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On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations
- Source :
- Journal of Differential Equations, Volume 415, 645-700 (2025)
- Publication Year :
- 2023
-
Abstract
- We consider a class of $n^{\text{th}}$-order linear ordinary differential equations with a large parameter $u$. Analytic solutions of these equations can be described by (divergent) formal series in descending powers of $u$. We demonstrate that, given mild conditions on the potential functions of the equation, the formal solutions are Borel summable with respect to the parameter $u$ in large, unbounded domains of the independent variable. We establish that the formal series expansions serve as asymptotic expansions, uniform with respect to the independent variable, for the Borel re-summed exact solutions. Additionally, we show that the exact solutions can be expressed using factorial series in the parameter, and these expansions converge in half-planes, uniformly with respect to the independent variable. To illustrate our theory, we apply it to an $n^{\text{th}}$-order Airy-type equation.<br />Comment: 47 pages, 3 figures, accepted for publication in Journal of Differential Equations. The exposition was improved based on the referee's comments
- Subjects :
- Mathematics - Classical Analysis and ODEs
34E05, 34E20, 34M25
Subjects
Details
- Database :
- arXiv
- Journal :
- Journal of Differential Equations, Volume 415, 645-700 (2025)
- Publication Type :
- Report
- Accession number :
- edsarx.2312.14449
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.jde.2024.09.041