2016 Fiscal Year Annual Research Report
Inversion and prediction problems in anomalous diffusion
| Project/Area Number |
16H06712
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| Research Institution | The University of Tokyo |
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Principal Investigator |
李 志遠 東京大学, 大学院数理科学研究科, 特任研究員 (00782450)
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| Project Period (FY) |
2016-08-26 – 2018-03-31
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| Keywords | anomalous diffusion / inverse problems / Carleman estimates |
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| Outline of Annual Research Achievements |
The Carleman estimates (CE) for the generalized time-fractional diffusion equations (TFDEs) were investigated. First, in the case of sub-diffusion, say, the largest fractional order is strictly less than 1/2, the CE for the TFDE was established by regarding fractional order terms as perturbation of the first order time-derivative, from which we further verified that a conditional stability for a lateral Cauchy problem. In the case of sup-diffusion where the largest order is rational number and less than 3/4, the CE for the TFDE was constructed. As an application, the conditional stability for an inverse source problem was proved as well as the stability for the lateral Cauchy problem. The fractional order 3/4 is the largest one which one can deal with based on the arguments of CE.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The Carleman estimates (CE) for the generalized time-fractional diffusion equations (TFDEs) were established in the following two cases:
1. the largest fractional order is strictly less than 1/2.
2. the largest order is rational number and less than 3/4.
The main idea is regarding the fractional order term as a perturbation of the first order time-derivative, which enables one to derive the Carleman estimate for the generalized time-fractional diffusion equations (TFDEs) in the framework of the Carleman estimate for the parabolic equations. Here it should be mentioned that the argument used in dealing with the above two cases cannot work for the general order case, and the fractional order 3/4 is the largest one which one can deal with based on the arguments of CE.
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| Strategy for Future Research Activity |
Because of no integration by parts in fractional calculus, it is not easy to follow the usual way to derive the Carleman estimates (CE) for time-fractional diffusion equations (TFDEs). The main idea I used is regarding the TFDE as a parabolic type equation by considering fractional order terms as a perturbation to the time-derivative. The CE for the TFDE heavily relies on CE for parabolic case. Especially, the fractional order 3/4 is the sharp case one can deal with is a direct conclusion by noting the powers of parameters in CE for parabolic equations. In the case where the fractional orders are irrational numbers between 1/2 and 3/4 or real numbers greater than 3/4, there are two possible ways:
1. Modifying the CE for parabolic equations
2. Using theory of pseudo-differential operators instead of integration by parts to derive the CE for TFDE.
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