Borel summability of divergent power series solutions to linear partial differential equations with time dependent coefficients and its applications

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K04898
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Aichi University of Education
• Principal Investigator: Kunio Ichinobe 愛知教育大学, 教育学部, 准教授 (20434417)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: k総和法 / ボレル総和法 / q-差分微分方程式 / 初期値問題 / 発散級数 / 漸近解析

Outline of Final Research Achievements

We considered linear partial differential equations in the complex domain, and obtained the following results. (1) We treated the local holomorphic Cauchy problem of partial differential equations with time dependent polynomial coefficients, and gave a sufficient condition that divergent power series solutions are k-summable in term of the global exponential growth estimate of the Cauchy data. (2) We treated the entire Cauchy problem of typical equations with some restriction, and gave a characterization of k-summable formal solutions.
We considered the Cauchy problem of typical q-difference-differential equations. (3) We gave a necessary and sufficient condition for the convergence of formal solutions. When the formal solutions diverge, we showed the existence and uniqueness of the asymptotic solutions. (4) When the Cauchy data is entire, we gave a characterization of k-summable formal solutions.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

複素領域における線形偏微分方程式の初期値問題に現れる発散級数解のボレル総和可能性の問題について、定数係数の場合は一応の決着を見ている。しかし、変数係数の場合はほとんど行われていなかった。本研究で得られた成果は、時間変数の多項式係数という制約がついているが、一般の変数係数の場合の解析に向けた大きな一歩となるものと考えられる。また、偏微分方程式のq-類似の結果を応用することにより、偏微分方程式の問題を進展することが出来たことは、q-類似を考えることの意義が大きいことを意味する。