Research on an estimation problem for the shape of time-varying domain via parabolic equations
| Project/Area Number |
18540155
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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 |
Basic analysis
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| Research Institution | Akita University |
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Principal Investigator |
KAWAKAMI Hajime Akita University, Faculty of engineering and resource science, associate professor (20240781)
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| Co-Investigator(Kenkyū-buntansha) |
TSUCHIYA Masaaki Kanazawa University, 金沢大学, professor emeritus (50016101)
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| Project Period (FY) |
2006 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥1,710,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥210,000)
Fiscal Year 2007: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
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| Keywords | inverse problem / identification of shape / parabolic equation / heat equation / mixed boundary condition / 未知形状 |
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| Research Abstract |
Our research program is concerned with an inverse problem of determining the shape of some time-varying unknown portion of the boundary of a multi-dimensional domain via a parabolic equation on the domain. Considering practical applications, we treat such a domain under weak regularity conditions and a parabolic operator of general type, and set a mixed boundary condition: Dirichlet's condition is imposed on the unknown portion and Robin's one on the other portions.
As an observable data, we take the boundary value on an accessible portion of the boundary of a solution to the parabolic equation. The correspondence between data and domains is generally nonlinear. Thus we first considered a linearized problem; then, for the shape of the unknown portion, we proved a unique identification theorem and provided a reconstruction algorithm from the data We also verified the convergence and stability of the algorithm. The result was reported in the symposium "Inverse Problems in Applied Sciences -towards breakthrough-", and published in the journal " Inverse Problems".
In the last term of the project, we considered the primary (not linearized) problem and obtained a unique identification theorem as follows. We first assume that the domain is Lipschitz and that the parabolic operator is non-degenerate and has bounded Lipschitz continuous coefficients. Secondarily we assume that the Robin boundary value does not vanish somewhere on the accessible portion at every observation time. Moreover we assume that the shape of the domain is known at the initial observation time or that the initial value of the solution is zero. Then, in the observation period, the shape of the unknown portion is uniquely determined by the data. The result was reported in the Annual Meeting of the Mathematical Society of Japan this spring. We are preparing to publish the details of the result.
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Report
(3 results)
Research Products
(8 results)
