| Project/Area Number |
10640128
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | TOKYO METOROPOLITAN UNIVERSITY |
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Principal Investigator |
NISHIOKA Kunio Tokyo Metropolitan Univ., Dept of Math., AP, 理学研究科, 助教授 (60101078)
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| Co-Investigator(Kenkyū-buntansha) |
HIRATA Masaki Dept of Math. AP, 理学研究科, 助手 (70254141)
SUMI Naoya Dept of Math. AP, 理学研究科, 助手 (50301411)
AOKI Nobuo Dept of Math.. P, 理学研究科, 教授 (60087020)
SATO Sadao Tokyo Electric Univ., Dept of Inform. AP, 理工学部, 助教授 (10170747)
SHIMURA Michio Toho Univ., Dept of Math.., P (90015868)
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| Project Period (FY) |
1998 – 1999
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| Project Status |
Completed (Fiscal Year 1999)
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| Budget Amount *help |
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1999: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1998: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Keywords | biharmonic operator / monopole / dipole / absorbing boundary / sticky boundary / reflecting boundary / Poisson分布 / 無限次元力学系 / 弱い意味での確率積分 / Girsanovの公式 / 4階非線形偏微分方程式 / 確率解 / カオス |
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| Research Abstract |
-△ィイD12ィエD1 is called a biharamonic operator, which we often encounter in the theory of elasticity and fluid dynamics. Following pioneer works of Hochberg (1978) and Funaki (1979), many attempts are done to study the operator from the view point of probability theory.
We consider 'particles' whose 'transition probability density' is the fundamental solution of ∂ィイD2tィエD2u = -△ィイD12ィエD1u, though it may have negative values. We call orbits of these 'particles' biharmonic pseudo process, or BPP in short. Our aim is to study relations between analytic boundary conditions of -△ィイD12ィエD1 and behaviors of BPP.
By Nishioka (1997), we know that BPP behave as a mixture of two sorts of particles which we call 'monopoles' and 'dipoles'. So we need two boundary conditions to solve a boundary value problem related to -△ィイD12ィエD1, since each boundary condition controls behavior of each sort of particles at the boundary. We set up one of 'absorbing', 'sticking', and various 'reflecting' barriers on monopoles and dipoles one by one. Then their combinations solve various kinds of boundary value problems to -△ィイD12ィエD1.
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