| 研究実績の概要 |
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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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
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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| 今後の研究の推進方策 |
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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