2020 Fiscal Year Final Research Report
Applications of micro-local analysis and wavelet analysis to wave equations with variable coefficients
| Project/Area Number |
16K05223
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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 |
Mathematical analysis
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| Research Institution | University of Tsukuba |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
梶谷 邦彦 筑波大学, 数理物質系(名誉教授), 名誉教授 (00026262)
石渡 聡 山形大学, 理学部, 准教授 (70375393)
久保 隆徹 お茶の水女子大学, 基幹研究院, 准教授 (90424811)
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| Project Period (FY) |
2016-04-01 – 2021-03-31
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| Keywords | 関数方程式論 / ウェーブレット / 数値解析 |
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| Outline of Final Research Achievements |
In this research, we studied partial differential equations, wavelet and Radon transform. We considered exact formulas and well-posedness of the Cauchy problem for wave equations with variable coefficients. As for the wavelet analysis, we designed some two-dimensional Parseval frames and dual frames. By numerical simulations we found that smooth frames in the frequency space give better reconstruction. Moreover, we also proposed some transforms concerned with the Radon transform and showed their properties and applications.
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| Free Research Field |
関数解析
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| Academic Significance and Societal Importance of the Research Achievements |
変数係数を持つ波動タイプの偏微分方程式に対する初期値問題の解の表現公式が得られれば、物理現象の法則となる解の性質等が引き出せ、数値実験もそのままの形で実行ができるため、理論的にも応用的にも非常に意義があり、波動現象の解明へと繋がることが期待できる。また、本研究で得られた2次元のパーセヴァルフレームや双対フレームは2次元の画像解析への応用が可能で、数値解析的な処理速度や画像の精度の向上が期待できる。
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