This article is concerned with the uniqueness of simultaneously determining the fractional order of the derivative in time, diffusion coefficient, and Robin coefficient, in one-dimensional time-fractional diffusion equations with derivative order α∈(0,1) and non-zero boundary conditions. The measurement data, which is the solution to the initial–boundary value problem, is observed at a single boundary point over a finite time interval. Based on the expansion of eigenfunctions for the solution to the forward problem and the asymptotic properties of the Mittag-Leffler function, the uniqueness of the fractional order is established. Subsequently, the uniqueness of the eigenvalues and the absolute value of the eigenfunction evaluated at x=0 for the associated operator are demonstrated. Then, the uniqueness of identifying the diffusion coefficient and the Robin coefficient is proven via an inverse boundary spectral analysis for the eigenvalue problem of the spatial differential operator. The results show that the uniqueness of three parameters can be simultaneously determined using limited boundary observations at a single spatial endpoint over a finite time interval, without imposing any constraints on the eigenfunctions of the spatial differential operator.

中文翻译：

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在时间分数扩散方程中分数阶和扩散系数的同时唯一性识别


本文关注的是，在具有导数阶数 α∈（0,1） 和非零边界条件的一维时间分数扩散方程中，同时确定导数的时间分数阶数、扩散系数和 Robin 系数的唯一性。测量数据是初始边界值问题的解，在有限的时间间隔内在单个边界点处观察。基于解正向问题的特征函数的展开和 Mittag-Leffler 函数的渐近性质，建立了分数阶的唯一性。随后，证明了特征值的唯一性和在 x=0 处为关联算子计算的特征函数的绝对值。然后，通过对空间微分算子特征值问题的逆边界谱分析，证明了识别扩散系数和 Robin 系数的唯一性。结果表明，在有限的时间间隔内，使用单个空间端点的有限边界观测可以同时确定三个参数的唯一性，而不会对空间微分算子的特征函数施加任何约束。