Development of a computer-assisted proof method to verify the existence of solutions for systems to large-scale nonlinear elliptic partial differential equations

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K17651
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Toyo University (2017-2018); Waseda University (2016)
• Principal Investigator: Sekine Kouta 東洋大学, 情報連携学部, 助教 (80732239)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: 計算機援用証明法 / 精度保証付き数値計算 / 楕円型偏微分方程式 / 数値解析

Outline of Final Research Achievements

In this study, we aimed at the development of a computer-assisted proof method for the solution of the boundary value problem of system for large-scale elliptic partial differential equations. In particular, the norm evaluation of the inverse operator of the linearized operator, which is the most difficult part of the computer-assisted proof method, is not practical because the existing method has large errors when applied to large-scale simultaneous partial differential equations.
We developed a method to obtain inverse operator norm evaluation of linearized operators corresponding to large-scale elliptic partial differential equations by using fractional operator of Laplacian.
As a result, the computer-assisted proof method of the solution of system for the large-scale elliptic partial differential equation that was difficult until now becomes possible, and it was actually applied to the Lotka-Volterra equation etc.

Free Research Field

精度保証付き数値計算

Academic Significance and Societal Importance of the Research Achievements

非線形偏微分方程式は様々な現象を記述し，現代科学への発展にはなくてはならないものである．しかし，非線形偏微分方程式は複雑であるため，その解が存在するかどうかすらわからない場合がある．そこで，計算機を利用した解の存在証明法は有効であることが知られている．解の存在性を示すことで，現象を表す偏微分方程式の妥当性を保証することが出来る．しかし，大規模な非線形偏微分方程式系となると複雑さは増し，今までの計算機援用証明法では解の存在を保証することができない例が多々存在した．
本研究成果で大規模な非線形偏微分方程式系に特化した手法を考案し，解の存在性を証明できる範囲の拡大に成功した．