2020 Fiscal Year Final Research Report
Basic research on quality assurance of numerical simulations by visualizing the regularity of solutions of differential equations
| Project/Area Number |
18K03436
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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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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Doshisha University |
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Principal Investigator |
Imai Hitoshi 同志社大学, 理工学部, 教授 (80203298)
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| Co-Investigator(Kenkyū-buntansha) |
藤原 宏志 京都大学, 情報学研究科, 准教授 (00362583)
磯 祐介 京都大学, 情報学研究科, 教授 (70203065)
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| Project Period (FY) |
2018-04-01 – 2021-03-31
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| Keywords | 品質 / 正則性 / 地図 / 数値計算 / 微分方程式 / 滑らかさ / 可視化 |
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| Outline of Final Research Achievements |
We developed numerical methods to investigate the location of singular points of one-variable functions and their smoothness at singular points. We found that the interpolated function oscillates violently near the boundary, and that the oscillation is localized by local averaging. For nonlinear ordinary differential equations with blow-up solutions, we developed highly accurate numerical methods for the blow-up time using the numerical limit, and succeeded in creating numerical regularity maps of the solution. For fractional differential equations with Helder continuous solutions, we succeeded in creating numerical regularity maps of the solution. They show that the property of the Caputo derivative changes whether the order is greater than or less than 1. We also found that the accuracy spike phenomenon occurs. We developed highly accurate numerical integration methods when the integrand has a sharp peak, and also developed methods for fast numerical computations.
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| Free Research Field |
数値解析学
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| Academic Significance and Societal Importance of the Research Achievements |
学術的意義は理論研究の発展や研究者人口の拡大への貢献にある。それには、本課題研究が基礎とする数値計算の汎用性、その数値計算による数学的性質の可視化が重要な役割を果たす。社会的意義は正確な数理モデル構築の貢献にある。数値計算の汎用性から、計算対象の問題や方程式は実用レベルの複雑なものが扱える。例えば本課題研究が想定した微分積分方程式は、世界最先端の近赤外光を使った癌治療法である光免疫療法に関連する。非整数階微分方程式は血糖値変化の数理モデルに現れる。本課題研究で提案した問題や解の数学的性質を数値計算で明らかにする研究は、このような実用問題に対して正確なモデルを構築する際に役立つ。
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