Studies on verified numerical computations for nonlinear hyperbolic partial differential equations

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K13453
• 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: University of Tsukuba
• Principal Investigator: Akitoshi Takayasu 筑波大学, システム情報系, 助教 (60707743)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 計算機援用証明 / 複素数値非線形熱方程式 / 非線形シュレディンガー方程式 / 一次元変数係数移流方程式 / Parameterization method / 厳密な数値求積 / 解の時間大域存在

Outline of Final Research Achievements

Mathematical problems obtained by modeling natural phenomena are called mathematical models. Mathematical models are often formulated as partial differential equations　(PDEs), and solving them mathematically and numerically to understand the behavior of unknown functions is a central research topic in the natural sciences. In this study, we have developed a computer-assisted proof method for a class of PDEs called hyperbolic PDEs, which appear in mathematical models of wave phenomena and quantum mechanics. Such a method proves that the solution of the initial boundary value problem exists in a neighborhood of the numerically computed approximate solution. This is called verified numerical computations, and is attracting attention as a modern approach to mathematical analysis of differential equations.

Free Research Field

数値解析

Academic Significance and Societal Importance of the Research Achievements

本研究成果は双曲型偏微分方程式を含むより広いクラスの偏微分方程式に対して、数値計算による証明手法を提供する。特に、解の挙動を無限次元力学系として捉え、各計算機援用証明手法により解の大域挙動を明らかにした研究成果は自然科学分野における数理モデルの開発や現象の解明に貢献している。物理の波動現象や量子力学をモデル化する際の偏微分方程式の解挙動を数学証明付きで理解することで、科学研究の進展や新たな技術・応用の開発に寄与し、社会の課題解決に役立つことが期待される。