Sign-change structure analysis for reaction-diffusion models based on rigorous numerical computation

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14601
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Waseda University
• Principal Investigator: Tanaka Kazuaki 早稲田大学, 理工学術院総合研究所, 次席研究員(研究院講師) (00801226)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 精度保証付き数値計算 / 計算機援用証明 / 符号変化構造解析

Outline of Final Research Achievements

The aim of this research was to conduct a sign-change structure analysis of reaction-diffusion models using guaranteed-accuracy numerical computation, thereby contributing to elucidating phenomena from a mathematical perspective. The PI established a method to post-validate the positivity of the true solution using the information on the eigenvalues of the operators describing the model and the results of guaranteed accuracy (inclusion information of solutions). This method was applied to the Allen-Cahn equation (a reaction-diffusion model that describes water state changes and phase separation phenomena) and successfully provided proof of the coexistence relationship of the liquid phase layers.

Free Research Field

数値解析

Academic Significance and Societal Importance of the Research Achievements

反応拡散モデルを含む微分方程式の解に対する精度保証付き数値計算の従来研究は「近似解の数値計算→精度保証」というプロセスの速度向上や、得られる結果の高精度化に焦点が当てられていたが、研究代表者はそこに「符号の保証」という新たな価値を与えた。この着眼点が評価され、精度保証という研究分野が発足した当初から40年以上の歴史を持ち、現在も最も権威があるとされる国際学会SCANの基調講演者として招待された。