| Project/Area Number |
18K13453
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | University of Tsukuba |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 計算機援用証明 / 複素数値非線形熱方程式 / 非線形シュレディンガー方程式 / 一次元変数係数移流方程式 / Parameterization method / 厳密な数値求積 / 解の時間大域存在 / 双曲型偏微分方程式 / 放物型偏微分方程式 / 分散型偏微分方程式 / 零点探索問題 / 簡易ニュートン写像 / 発展作用素 / 精度保証付き数値計算 / 無限次元力学系 / 解の時間大域挙動 / スペクトル法 / Lyapunov-Perronの方法 / C0半群 / 解の数値的検証 / 数値解析 |
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| Outline of Final Research Achievements |
Mathematical problems obtained by modeling natural phenomena are called mathematical models. Mathematical models are often formulated as partial differential equations (PDEs), and solving them mathematically and numerically to understand the behavior of unknown functions is a central research topic in the natural sciences. In this study, we have developed a computer-assisted proof method for a class of PDEs called hyperbolic PDEs, which appear in mathematical models of wave phenomena and quantum mechanics. Such a method proves that the solution of the initial boundary value problem exists in a neighborhood of the numerically computed approximate solution. This is called verified numerical computations, and is attracting attention as a modern approach to mathematical analysis of differential equations.
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| Academic Significance and Societal Importance of the Research Achievements |
本研究成果は双曲型偏微分方程式を含むより広いクラスの偏微分方程式に対して、数値計算による証明手法を提供する。特に、解の挙動を無限次元力学系として捉え、各計算機援用証明手法により解の大域挙動を明らかにした研究成果は自然科学分野における数理モデルの開発や現象の解明に貢献している。物理の波動現象や量子力学をモデル化する際の偏微分方程式の解挙動を数学証明付きで理解することで、科学研究の進展や新たな技術・応用の開発に寄与し、社会の課題解決に役立つことが期待される。
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