Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation
Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation
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一维分数扩散方程反问题的唯一性
DOI：
10.1088/0266-5611/25/11/115002
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发表时间：
2009-11-01
期刊：
影响因子：
2.1
通讯作者：
Yamazaki, Tomohiro
中科院分区：
文献类型：
--
作者：
Cheng, Jin;Nakagawa, Junichi;Yamazaki, Tomohiro
考虑一维分数扩散方程：偏导数（α）(t) u(x, t) =偏导数/偏导数x (p(x)偏导数u/偏导数x (x, t)), 0 < x < l，其中0 < α < 1，偏导数（α）(t)表示在α阶时间上的Caputo导数。我们附加了齐次诺依曼边界条件在x = 0,1和狄拉克函数给出的初值。我们证明了alpha和p(x), 0 < x < l是由数据u(0, t), 0 < t < t唯一决定的，该唯一性结果为实验确定许多重要的异常扩散现象的alpha阶提供了理论背景，例如在环境工程中。该证明是基于初值/边值问题弱解的特征函数展开式和Gel’fand- levitan理论。
We consider a one-dimensional fractional diffusion equation: partial derivative(alpha)(t) u(x, t) = partial derivative/partial derivative x (p(x) partial derivative u/partial derivative x (x, t)), 0 < x < l, where 0 < alpha < 1 and partial derivative(alpha)(t) denotes the Caputo derivative in time of order alpha. We attach the homogeneous Neumann boundary condition at x = 0, l and the initial value given by the Dirac delta function. We prove that alpha and p(x), 0 < x < l, are uniquely determined by data u(0, t), 0 < t < T. The uniqueness result is a theoretical background in experimentally determining the order alpha of many anomalous diffusion phenomena which are important, for example, in environmental engineering. The proof is based on the eigenfunction expansion of the weak solution to the initial value/boundary value problem and the Gel'fand-Levitan theory.