2022 Fiscal Year Final Research Report
Analysis of a global-in-time solution for reaction-diffusion system using verified numerical computation
| Project/Area Number |
18K13462
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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 | Chuo University (2021-2022)
Waseda University (2018-2020) |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | 計算機援用証明 / 解の精度保証付き数値計算法 / 放物型偏微分方程式 |
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| Outline of Final Research Achievements |
In this research, we mainly aim to improve numerical verification method for solutions of parabolic partial differential equations including reaction-diffusion systems, and to establish a method for verifying the existence of special solutions such as global-in-time solutions and blow-up solutions. First, for improving the verification method, we obtained the best value of the error constant of the semi-discrete approximation of the parabolic equation.The improved method were finally able to clarify the range of the explosion time of the blow-up solution, which could not be clarified by previous mathematical methods, of a parabolic equation.
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| Free Research Field |
偏微分方程式の解の精度保証付き数値計算
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| Academic Significance and Societal Importance of the Research Achievements |
一般的な非線形偏微分方程式の解を解析的に解くことは難しい. しかし解の精度保証付き数値計算法を用いれば偏微分方程式の解の厳密な存在範囲を明確に示すことができる. そのため方程式の解の存在だけでなく, 数値シミュレーション結果の妥当性を保証するといった工学面に対する応用も可能である. この解の精度保証付き数値計算法の改良かつその手法の適応範囲の拡大が本研究の主な目的である. 本研究の最も重要な成果はその計算手法を用いてある放物型方程式の解の爆発時間の範囲を得たことである. それは既存の数学的手法では得られなかった現象(爆発現象)の一端が解明できたことを意味する.
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