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| 担当者 | 大島 利雄, 坂井 秀隆 |
|---|
2024年01月24日(水)
10:30-12:00 数理科学研究科棟(駒場) 122号室
Gergő Nemes 氏 (東京都立大学)
On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations
(English)
Gergő Nemes 氏 (東京都立大学)
On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations
(English)
[ 講演概要 ]
We will consider a class of $n$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 shall 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 will 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, 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 $n$th-order Airy-type equation.
Related preprint: https://arxiv.org/abs/2312.14449
We will consider a class of $n$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 shall 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 will 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, 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 $n$th-order Airy-type equation.
Related preprint: https://arxiv.org/abs/2312.14449
