - 1934 |

1 | Transseries, analysable functions and Dulac's conjecture - Ecalle - 1993 |

1 | Birkhooe invariants and eoeective calculations for meromorphic linear dioeerential equations - Jurkat, Lutz, et al. - 1976 |

1 | Studies in Applied Mathematics 85:129-181 - Kruskal, Segur - 1991 |

1 |
Astérisque
- Ramis
- 1978
(Show Context)
Citation Context ...ons on w we obtain sharper asymptoticity classes: e.g. if fk ∼ α n (n!) β and we consider the functions f for which wf(k) ≤ γ n (n!) β , γ > α we obtain the familiar Gevrey or Gevrey-Roumieux classes =-=[18]-=-. It is natural to take wf as a measure of the separation between a function f and a formal series, ˜ f. We will say that f is closer to ˜ f than g iff or, on occasions if a weaker condition holds: wf... |