This article investigates the uniqueness of simultaneously determining the diffusion coefficient and initial value in a time-fractional diffusion equation with derivative order . By additional boundary measurements and a priori assumption on the diffusion coefficient, the uniqueness of the eigenvalues and an associated integral equation for the diffusion coefficient are firstly established. The proof is based on the Laplace transform and the expansion of eigenfunctions for the solution to the initial value/boundary value problem. Furthermore, by using these two results, the simultaneous uniqueness in determining the diffusion coefficient and initial value is demonstrated from the Liouville transform and Gelfand–Levitan theory. The result shows that the uniqueness in simultaneous identification can be achieved, provided the initial values non-orthogonality to the eigenfunction of differential operators, which incorporates only one diffusion coefficient rather than scenarios involving two diffusion coefficients.

中文翻译：

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时间分数扩散方程中扩散系数和初值辨识的同时唯一性


本文研究了在具有导数阶数的时间分数扩散方程中同时确定扩散系数和初始值的唯一性。通过额外的边界测量和对扩散系数的先验假设，首先建立了扩散系数的特征值的唯一性和相关的积分方程。证明基于拉普拉斯变换和特征函数展开来解决初始值/边界值问题。此外，利用这两个结果，从Liouville变换和Gelfand-Levitan理论中证明了确定扩散系数和初始值的同时唯一性。结果表明，只要初始值与微分算子的本征函数非正交，就可以实现同时识别的唯一性，仅包含一个扩散系数，而不是涉及两个扩散系数的情况。