On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912047572189184 |
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| author | Nemes, Gergő |
| author_facet | Nemes, Gergő |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14449 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations Nemes, Gergő Classical Analysis and ODEs 34E05, 34E20, 34M25 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. |
| title | On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations |
| topic | Classical Analysis and ODEs 34E05, 34E20, 34M25 |
| url | https://arxiv.org/abs/2312.14449 |