Mathematical Analysis of partial differential equations related to a variational problem via the discrete Morse Semiflows

通过离散莫尔斯半流对与变分问题相关的偏微分方程进行数学分析

• 批准号: 11640159
• 负责人: OMATA Seiro
• 金额: $ 2.3万
• 依托单位: Kanazawa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640159/
• 关键词: Variational problem; Nonlinear partial differential equations; Numerical Analysis; Minimizing methods; Free boundary problem

We mainly investigated partial differential equations related to a variational problem via the discrete Morse semiflows. Our main interest is on sets of singular points of a solutions. Such sets has sometimes big energy concentrate on it. So, we can cosider that our purpose is on treating the energy concentration phenomena on the singularity of solutions. In this stand point of view, we treated the following type of problems :(1) Develop a prallel machine for solving mininizing problems,(2) Develop a Numerical method via a minimization process,(3) Develop a method to solve both parabolic and hyperbolic equations via minimizing.For these problems, we have developped a 8-CPU parallel computer for solving minimizing problems. By use of this, we did a numerical copmutations to catch the structure of singularities for eikonal equation, Ginzburg-Landau system, and smestics liquid crystal problems. Basic method due to discrete Morse semiflow for parabolic and hyperbolic problems.We also solved the asymptotic behavior of solitary wave solutions for BBM-Burgers equations.Moreover we developped a software to solve hyperbolic free boundary problems. This is based on the smoothing method of a equation and we can get good results even when the free boundary changes its topology.We summed up these results into 8 papers (appeared or in press) and 2 preprint (submitted).

我们主要通过离散的Morse半流量研究了与变异问题有关的部分微分方程。我们的主要兴趣是解决方案的奇异点集。这样的集合有时会集中在它上。因此，我们可以选择我们的目的是在解决方案的奇异性上处理能量浓度现象。从这种立场角度来看，我们处理了以下类型的问题：（1）开发一种用于解决最小化问题的Prallal机器，（2）通过最小化过程开发一种数值方法，（3）开发一种方法来通过最小化解决抛物线寄生虫方程。对于这些问题，我们已经开发了一个8 cpu shore Computer以解决最小化的问题。通过使用此功能，我们进行了数值的交流，以捕获用于艾科纳尔方程，金茨堡 - 兰道系统和Smestics液晶问题的奇异性结构。由于抛物线和双曲线问题的离散莫尔斯半节，基本方法。我们还解决了BBM-Burgers方程的孤立波解决方案的渐近行为。此外，我们开发了一个软件来解决双曲自由边界问题。这是基于方程式的平滑方法，即使自由边界更改其拓扑，我们也可以得到良好的结果。我们将这些结果概括为8篇论文（出现或在印刷中）和2个预印本（已提交）。

项目成果

K.Kikuchi,S.Omata: "A free boundary problem for a one dimensional hyperbolic equation"Adv.Math.Sci.Appl.. 10 No.1. 775-786 (1999)

K.Kikuchi，S.Omata：“一维双曲方程的自由边界问题”Adv.Math.Sci.Appl.. 10 No.1。

H.Imai,K.Kikuchi,K.Nakane,S.Omata,T.Tachikawa: "A Numerical Approach to the Asymptotic Bbehavior of Solutions of a One-Dimensional Free Boundary Problem of Hyperbolic Type"Japan Journal of Industrial and Applied Mathematics. 18(1). 43-58 (2001)

H.Imai，K.Kikuchi，K.Nakane，S.Omata，T.Tachikawa：“双曲型一维自由边界问题解的渐近行为的数值方法”日本工业与应用数学杂志。

H.Imai, K.Kikuchi, K.Nakane, S.Omata and T.Tachikawa: "A Numerical Approach to the Asymptotic Bbehavior of Solutions of a One-Dimensional Free Boundary Problem of Hyperbolic Type"Japan Journal of Industrial and Applied Mathematics. 18, No.1(to appear). 43

H.Imai、K.Kikuchi、K.Nakane、S.Omata 和 T.Tachikawa：“双曲型一维自由边界问题解的渐近行为的数值方法”日本工业与应用数学杂志。

S.Omata, T.Okamura and K.Nakane: "Numerical analysis for the discrete Morse semiflow related to the Ginzburg Landau functional"Nonlinear Analysis. 37, No.5. 589-602 (1999)

S.Omata、T.Okamura 和 K.Nakane：“与 Ginzburg Landau 泛函相关的离散莫尔斯半流的数值分析”非线性分析。

S.Kinami, M.Mei and S.Omata: "Asymptotic Toward Diffusion Waves of the Solutions for Benjamin-Bona-Mahony-Burgers Equations"Applicable Analysis. 75, (3-4). 317-340 (2000)

S.Kinami、M.Mei 和 S.Omata：“Benjamin-Bona-Mahony-Burgers 方程解的渐近扩散波”应用分析。