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Classical Analysis
Seminar information archive ～05/01｜Next seminar｜Future seminars 05/02～
2024/01/24
10:30-12:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Gergő Nemes (Tokyo Metropolitan University)
On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations (English)
Gergő Nemes (Tokyo Metropolitan University)
On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations (English)
[ Abstract ]
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
