Mathematical analysis for nonlinear partial differential equations with singular solutions

具有奇异解的非线性偏微分方程的数学分析

• 批准号: 15340041
• 负责人: OMATA Seiro
• 金额: $ 7.04万
• 依托单位: Kanazawa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340041/
• 关键词: variational problem; nonlinear PDEs; singularity; numerical analysis; minimizing method; free boundary problem; 特異点数値解析

The aim of this research was to solve nonlinear Partial Differential Equations whose solution is expected to have singularities depending on time. The candidates of singularities are defects in harmonic mapping, vortex in Ginzburg-Landau problem and free boundaries.We have solved the following problems;(1)On a Soap film vibration with free boundary, we have established the method to treat wave type free boundary problems(2)We developed a numerical method via the discrete Mores flow for volume constraint conditions(3)We constructed a weak solution to a hyperbolic equation with volume constraint(4)We constructed a weak solution to a parabolic equation with volume constraint and showing Hoelder continuity of thesolutionMoreover we have developed solvers for parallel machine with minimizing algorithm via the discrete Morese flows. This works very well especially for volume constraint problems. This is also very nice for a weak connected parallel machines, because it uses direct method of variational principle.Finally, we would like to express pur special thanks to all participants of this project.

本研究的目的是求解非线性偏微分方程，其解预计具有随时间变化的奇点。奇异点的候选是调和映射中的缺陷、Ginzburg-Landau问题中的涡流和自由边界。我们解决了以下问题：(1)在具有自由边界的皂膜振动上，建立了处理波型自由边界的方法问题(2)我们通过体积约束条件的离散莫尔斯流开发了一种数值方法(3)我们构造了具有体积约束的双曲方程的弱解(4)我们构造了具有体积约束的抛物线方程的弱解并显示霍尔德连续性此外，我们还开发了并行机求解器，通过离散 Morese 流实现最小化算法。这对于体积约束问题尤其有效。这对于弱连接并行机来说也是非常好的，因为它使用了变分原理的直接方法。最后，我们要对这个项目的所有参与者表示特别的感谢。

项目成果

S.Omata: "A numerical treatment of thin film motion with free boundary"Adv.Math.Sci.Appl., 14,. 14(to appear). (2004)

S.Omata：“自由边界薄膜运动的数值处理”Adv.Math.Sci.Appl.，14，。

Bubble Motion on water surface

水面上的气泡运动

• 发表时间: 2005
• 期刊: Gakuto Intern. Ser. Math. Sci. Appl. 23
• 作者: T.Yamazaki;S.Omata et.el.
• 通讯作者: S.Omata et.el.

A numerical computation to the American option pricing via the discrete Morse flow

基于离散莫尔斯流的美式期权定价数值计算

• 发表时间: 2003
• 期刊: Theoretical and Applied Mechanics 52
• 作者: U.C.Ji;N.Obata;N.Obata;S.Omata et.el.
• 通讯作者: S.Omata et.el.

S.Jimbo, J.Zhai: "Instability in a geometric parabolic equation on convex domain"J.Differential equations. 188. 447-460 (2003)

S.Jimbo，J.Zhai：“凸域上几何抛物线方程的不稳定性”J.微分方程。

Numericalsolution of film vibration with obstacle

有障碍薄膜振动的数值求解

• 发表时间: 2006
• 期刊: Adv. Math. Sci. Appl. 16, 1
• 作者: H.Yoshiuchi;S.Omata;K.Svadlenka;K.Ohara
• 通讯作者: K.Ohara