Studies on verified numerical computations for nonlinear parabolic partial differential equations

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K17596
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: University of Tsukuba (2016-2017); Waseda University (2015)
• Principal Investigator: Takayasu Akitoshi 筑波大学, システム情報系, 助教 (60707743)
• Research Collaborator: OISHI SHIN'ICHI 早稲田大学, 理工学術院, 教授 ; KUBO TAKAYUKI 筑波大学, 数理物質系, 講師 ; MATSUE KANAME 九州大学, マス・フォア・インダストリ研究所, 助教 ; MIZUGUCHI MAKOTO 早稲田大学, 理工学術院, 次席研究員
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 精度保証付き数値計算 / 非線形放物型偏微分方程式 / 解の数値的検証 / 爆発問題 / ケラー・シーゲル方程式系

Outline of Final Research Achievements

Partial differential equations (PDEs), which expresses some relations using derivatives of unknown functions, frequently occur when natural phenomena in the world are modeled as mathematical problems. In the field of natural science, it is a subject of research to solve such PDEs mathematically or numerically to specify the unknown function. In this study, we have developed a computer-assisted method to numerically verify the existence/non-existence of solutions to nonlinear parabolic PDEs, which appear in the combustion theory of solid fuel and in the mathematical model of biological growth. Such a method validates whether the exact solution exists in the neighborhood of a numerically computed approximate solution. This is called verified numerical computations and is gaining attention as a modern approach to mathematical analysis of differential equations.

Free Research Field

数値解析