| Project/Area Number |
15K04898
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Aichi University of Education |
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Principal Investigator |
Kunio Ichinobe 愛知教育大学, 教育学部, 准教授 (20434417)
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| Project Period (FY) |
2015-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2017: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
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| 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 / k-総和法 / 発散級数解 / 偏微分方程式 / q差分微分方程式 / q超幾何級数 / 整関数 |
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| 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.
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| 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。
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